Keywords: analytic theory of differential equations

Je tiens à exprimer toute ma reconnaissance envers les personnes qui m'ont permis de mener à bien mon parcours doctoral.

Je remercie tout particulièrement mon directeur de thèse, Raphaël Côte, sans qui mes travaux n'auraient pas vu le jour. Ses encouragements et conseils, sa disponibilité, ses intuitions brillantes et sa confiance dans l'évolution de ma thèse ont constitué des richesses colossales à mon égard.

Je remercie chaleureusement les rapporteurs de cette thèse, Rémi Carles et Stephen Gustafson, pour leur lecture attentive du manuscrit et leurs observations ainsi que les autres membres du jury qui m'ont fait l'honneur d'accepter d'évaluer ce travail.

Cette thèse s'accompagne de l'achèvement de mes études à l'Université de Strasbourg. Si je suis arrivé à ce stade, c'est indéniablement grâce à la contribution des différents enseignants qui ont croisé mon parcours d'études. Aussi, j'aimerais témoigner plus que jamais de ma satisfaction d'avoir bénéficié de savoirs inestimables, ne serait-ce que d'une infime partie, et de visions très variées, éclairantes, des mathématiques.

Je suis en outre très heureux d'avoir eu moi-même la possibilité d'enseigner les mathématiques à l'Université et je ne peux que regretter que les séances de cours et de travaux dirigés avec les étudiants, même à distance, se soient déroulées si rapidement. J'ai ici une pensée envers les collègues et le personnel administratif et technique qui ont oeuvré au bon déroulement de ces enseignements.

Cette thèse n'aurait sans doute pas eu le même goût sans les échanges avec les autres doctorants en mathématiques, certains parmi lesquels m'étaient connus de plus longue date. Je suis convaincu que plusieurs de nos chemins se recroiseront à l'avenir.

Je souhaite enfin saluer mes proches qui m'ont soutenu tout au long de ma formation doctorale, à la fois de près et de loin. v Résumé Cette thèse est consacrée aux propriétés qualitatives des multi-solitons de trois équations aux dérivées partielles non-linéaires dispersives focalisantes.

Les premiers résultats nouveaux concernent l'équation de Schrödinger non-linéaire (NLS). Nous construisons des multi-solitons réguliers, à valeurs dans des espaces de Sobolev , où

> 1 est directement relié à la régularité de la non-linéarité. Nous démontrons également un résultat d'unicité sous condition pour les multi-solitons de (NLS) dans les cas stable et 2 -critique.

Par ailleurs, nous énonçons une propriété de Liouville au voisinage des multi-solitons de l'équation de Korteweg-de Vries généralisée (gKdV). Il s'agit d'un résultat de rigidité qui repose sur le concept de non-dispersion. En particulier dans les cas intégrables, ce concept permet de caractériser les multi-solitons. Dans une autre mesure, nous étudions le comportement ponctuel des multi-solitons de (gKdV).

Enfin, nous nous intéressons à l'équation de Klein-Gordon non-linéaire (NLKG). Pour cette dernière, nous construisons une famille à paramètres de -solitons à décroissance exponentielle en temps. En ce qui concerne la question de l'unicité, nous classifions les multi-solitons dans une certaine classe à décroissance polynomiale et obtenons la classification générale dans le cas où l'on considère une seule onde solitaire.

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Nom : Prénom :

= - 2 + 2 (KdV)
a vu le jour au dix-neuvième siècle avec l'observation, dans des eaux peu profondes, de vagues qui prennent la forme d'ondes solitaires se propageant au cours du temps sans se déformer. La structure de (KdV) présente l'intérêt d'admettre une famille complète de solutions, indexée par deux paramètres, la vitesse 0 > 0 et la position 0 ∈ R, qui correspondent aux ondes solitaires précédentes. Les éléments 0 , 0 de cette famille sont appelés solitons et prennent la forme explicite suivante :

0 , 0 ( , ) = 3 0 2 cosh 2 √ 0 2 ( -0 -0 ) .
Le soliton 0 , 0 est donc une solution, qui à chaque temps , a un profil de courbe en cloche centrée en 0 + 0 et qui se déplace au cours du temps vers la droite à vitesse 0 , sans changer de forme ni d'amplitude. au temps 1 au temps 2 > 1 0 1 + 0 0 2 + 0 vitesse 0 Dans la même optique, nous considérons les équations de Korteweg-de Vries généralisées suivantes qui admettent encore un nombre important d'applications à la physique [START_REF] Lamb | Elements of Soliton Theory[END_REF][START_REF] Miura | The Korteweg-de Vries equation: A survey of results[END_REF] :

= - 2 + (gKdV)
avec > 1 et où est une fonction des variables unidimensionnelles et et à valeurs réelles.

Un cadre naturel d'étude et d'appartenance des solutions de (gKdV) est l'espace de Sobolev 1 (R). Le problème de Cauchy associé à (gKdV) est en effet localement bien posé dans 1 (R) selon Kenig, Ponce et Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF].

Il est remarquable d'observer que (gKdV) admet une famille de solitons, encore indexée par R * + × R et qui généralise la famille de solitons de (KdV) de la manière suivante : notant l'unique solution positive (à translation près) dans 1 (R) [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF][START_REF] Kwong | Uniqueness of positive solutions of Δ -+ = 0 in R[END_REF] de l'équation elliptique stationnaire associée à (gKdV) + = , qui admet l'expression

( ) = + 1 2 cosh 2 -1 2 1 -1 , et 0 ( ) := 1 -1 0 ( √ 0 )
, les éléments 0 , 0 définis par 0 , 0 ( , ) = 0 ( -0 -0 ) sont des solutions globales de (gKdV) qui sont les solitons de (gKdV).

Les solitons sont des objets très particuliers, dont l'existence est liée à la propriété focalisante de (gKdV), due au signe de la non-linéarité : il y a un certain effet compensatoire entre ce terme non-linéaire et la partie linéaire de l'équation fondamentale qui est dispersive en raison de la dérivée seconde.

Nous rappelons aussi que les quantités suivantes sont conservées (au moins formellement) pour les solutions de (gKdV) :

• (la masse ou la norme 2 ) ∫ R 2 ( , )

• (l'énergie) ∫ R 1 2 2 - 1 + 1 +1 ( , ) .
La norme 1 n'est en revanche pas constante au cours du temps en général. En outre, la norme ( ) , où ( ) := 1 2 -2 -1 est conservée par l'invariance d'échelle de (gKdV), selon laquelle pour tout > 0, ( , ) ↦ → 2 -1 3 , est solution de (gKdV) dès lors que l'est. Il s'agit de noter que la dynamique des solutions est liée intimement au signe de ( ), ce qui donne lieu à la distinction des cas :

•2 -sous-critique, correspondant à ( ) < 0 (ou 1 < < 5), où toutes les solutions 1 de (gKdV) sont globales en temps et uniformément bornées dans 1

• 2 -critique, correspondant à ( ) = 0 (ou = 5), où il existe des solutions qui explosent en temps fini [START_REF] Martel | Blow up in finite time and dynamics of blow up solutions for the 2 -critical generalized KdV equation[END_REF][START_REF] Martel | Nonexistence of blow-up solution with minimal 2 -mass for the critical gKdV equation[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. II: Minimal mass dynamics[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. III: Exotic regimes[END_REF] • 2 -surcritique, correspondant à ( ) > 0 (ou > 5), où il existe des solutions qui explosent en temps fini au moins pour tout ∈ (5, * ) pour un certain réel * > 5 [START_REF] Lan | Blow-up solutions for 2 -supercritical gKdV equations with exactly k blow-up points[END_REF].

Il s'avère que les critères connus d'existence de solutions qui explosent en temps fini lorsque ≥ 5 reposent sur l'existence et les propriétés des solitons.

Notons enfin que les cas particuliers de (gKdV) qui correspondent à = 2 (KdV) et = 3 (mKdV) (ou équation de Korteweg-de Vries modifiée) sont des équations complètement intégrables : elles admettent une infinité de lois de conservation [START_REF] Peter | Integrals of nonlinear equations of evolution and solitary waves[END_REF][START_REF] Miura | The Korteweg-de Vries equation: A survey of results[END_REF].

Un autre exemple important d'équation aux dérivées partielles dispersive non-linéaire qui admet des solitons est fourni par l'équation de Schrödinger :

= Δ + | | -1 ,
(NLS) où 1 < < 1 + 4 ( -2) + , ( , ) appartient à R × R et est à valeurs dans C. Dans le contexte de (NLS), le problème de Cauchy est encore localement bien posé dans 1 (R ) selon un résultat de Ginibre et Velo [START_REF] Ginibre | On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case[END_REF]. De plus, les quantités suivantes sont conservées par les solutions 1 de (NLS) :

• la masse ou norme 2

∫ R | ( , )| 2 • l'énergie ∫ R 1 2 |∇ ( , )| 2 - 1 + 1 | ( , )| +1
• le moment Im ∫ R ∇ ( , ) ( , ) .

L'équation adimensionnée (NLS) que nous considérons est aussi au coeur d'un certain nombre de modélisations de phénomènes physiques et tout particulièrement optiques. Pour ne citer qu'un exemple, l'enveloppe complexe du champ électrique associé à un faisceau d'onde de lumière stationnaire qui se propage le long de l'axe des dans un milieu d'indice de réfraction non-linéaire = 0 + 2 avec > 0 est bien décrite par l'équation

2 + 2 2 = -2 0 | | 2 , (1.1)
où est le nombre d'onde associé [START_REF] Kelley | Self-Focusing of Optical Beams[END_REF]. D'un point de vue mathématique, il revient bien sûr au même d'étudier cette dernière équation ou d'étudier (NLS) cubique (c'est-à-dire avec = 3).

En dimension 1, (NLS) cubique se réécrit = et possède des solitons qui sont des solutions explicites particulières ayant une expression de la forme

( , ) := √ 2 -4 ( 2 -2 ) -2 + cosh(2 ( -0 ) + 8 ) , où ∈ R, > 0, 0 ∈ R et ∈ R.
Les solitons précédents sont ainsi caractérisés par les quatre paramètres de taille , de vitesse , de position 0 et de phase . Dans le contexte physique de l'équation (1.1), la variable remplace et le soliton précédent ( , ) correspond à une onde solitaire qui se propage sans se déformer, dans la direction donnée par un angle =arctan(4 ) avec l'axe des . direction de propagation de l'onde solitaire En dimension supérieure, on sait encore démontrer, pour tout > 0, l'existence (travaux de Berestycki et Lions [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF]) et l'unicité (travaux de Kwong [START_REF] Kwong | Uniqueness of positive solutions of Δ -+ = 0 in R[END_REF]) de solutions fondamentales qui vérifient le problème stationnaire Δ + = , mais on ne sait pas expliciter de telles solutions. Par conséquent, nous n'avons pas de formule explicite pour les solitons construits à partir d'une telle solution fondamentale et des propriétés d'invariance de (NLS) par translation, par rotation et par transformation galiléenne.

Pourquoi les multi-solitons ?

Il convient de souligner que les solitons sont des objets centraux dans la compréhension et l'analyse du comportement des solutions des équations aux dérivées partielles dispersives non-linéaires focalisantes.

Nous avons déjà vu par exemple que les solitons permettent de dégager des conditions pour assurer l'existence de solutions de (gKdV) qui explosent en temps fini dans le cas 2 -critique [START_REF] Martel | Blow up in finite time and dynamics of blow up solutions for the 2 -critical generalized KdV equation[END_REF]. Mais ces solutions très particulières ont aussi un rôle essentiel dans l'étude de la dynamique de solutions générales, définies en temps long.

Considérons par exemple la question de la stabilité des solitons qui est un pas naturel pour comprendre la dynamique des équations dispersives non-linéaires focalisantes. Il s'agit en fait d'étudier le comportement de solutions qui sont, à l'instant initial, proches d'un soliton. La notion de stabilité orbitale que l'on définit dès à présent est appropriée au contexte des ondes solitaires. Definition 1.1. Soit : [0, ) → 1 une solution d'une équation aux dérivées partielles ( ) et soit un groupe qui agit sur l'ensemble { ( ) | ∈ [0, )}. On dit que est -orbitalement stable (ou orbitalement stable s'il n'y a pas d'ambiguité) si pour tout > 0, il existe > 0 tel que pour toute solution de ( ) définie sur [0, ) et à valeurs dans 1 ,

(0) -(0) 1 ≤ ⇒ ∀ ∈ [0, ), inf ∈ ( ) -• ( ) 1 ≤ .
Mais surtout, l'introduction du concept de soliton et son étude tirent essentiellement leur motivation de la conjecture de résolution en solitons, selon laquelle toute solution générique d'une équation aux dérivées partielles non-linéaire dispersive focalisante se décompose en temps long comme la somme de solitons (ou, plus largement, de solutions localisées) et d'un terme dispersif linéaire.

Rappelons qu'à l'origine de cette conjecture, un phénomène d'interaction et de collision élastique a été observé et étudié numériquement par Zabusky et Kruskal [112] : des solutions particulières de (KdV) dont le profil correspond à celui de la superposition de plusieurs ondes solitaires de vitesses deux à deux distinctes avant collision conservent un profil similaire après collision. Les caractéristiques physiques comme l'amplitude des « pulses » des ondes solitaires observées restent les mêmes, seul l'ordre d'apparition en espace de ces pulses est inversé. Profil typique de la solution observé après collision La méthode de scattering inverse développée par Gardner, Greene, Kruskal et Miura [START_REF] Gardner | Kortewegde Vries equation and generalizations. VI. Methods for exact solutions[END_REF] confirme et justifie l'observation de Zabusky et Kruskal d'un point de vue théorique. Les résultats obtenus par cette méthode vont pleinement dans le sens de la conjecture de résolution en solitons.

Dans le cas de l'équation intégrable (KdV), un résultat de résolution en solitons a été notamment élaboré par Eckhaus et Schuur [START_REF] Eckhaus | The emergence of solutions for the generalized Korteweg-de Vries equation from arbitrary initial conditions[END_REF] puis raffiné par Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF] à partir de la méthode précédente. Il concerne les solutions de (KdV) génériques, au sens où la donnée initiale correspondante est suffisamment régulière et à décroissance suffisamment rapide. Théorème 1.2 (Eckhaus et Schuur [START_REF] Eckhaus | The emergence of solutions for the generalized Korteweg-de Vries equation from arbitrary initial conditions[END_REF]; Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF]). Soit = 2 et soit 0 ∈ C 4 (R) une fonction telle qu'il existe 0 > 0 tel que pour tout = 0, . . . , 4 et ∈ R, 0 ( ) ≤ 0 | | -11 .

(1.2)

Soit la solution globale de (KdV) ayant pour condition initiale 0 . Alors il existe une solution qui est un multi-soliton ou la solution nulle telle que pour tout > 0, il existe ≥ 0 tel que pour tout > 0 ( ) -( ) ∞ ( > ) + ( ) -( ) 2 ( > ) ≤ - 1 3 .

(1.3)

Un autre résultat de résolution en solitons et breathers a été donné pour des solutions « génériques » de (mKdV) par Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF] et Chen et Liu [START_REF] Chen | Soliton resolution for the modified KdV equation[END_REF] (voir Chapitre 3).

En vue d'éclairer d'une certaine manière l'étude de la décomposition en temps long des solutions à l'aide des solitons, nous introduisons le concept de multi-soliton.

Définition 1.3. Considérons

≥ 1 solitons 1 , . . . , d'une équation aux dérivées partielles non-linéaire ( ) ayant des vitesses deux à deux distinctes. Un multi-soliton en +∞ (resp. en -∞) associé aux , = 1, . . . , , est une solution de ( ) définie au voisinage de +∞ (resp. -∞), à valeurs dans 1 (R ) et telle que

( ) - =1 ( ) 1 → 0, quand → +∞ (resp. quand → -∞). (1.4)
Typiquement pour (gKdV), nous prenons pour = 1, . . . , , ( , ) := ( --), où 0 < 1 < • • • < et 1 , . . . , ∈ R. Pour (NLS), il est classique de travailler avec solitons (au lieu de solitons), et de considérer pour tout = 1, . . . , , ( , ) := ( -0 -)

1 2 • + - | | 2 4
+ , où les différents paramètres qui interviennent dans l'écriture sont > 0, ∈ R, 0 ∈ R et ∈ R tels que pour tout ≠ , ≠ .

Se posent d'emblée un certain nombre de questions en ce qui concerne les multi-solitons. D'abord, ces objets existent-ils toujours ? Peut-on les construire, les caractériser, en donner des formules explicites ? Ces solutions particulières sont-elles régulières, uniques, stables... ? Peut-on en préciser le comportement en temps, en espace ?

Notre objectif est de répondre, dans la mesure du possible, aux questions précédentes. En vérité, on retrouve les multi-solitons dans différents contextes. A présent, on sait démontrer leur existence via une preuve constructive. En revanche, beaucoup d'autres propriétés qualitatives des multi-solitons sont encore assez peu comprises. Il convient de noter toutefois que le cadre des équations de Korteweg-de Vries généralisées (dont la structure particulière permet aux méthodes usuelles d'analyse non-linéaire d'aboutir) offre une bonne compréhension des multi-solitons ; la classification complète, dans chacun des cas 2 -sous-critique, critique et sur-critique a été dressée. Cela dit, même dans le contexte de (gKdV), certaines questions demeurent sans réponse à ce jour. Aussi, nous souhaitons dégager de nouveaux résultats concernant les multi-solitons de (gKdV) et (NLS), ainsi que pour les équations de Klein-Gordon non-linéaires (NLKG) que nous introduisons dans la partie 1.4. D'autres problèmes qui dépassent le cadre de cette thèse restent encore ouverts aujourd'hui. Par exemple, il serait intéressant de pouvoir décrire le comportement en -∞ des multi-solitons en +∞ qui sont définis pour tout temps ∈ R. Une approche naturelle serait évidemment de relier la décomposition en somme de solitons en +∞ avec une éventuelle autre décomposition en -∞ pour de telles solutions, et pour cela, d'essayer de mieux comprendre le phénomène de collision des différentes ondes solitaires en jeu.

En appliquant la méthode de scattering inverse à (KdV), on obtient à ce propos l'existence de solutions qui sont à la fois des multi-solitons en +∞ et en -∞ : précisément, étant donnés solitons 1 , . . . , associés à des paramètres de vitesses 0 < 1 < • • • < et de positions 1 , . . . , , il existe une solution telle que

( ) - =1 ( ) 1 → 0, quand → +∞ et ( ) - =1 ( ) (• -) 1 → 0, quand → -∞,
où ∈ R est quantifiable à l'aide des vitesses . En outre, cette solution est explicite et peut s'écrire (voir par exemple [START_REF] Gardner | Kortewegde Vries equation and generalizations. VI. Methods for exact solutions[END_REF], [START_REF] Miura | The Korteweg-de Vries equation: A survey of results[END_REF] ou [START_REF] Hirota | Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[END_REF]) :

= 6 2 2 ln det , où ( , ) est la matrice carrée de taille dont l'élément générique ( , ) ( , ) est égal à

, + 2 ( ) 1 4
√ + √ En utilisant la méthode de scattering inverse, Zakharov et Shabat [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF] construisent un multisoliton pour (NLS) en dimension = 1. Là encore, le multi-soliton obtenu se découple en des solitons de mêmes fréquences et mêmes vitesses lorsque → ±∞ ; les seuls paramètres modifiés sont la position et la phase.

Hormis dans les cas intégrables précédents où la collision entre les différents solitons est élastique (au sens où les vitesses des différentes ondes solitaires en lesquelles se découplent les multisolitons restent inchangées), le comportement asymptotique en -∞ d'un multi-soliton en +∞ qui est de plus global n'est pas encore bien connu. Les estimées asymptotiques dont on dispose en +∞ pour les multi-solitons définis au voisinage de +∞ ne permettent guère de préciser le phénomène de collision qui se produit.

Cependant, d'après un résultat de collision rigoureux dû à Martel et Merle [START_REF] Martel | Description of two soliton collision for the quartic gKdV equation[END_REF][START_REF] Martel | Inelastic interaction of nearly equal solitons for the quartic gKdV equation[END_REF][START_REF] Martel | On the nonexistence of pure multi-solitons for the quartic gKdV equation[END_REF] qui fournit des estimées très précises des différents paramètres impliqués, nous savons qu'un 2-soliton en +∞ de (gKdV) avec = 4 ne peut pas se comporter comme une somme de solitons en -∞. Notons en outre que Muñoz [START_REF] Muñoz | On the inelastic 2-soliton collision for gKdV equations with general nonlinearity[END_REF] a généralisé ce premier résultat de collision inélastique à des non-linéarités sous-critiques plus générales pour (gKdV).

Bien que nous n'en faisions pas l'étude ici, signalons aussi, outre les multi-solitons, l'existence d'autres solutions particulières en lien avec la conjecture de résolution en solitons. Aussi, Côte [START_REF] Côte | Large data wave operator for the generalized Korteweg-de Vries equations[END_REF] a démontré, pour des fonctions suffisamment décroissantes, l'existence de solutions de (gKdV) définies en temps long et telles que

( ) -( ) 1 → 0, lorsque → +∞,
où { ( )} est le groupe linéaire d'Airy : de telles solutions dispersent donc complètement, en se comportant comme des solutions linéaires en temps long. Par ailleurs, Côte a également construit, à nouveau pour une certaine classe de fonctions , des solutions de (gKdV) pour = 4 [START_REF] Côte | Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior[END_REF] et = 5 [START_REF] Côte | Construction of solutions to the 2 -critical KdV equation with a given asymptotic behaviour[END_REF] qui se comportent asymptotiquement comme une somme de solitons et d'un terme dispersif linéaire :

( ) - =1 ( ) -( ) 1 → 0, lorsque → +∞.
Remarquons que la solution précédente possède en quelque sorte une dynamique plus complexe que les multi-solitons puisqu'elle se découple en temps long en une partie non-linéaire décrite par la somme des solitons qui se propagent vers la droite et aussi en une partie linéaire qui se propage vers la gauche.

Des outils d'analyse non-linéaire adaptés à l'étude des multi-solitons

Les outils classiques d'analyse linéaire comme les transformations de Fourier ou de Laplace tombent rapidement en défaut lorsqu'il s'agit d'étudier des équations non-linéaires comme (gKdV) et (NLS). Il est usuel de recourir alors à des méthodes plus appropriées de point fixe, de compacité, d'énergie, et autres fondées sur l'étude de l'évolution de certaines quantités dans le temps. Mais il est souvent utile, voire indispensable, de considérer un cadre linéaire proche du problème non-linéaire qui fait l'objet d'étude. L'intérêt est de pouvoir manipuler des opérateurs linéaires et, si possible, de considérer leur spectre qui fournit des renseignements précieux quant à la dynamique générale des solutions de l'équation étudiée. Aussi, lorsqu'on travaille plus particulièrement avec des solitons, il s'avère utile d'introduire les opérateurs linéarisés au voisinage de chacun de ces solitons.

Par exemple, pour tout > 0, l'équation linéarisée de (gKdV) autour de + s'écrit = ( ) où est l'opérateur linéaire 1 (R) → 1 (R) d'expression

= -2 + - -1
.

L'article de Pego et Weinstein [START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF] s'attache à la théorie spectrale de l'opérateur ; celle-ci est le point de départ pour démontrer que est strictement coercif en restriction à des sous-espaces bien choisis de 1 (R). Ces sous-espaces dépendent du régime 2 -sous-critique, critique ou surcritique considéré, autrement dit de la valeur de . Les cas 2 -sous-critique et critique sont traités dans l'article [START_REF] Michael | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] de Weinstein. Dans le cas 2 -surcritique, l'existence de vecteurs propres + et -associés aux valeurs propres respectives 0 > 0 et -0 [START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF] suppose une propriété de coercivité différente, démontrée par Côte, Martel et Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] (à partir des idées de Duyckaerts et Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF]) ; les directions instables + et - doivent notamment être contrôlées. Ces propriétés de coercivité remarquables font l'objet de la proposition suivante. Proposition 1.4. Soit > 0. Si 1 < < 5, alors il existe > 0 tel que pour tout ∈ 1 (R),

si ∫ R = ∫ R = 0, alors ∫ R ( ) ≥ 2 1 .
Si = 5, alors il existe > 0 tel que pour tout ∈ 1 (R),

si ∫ R 3 = ∫ R = 0, alors ∫ R ( ) ≥ 2 1 .
Si > 5, alors il existe > 0 tel que pour tout ∈ 1 (R),

si ∫ R + = ∫ R - = ∫ R = 0 alors ∫ R ( ) ≥ 2 1 .
Pour (NLS), considérant > 0, l'équation linéarisée de (NLS) autour de ( + ) s'écrit = L où

L : 1 (R , C) → 1 (R , C) = 1 + 2 ↦ → -Δ + - -1 + ( -1) -1 1 .
L'opérateur L se décompose de façon immédiate à l'aide de deux opérateurs +, , -, : 1 (R , R) → 1 (R , R) comme suit : L ( ) = +, 1 + -, 2 .

Les opérateurs +, et -, sont aussi coercifs en restriction à des sous-espaces de 1 (R , R) convenables qui dépendent du signe de ∫ R ( ) 2 , ou encore de la valeur de . Remarquons d'ailleurs qu'en dimension 1, l'opérateur -, s'identifie à l'opérateur linéaire de (gKdV) qui correspond à = .

Définissons l'énergie linéarisée autour de : pour tout ∈ 1 (R ), on pose :

( ) := Re ∫ R L , = ∫ R |∇ | 2 + | | 2 - -1 | | 2 + ( -1) -1 2 1 .
Les propriétés de coercivité des opérateurs linéarisés sont centrales dans la théorie non-linéaire de (gKdV) et de (NLS) autour des solitons, et particulièrement dans l'étude des propriétés qualitatives des solitons et des multi-solitons. Une des utilisations importantes de la coercivité concerne la stabilité orbitale des solitons et des multi-solitons ; on pourra consulter par exemple [3, Definition 4.1], respectivement [START_REF] Michael | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF], pour une définition de cette notion pour (gKdV), respectivement pour (NLS).

Rappelons que, dans le cas 2 -sous-critique, la coercivité de est un élément clé de la preuve de la stabilité orbitale des solitons de (gKdV) [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Michael | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF]. La considération des opérateurs linéarisés montre aussi que les solitons sont instables dans les cas critique et surcritique [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF].

Dans le cas des équations de Schrödinger avec non-linéarité de type puissance, les travaux successifs et complémentaires concernant l'étude des opérateurs linéarisés et de la stabilité des solitons dus à Weinstein [START_REF] Michael | Modulational stability of ground states of nonlinear dispersive Schrödinger equations[END_REF][START_REF] Michael | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF], Grillakis [START_REF] Grillakis | Analysis of the linearization around a critical point of an infinite dimensional hamiltonian system[END_REF], Grillakis, Shatah et Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], Maris [START_REF] Mihai | Existence of nonstationary bubbles in higher dimension[END_REF], Schlag [START_REF] Schlag | Spectral theory and nonlinear partial differential equations: A survey[END_REF], Duyckaerts et Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF], Duyckaerts et Roudenko [START_REF] Duyckaerts | Threshold solutions for the focusing 3D cubic Schrödinger equation[END_REF] et Côte, Martel et Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] permettent d'obtenir : Proposition 1.5. Soit > 0. Si 1 < < 1 + 4 , alors il existe + > 0 tel que pour tout = 1 + 2 ∈ 1 (R, C),

( ) ≥ + 2 1 - 1 + ∫ R 1 2 + =1 ∫ R 1 2 + ∫ R 2 2
(1.5) et de plus est orbitalement stable. Si 1 + 4 < < +2 -2 , alors L possède un vecteur propre associé à une valeur propre 0 > 0, il existe + > 0 tel que pour tout = 1 + 2 ∈ 1 (R, C),

( ) ≥ + 2 1 - 1 + ∫ R 1 2 2 - 1 + =1 ∫ R 1 2 + ∫ R 2 1 2 + ∫ R 2 2
(1.6) et de plus est instable.

Dans le cadre de l'étude des équations de Schrödinger avec non-linéarités plus générales, mentionnons qu'il est possible d'adapter les résultats de coercivité et de stabilité précédents (on pourra se reporter aux propositions 2.15 et 2.16 au Chapitre 2).

Pour l'étude plus spécifique des multi-solitons dans les cas non-intégrables où la méthode de scattering inverse ne s'applique pas, on recourt à des versions dépendantes du temps et localisées en espace de la coercivité des opérateurs linéarisés précédents, en se souvenant, par exemple pour (gKdV), que le multi-soliton se comporte essentiellement comme le ème soliton au voisinage de + pour tout = 1, . . . , . Concrètement, afin de concentrer l'étude autour du point + en espace et ainsi d'observer le comportement de la solution au voisinage du ème soliton, cela suggère d'introduire des fonctions « cut off » .

Une telle famille de fonctions est typiquement générée de la manière suivante (voir aussi par exemple [80, paragraphe 2.2] ou [63, paragraphe 3.1]). On considère une fonction : R → R de classe C ∞ , strictement croissante et telle que

lim -∞ = 0, lim +∞ = 1, ∀ ∈ R, ( ) = 1 -(-), et ∀ ∈ R, | (3) ( )| ≤ 0 ( ),
avec 0 > 0 suffisamment petit, dépendant des paramètres . On précise qu'une telle fonction existe effectivement. Pour tout ≥ 2, on définit alors

( , ) = - -1 + 2 - -1 + 2
.

-1 + -1 + +1 + +1
Illustration : « filtration » des ondes solitaires par la fonction ( )

( ) ( ) En posant 1 ( , ) = 1 -2 ( , ), ( , ) = ( , ), ( , ) = ( , ) -+1 ( , ), = 2, . . . , - 1
on se ramène à l'étude de quantités comme « l'énergie localisée au voisinage du ème soliton », soit pour (gKdV) l'intégrale

∫ R ( ) 2 + 2 - -1 2 ( , ) ( , ) .
Par exemple, ∫ R 2 ( , ) ( , ) correspond à la masse de la solution en restriction à un certain voisinage de + (à une précision O( -)) et on peut établir un lien entre cette quantité et la masse du ème soliton.

La structure particulière de (gKdV) permet d'obtenir la presque-monotonie de certaines quantités, définies à partir de et de ; en particulier,

∫ R 2 ( , ) ( , ) ≤ -.
Notons que les termes d'interaction entre les différents solitons sont essentiellement contrôlés par des inégalités de la forme

∀ ≠ , ∫ R ( ) ≤ -,
où > 0 dépend des vitesses (et des fréquences pour (NLS)) des solitons et ; il est ici important de supposer que les vitesses sont deux à deux distinctes.

A l'aide des propriétés de coercivité et de monotonie comme ce qui précède, les résultats de stabilité de Martel, Merle et Tsai [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of solitons for subcritical gKdV equations[END_REF] des multi-solitons de (gKdV) ont ainsi pu voir le jour. En procédant de façon analogue, mais en prenant des fonctions décroissantes, l'existence et l'unicité des multi-solitons de (gKdV) ont été démontrées dans les cas 2 -sous-critique et 2 -critique [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF].

Existence et construction de multi-solitons pour (NLS) et (gKdV)

Première construction dans un cas non-intégrable

Dans son article [START_REF] Merle | Construction of solutions with exactly blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] publié en 1990, Merle construit pour la première fois un multi-soliton dans le cadre d'une équation aux dérivées partielles non-intégrable, celle de Schrödinger non-linéaire 2 -critique

= Δ + | | 4 . (cNLS)
A titre d'illustration, notons qu'en dimension = 2, l'équation précédente consiste en un modèle d'approximation satisfaisant pour décrire la propagation d'un faisceau laser planaire le long d'une seule direction dans R 3 .

L'équation (cNLS) présente la caractéristique d'admettre la loi d'invariance pseudo-conforme suivante : si est une solution de (cNLS), alors ( ,

) ↦ → 1 | | 2 | | 2 4
1 , est également solution de (cNLS). On observe en outre que les solutions de (cNLS) de la forme

, 0 ( , ) = 1 ( -) 2 - | | 2 4 - -0 - , (1.7)
où > 0 et vérifie -Δ + -4 +1 = 0 explosent en temps fini au point 0 . En fait, on peut montrer réciproquement que toutes les solutions de (cNLS) de masse minimale 2 qui explosent en temps fini sont de la forme précédente [START_REF] Merle | On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass[END_REF][START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF].

Merle [START_REF] Merle | Construction of solutions with exactly blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] obtient l'existence de multi-solitons associés à solitons comme corollaire d'un résultat d'existence de solutions explosant en temps fini en points 1 , . . . , , construites à partir de solutions , définies comme ce qui précède (1.7), en utilisant la loi d'invariance pseudoconforme de (cNLS).

Une méthode de construction typique de multi-solitons à valeurs dans 1

L'existence des multi-solitons pour les équations (gKdV) est garantie pour toute valeur de > 1.

Théorème 1.6 (Merle [85] ; Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] ; Côte, Martel et Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]).

Soit > 1, soit ∈ N \ {0, 1} et soient solitons 1 , 1 , . . . ,
, de (gKdV) associés à des paramètres de vitesse

1 , . . . , > 0 deux à deux distincts. Il existe ∈ R et ∈ C ( [ , +∞), 1 (R)) une solution de (gKdV) telle que ( ) - =1 , ( ) 1 → 0, lorsque → +∞.
Se fondant sur les idées de Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] qui démontre l'existence de multi-solitons pour (gKdV) dans les cas 2 -sous-critique et 2 -critique, le principe de la construction de multi-solitons à valeurs dans 1 , qui a été étendue par Côte, Martel et Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] au cadre 2 -surcritique, est aujourd'hui bien connu.

La preuve de l'existence de multi-solitons dans 1 a ainsi été établie pour un certain nombre d'équations aux dérivées partielles dispersives non-linéaires, dans chacun des cas stable et instable : pour (NLS) [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF][START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Merle | Construction of solutions with exactly blow-up points for the Schrödinger equation with critical nonlinearity[END_REF], pour les équations de Klein-Gordon non-linéaires [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF], pour l'équation de Hartree [START_REF] Krieger | Two-soliton solutions to the threedimensional gravitational Hartree equation[END_REF], pour le système des « water-waves » [START_REF] Ming | Multi-solitons and related solutions for the water-waves system[END_REF], et récemmment pour les équations de Zakharov-Kuznetsov [START_REF] Valet | Asymptotic K-soliton-like solutions of the Zakharov-Kuznetsov type equations[END_REF].

Rappelons les étapes essentielles de la construction dans le cas où tous les solitons considérés sont stables (c'est-à-dire lorsque 1 < < 5). Considérant une suite strictement croissante de réels ( ) qui tend vers +∞ et ( ) une suite de solutions telles que = ( ), où

= =1
, on démontre l'existence de 0 ∈ R (indépendant de ) tel que pour tout suffisamment grand, Enfin, la solution de l'équation considérée avec condition initiale 0 en 0 est un multi-soliton qui vérifie la propriété de décroissance exponentielle. Cela peut se justifier notamment par des arguments de dépendance continue par rapport à la donnée initiale des solutions à valeurs dans , ∈ [0, 1], sur des intervalles de temps compacts et par des arguments plus élémentaires de passage à la limite. Par exemple, Martel procède de la sorte dans [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] pour (gKdV), en utilisant les résultats de dépendance continue des solutions 1 sur les compacts de Kenig, Ponce et Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF]. Martel et Merle [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF] utilisent les résultats de Tsutsumi [START_REF] Tsutsumi | 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF] de dépendance continue (par rapport à la donnée initiale) pour les solutions globales à valeurs dans 2 en ce qui concerne l'équation (NLS) associée à la non-linéarité de type puissance avec 1 < < 1 + 4 et aussi ceux de Cazenave et Weissler [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in[END_REF] pour les non-linéarités plus générales.

∀ ∈ [ 0 , ], ( ) -( ) 1 ≤ -. ( 1
Dans le cas où l'un des solitons au moins n'est pas stable, le contrôle des directions instables nécessite la mise en place d'un argument supplémentaire, comprendre un argument topologique qui repose sur l'utilisation du théorème de Brouwer. On pourra se référer aux travaux de Côte, Martel et Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF], novateurs en cela.

Dans le même état d'esprit, la construction de trains infinis de solitons a été réalisée pour (NLS) par Le Coz et Tsai [START_REF] Le | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] et Le Coz, Li et Tsai [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF] dans le cas 1 -sous-critique, moyennant certaines hypothèses de vitesses relatives élevées et de grandes fréquences pour les solitons. L'approche est en revanche différente de celle initiée par Martel puisque ces articles exploitent la formule de Duhamel, les inégalités de Strichartz et travaillent bien davantage avec les normes .

Une autre interaction possible entre les ondes solitaires

Dans le cadre de la définition 1.3, les interactions entre solitons sont exponentiellement petites ; on parle d'interaction « faible ». Récemment, on a commencé à étudier des solutions pour lesquelles les interactions entre solitons sont algébriquement petites ; dans ce contexte, on dit que l'interaction est « forte ». Nous détaillons ici quelques résultats à ce sujet.

Pour (cNLS) en dimension 2, Martel et Raphaël [83, Theorem 1] démontrent l'existence de solutions globales qui se comportent asymptotiquement lorsque → +∞ comme la somme de ondes solitaires centrées en les sommets d'un -gone régulier. Ces solutions ont une dynamique en temps long différente des multi-solitons de la définition 1.3 du paragraphe 1.1.2 ; elles ont la particularité notable d'exploser en temps infini : précisément, la norme 2 du gradient de la solution évolue ici en log . De surcroît, les ondes solitaires ne sont pas découplées en temps long comme dans le cas d'une interaction faible prévue par la définition 1.3. Dit autrement, les solutions obtenues par invariance pseudo-conforme de (cNLS) à partir des solutions construites concentrent les « bulles » en un même point au temps limite d'explosion.

Un autre exemple d'interaction forte apparaît en dimension 1 dans le contexte des équations (gKdV). Nguyen [START_REF] Vinh | Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation[END_REF] construit pour 2 < < 5 et > 5 un 2-soliton en +∞ pour lequel les deux ondes solitaires associées se propagent à la même vitesse et dont la distance relative est en 2 log + cte. Pour une certaine constante que l'on peut expliciter en fonction de la puissance de la non-linéarité, il existe une solution telle que

( ) - • --log( ) - • -+ log( ) 1 → 0 lorsque → +∞,
où = -1 en régime sous-critique et = 1 dans le cas surcritique. Prolongeant les résultats précédents et les méthodes mises à l'oeuvre dans les preuves correspondantes, l'article [START_REF] Vinh | Existence of multi-solitary waves with logarithmic relative distances for the NLS equation[END_REF] révèle l'existence de solutions de l'équation de Schrödinger (NLS) qui se comportent comme une somme de deux solitons séparés d'une distance qui évolue également en 2 log lorsque → +∞, dans les cas 2 -sous-critique et 2 -surcritique. En outre, pour le système d'équations de Schrödinger cubiques unidimensionnelles couplées, Martel et Nguyen [START_REF] Martel | Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system[END_REF] obtiennent des 2-solitons symétriques avec distance relative logarithmique (précisément en log + 1 2 log log ). Dans le même papier, les deux auteurs exhibent également une nouvelle dynamique qui n'apparaît notamment pas dans le cas de l'équation intégrable (NLS) cubique : deux solitons de vitesses différentes (donc d'amplitudes différentes) peuvent interagir de manière forte en O(log ).

Des solutions composées de deux ou plusieurs solitons interagissant de manière forte ont aussi fait récemment l'objet d'étude pour l'équation des ondes critique par Jendrej [START_REF] Jendrej | Construction of two-bubble solutions for energy-critical wave equations[END_REF] et Jendrej et Martel [START_REF] Jendrej | Construction of multi-bubble solutions for the energycritical wave equation in dimension 5[END_REF].

Les dynamiques logarithmiques précédentes semblent très particulières et plutôt exceptionnelles. En ce qui concerne les questions relatives à la classification, mentionnons que l'article [START_REF] Jendrej | Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations[END_REF] de Jendrej signale pour (gKdV) 2 -surcritique que des 2-solitons associés à deux ondes solitaires de même amplitude ne peuvent exister que si ces dernières sont séparées d'une distance adéquate en O(log ). De même, pour l'équation des wave maps (à valeurs dans des variétés plus générales que R ou C), on pourra consulter l'article [START_REF] Jendrej | Uniqueness of two-bubble wave maps[END_REF] qui fournit un résultat d'unicité dans une certaine classe pour les solutions particulières issues de l'interaction forte entre les ondes solitaires.

Ces solutions obtenues récemment font l'objet de constructions souvent assez délicates. Des résultats d'existence analogues, inspirés des travaux ci-dessus, sont à ce jour au stade de conjecture pour les équations de Zakharov-Kuznetsov [START_REF] Valet | Etude de comportements en temps long de solutions des équations de Zakharov-Kuznetsov[END_REF]. Plus généralement, beaucoup de propriétés concernant les solutions de la nature décrite sont à explorer dans les différentes situations relatées. Toutefois, nous ne nous focaliserons pas davantage sur ces solutions « exceptionnelles » dont l'existence repose sur l'interaction forte entre les solitons. Aussi, dans cette thèse, nous limiterons notre étude au cadre de la définition 1.3.

Propriétés des multi-solitons des équations de Schrödinger nonlinéaires 1.2.1 Etude de la régularité des solutions

Afin d'étudier la régularité des multi-solitons de (NLS), nous nous appuyons considérablement sur les résultats d'existence et d'unicité de solutions générales de l'équation dans , pour tout ≥ 1. L'étude préalable du problème de Cauchy pour (NLS) dans s'avère donc indispensable.

Problème de Cauchy pour (NLS) dans

Le problème de Cauchy pour (NLS) a été étudié à l'aide d'arguments de points fixes appliqués à différents espaces bien choisis. Après les résultats qui ont été obtenus dans 1 (R ) [START_REF] Cazenave | The Cauchy problem for the nonlinear Schrödinger equation in 1[END_REF][START_REF] Ginibre | On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case[END_REF] et, plus spécifiquement pour 1 < < 1 + 4 dans 2 (R ) [START_REF] Nakamura | Nonlinear Schrödinger equations in the Sobolev space of critical order[END_REF][START_REF] Tsutsumi | 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF], Cazenave et Weissler [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in[END_REF] ont montré que, sous des hypothèses -sous-critiques convenables, (NLS) est localement bien posé dans ( > 1) avec dépendance continue dans par rapport à la donnée initiale sur certains compacts. Notons que le résultat de dépendance continue par rapport à la donnée initiale a été affiné par Dai, Yang et Cao [START_REF] Dai | Continuous dependence of Cauchy problem for nonlinear Schrödinger equation in[END_REF]. En s'appuyant sur les énoncés de Cazenave et Weissler et de Kato, on peut démontrer que la dépendance continue dans (R ) est valide sur tout compact [START_REF] Dai | Continuous dependence of Cauchy problem for nonlinear Schrödinger equation in[END_REF]. En reprenant les notations du théorème 1.7, on a : Le théorème précédent présente un intérêt majeur : comme le problème de Cauchy est localement bien posé dans (R ) avec dépendance continue sur tout compact, beaucoup de calculs que nous réalisons (comme les dérivées d'intégrales à paramètres de fonctions ) et qui sont présentés formellement peuvent être justifiés par des arguments de régularisation.

Théorème 1.7. Si ≥ 2 , ou si 0 ≤ < 2 et 1 < < 1 + 4 -2 , alors pour tout 0 ∈ (R ), il existe ( 0 ) > 0 et une unique solution ∈ C ((- ( 0 ), ( 0 )), (R )) de (NLS) telle que (0) = 0 . De plus, il existe ∈ (0, ( 0

Multi-solitons

Nous étudions ici la question naturelle de la régularité des multi-solitons. Peut-on espérer en particulier l'appartenance des multi-solitons construits à valeurs dans 1 à des espaces de Sobolev supérieurs ? Les travaux de Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] apportent une réponse affirmative dans le cas de (gKdV), unidimensionnelle. Il est démontré dans [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] que la convergence exponentielle (1.8) a lieu au sens de la norme , pour tout ∈ N, avec une constante devant l'exponentielle qui dépend de et un taux de décroissance exponentielle indépendant de . Notons que des estimées analogues ont été obtenues très récemment pour les équations de Zakharov-Kuznetsov par Valet dans [START_REF] Valet | Asymptotic K-soliton-like solutions of the Zakharov-Kuznetsov type equations[END_REF] en faisant l'usage d'un argument qui utilise la monotonie typique des équations (gKdV).

Un résultat important de cette thèse est la construction pour (NLS) avec non-linéarité générale

= Δ + (| | 2 ) (NLS)
d'un multi-soliton à valeurs dans (R ) où l'indice de régularité > 1 est intimement lié à la régularité de la fonction : ↦ → (| | 2 ).

Plus exactement, dans le cas particulier de la non-linéarité de type puissance, le théorème démontré au chapitre 2 prend la forme suivante. Théorème 1.9 (Côte, F.).

Supposons ≥ 3. Soit 0 = -1 ≥ 2 ou 0 = +∞ si est un entier impair. Il existe > 0, 1 > 0 et ∈ C ( [ 1 , +∞), 0 (R )) solutions de (NLS) avec non-linéarité de type puissance : ↦ → -1 2 tels que pour tout entier positif ≤ 0 , il existe une constante ≥ 1 telle que ∀ ≥ 1 , ( ) -( ) ≤ -2 +1 .
Par ailleurs, si est un entier impair, alors pour tout ≥ 0,

∀ ≥ 1 , ( ) -( ) ≤ -.
Précisons que la solution précédente est a priori différente de celle du théorème 1.6, obtenue dans 1 .

Le théorème 1.9 qui concerne la non-linéarité particulière de type puissance se limite en réalité aux dimensions inférieures ou égales à 3 en raison de l'hypothèse 1 -sous-critique < 1 + 4 ( -2) + faite pour assurer l'existence de solitons.

Nous disposons d'un résultat de régularité analogue au théorème 1.9 pour des non-linéarités générales. Sous l'hypothèse que est un élément de 0 +1,∞ (C), que l'on a 0 > 2 et que chaque opérateur linéarisé autour des solitons vérifie une propriété de coercivité adéquate qui traduit en quelque sorte la stabilité ou l'instabilité du soliton considéré (voir Proposition 1.5), nous démontrons que la conclusion du théorème 1.9 est encore vraie dans ce cadre plus général.

La démonstration des résultats de régularité précédents, que l'on trouvera au chapitre 2, est écrite pour des non-linéarités quelconques. Soulignons qu'elle ne consiste pas en un travail direct sur le multi-soliton 1 qui a été construit dans [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF] ou [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]. Elle repose en revanche sur la preuve d'estimées -uniformes satisfaites par une suite ( ) de solutions bien choisie, qui était déjà considérée dans [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF] et [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] et sur la mise en place d'un argument usuel de compacité. La conclusion de la preuve se fait à l'aide de la dépendance continue sur tout compact des solutions de (NLS) par rapport aux données initiales décrite par le théorème 1.8. Prenant comme point de départ les estimées

, : ↦ → ∫ R        | |= | | 2 - | |= -1 -1 Re 2 2 (| | 2 )        ( ) ,
qui s'écrit de la même manière en toute dimension , peut être contrôlée par ( ) -( ) -1 . L'introduction de cette fonctionnelle permet d'éliminer, lorsqu'on la dérive, les termes que l'on ne sait pas contrôler dans la dérivée de

( ) 2 , à savoir ∫ R Im 2 ( ) 2 (| | 2 )
pour les multi-indices ∈ N de longueur . De la sorte, la dérivée première de , ne contient ni terme quadratique faisant intervenir avec | | = ni terme avec | | > , ce qui est essentiel pour pouvoir appliquer l'hypothèse de récurrence. En pratique, le contrôle des différents termes qui apparaissent dans , est réalisé à l'aide d'outils classiques d'analyse fonctionnelle, notamment l'inégalité de Hölder, les injections de Sobolev et les inégalités d'interpolation de Gagliardo-Nirenberg.

Notons que si pour (gKdV), Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] parvient à montrer que toutes les quantités ( ) -( ) décroissent exponentiellement en temps long avec la même vitesse, nous obtenons, dans le cadre de (NLS), des taux de décroissance exponentielle qui sont divisés par deux pour ces mêmes quantités lorsque l'on passe de à + 1. En effet, il ne nous est guère possible de considérer autant d'intégrations par parties, dans la mesure où la non-linéarité n'est pas nécessairement indéfiniment dérivable et où surtout l'algèbre associée à (NLS) n'est pas aussi favorable. Plus précisément, pour (gKdV), les termes qui apparaissent du type +1 peuvent être intégrés tandis que pour (NLS), les termes analogues sont de la forme Im +1 Re ( ) (en dimension 1) et ne peuvent ainsi pas être traités de la même façon.

Ainsi, nous notons la dépendance en dans les estimées du théorème 1.9 liée à la perte en ce qui concerne la vitesse de décroissance exponentielle lorsqu'il s'agit de passer de la preuve de l'estimée à celle de l'estimée +1 . Le taux de décroissance exponentielle de la norme de ( ) -( ), correspondant à 2 +1 et obtenu finalement par interpolation, est optimal pour ≥ 2 (qui est le cas intéressant du théorème précédent).

Rajoutons que notre preuve est aussi plus technique en raison de la dimension. Comme les éléments de 1 (R ) ne sont pas nécessairement dans ∞ (R ) pour ≥ 2, nous recourons à un argument de type « bootstrap » afin de démontrer les estimées -uniformes souhaitées. L'hypothèse 0 > 2 est déjà utile à ce niveau.

Par ailleurs, se pose la question de l'hypothèse optimale à fournir quant à la non-linéarité de sorte à obtenir un multi-soliton dans 0 (R ) tel que ( ) -( ) 0 soit à décroissance exponentielle.

En l'occurrence, justifions qu'il convient de supposer que soit dans l'espace 0 +1,∞ (C). Rappelons qu'il suffisait de prendre de classe C 1 (considérée comme fonction R 2 → R 2 ) pour avoir le résultat avec 0 = 1. D'autre part, dans notre cadre d'étude de (NLS) où l'on exige 0 > 2 , le problème de Cauchy est localement bien posé dans 0 sous l'hypothèse que est de classe C 0 (Kato, [START_REF] Kato | On nonlinear Schrödinger equations. II, solutions and unconditional wellposedness[END_REF]). Au minimum, il est donc souhaitable de faire cette dernière hypothèse. Cela dit, afin d'obtenir les inégalités voulues, il est incontournable de savoir contrôler les quantités de la forme 0 0 -( ) -0 0 -( ), où = 0, . . . , 0 . Il semble donc naturel de supposer que 0 0 -soit localement lipschitzienne sur C (ou, autrement dit, appartienne à 1,∞ (C)). D'autre part, le contrôle de l'intégrale qui fait intervenir typiquement les dérivées d'ordre maximal 0 des deux quantitéset nécessite une intégration par parties de sorte que la quantité apparaisse avec une dérivée d'ordre 0 -1, ceci afin de pouvoir appliquer l'hypothèse de récurrence. Pour un multi-indice = ( 1 , . . . , ) donné de longueur 0 et de composante ≠ 0, pour une fonction ℎ donnée de classe C 1 et pour = 0, . . . , 0 , nous pouvons écrire

∫ R ℎ( ) 0 0 -( ) = - ∫ R - ℎ( ) 0 0 -( ) (1.9)
(avec le ième vecteur de la base canonique de R ) sous réserve que la dérivée de ↦ → 0 0 -( ) appartienne à un certain espace de Lebesgue convenable. Ce point est assuré dès lors que cette dérivée est bornée.

Notre démonstration du résultat de régularité s'appuie manifestement sur l'hypothèse 0 > 2 qui, remarquons-le, est automatique en dimension ≤ 3. Cette dernière, utilisée dans l'argument bootstrap afin de travailler avec des éléments de ∞ (R ) comme évoqué plus haut, apparaît aussi naturellement lorsqu'il y a lieu de contrôler les différents termes dans l'expression de , .

Probablement qu'une autre méthode, voire d'autres outils, seraient à mettre en oeuvre pour répondre à la question de la régularité dans le cas où 0 ≤ 2 . Pour l'heure, il semblerait que les inégalités de Strichartz usuelles associées à (NLS) ne permettent pas de répondre positivement au problème (sauf si les vitesses relatives sont grandes), mais nous pouvons raisonnablement penser que des estimées de ce type, qui concerneraient plus particulièrement l'équation linéarisée autour d'une somme de solitons, bien qu'inexistantes actuellement, pourraient s'avérer utiles dans ce cadre. D'ailleurs, dans une autre perspective, le fait d'avoir à disposition de telles estimées serait de bon présage pour obtenir ou réobtenir des résultats de stabilité ou de stabilité asymptotique en ce qui concerne les multi-solitons de (NLS).

Question de l'unicité des multi-solitons

Sous l'impulsion de [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] qui fournit l'existence et l'unicité des multi-solitons de (gKdV) dans les cas2 -sous-critique et critique puis de [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] qui donne l'ingrédient nécessaire à la construction dans le cas 2 -surcritique, les travaux de Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF] permettent d'apporter une réponse complète à la question de la classification des multi-solitons de (gKdV). Au contraire, pour (NLS) et pour la plupart des autres équations aux dérivées partielles non-linéaires dispersives focalisantes, la classification exhaustive des multi-solitons n'est pas encore comprise à ce jour. Notons que les travaux de Valet [START_REF] Valet | Asymptotic K-soliton-like solutions of the Zakharov-Kuznetsov type equations[END_REF] attachés à l'étude des multi-solitons de l'équation de Zakharov-Kuznetsov (ZK) fournissent aussi l'unicité dans les cas 2 -sous-critique et 2 -critique, mais des informations cruciales liées au spectre de l'opérateur linéarisé pour (ZK) manquent encore pour démontrer l'existence, voire espérer une classification des multi-solitons, en régime surcritique. De même que pour (gKdV) [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF], il est connu de Combet [START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF] l'existence d'une famille à paramètres de solutions pour (NLS) dans le cas 2 -surcritique en dimension 1. En particulier, il n'y a pas unicité pour les multi-solitons associés à solitons donnés dans ce cadre, mais pour autant, il n'est pas clair que l'on puisse obtenir tous les multi-solitons de la manière décrite dans [START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF], à la différence de (gKdV). Par ailleurs, un premier résultat d'unicité pour (NLS) a vu le jour dans les papiers [START_REF] Le | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] et [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF]. Au même titre que pour la partie existence, ce résultat exige des vitesses relatives suffisamment élevées ; de plus, l'unicité démontrée par les trois auteurs Le Coz, Li et Tsai est valide dans une classe de solutions telles que ( ) -( ) est à décroissance exponentielle en norme , Strichartz ou 1 .

En nous inspirant de [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] et de [START_REF] Martel | Stability in 1 of the sum of solitary waves for some nonlinear Schrödinger equations[END_REF], nous obtenons un autre résultat d'unicité pour (NLS), au sens de la norme 1 , qui concerne les cas stable et 2 -critique. C'est aussi un résultat d'unicité « avec condition », mais au sens où l'unicité a lieu dans la classe des multi-solitons tels que ( ) -( ) 1 décroît plus rapidement qu'une certaine puissance de 1 pour assez grand. Exposons le résultat d'unicité obtenu lorsqu'on considère la non-linéarité particulière de type puissance.

Théorème 1.10 (Côte, F.). Soit ≤ 2 et supposons que ∈ 3, 1

+ 4 . Il existe ∈ N et une unique solution ∈ C ( [ 1 , +∞), 1 (R )) de (NLS) telle que ( ) -( ) 1 = O 1 , lorsque → +∞. (1.10)
Une conséquence directe, importante et rassurante, est que les multi-solitons des théorèmes 1.6 et 1.9 coïncident (et nous pouvons prendre en outre 0 = 1 ).

Là encore, le théoreme 1.10 possède en dimension ≤ 3 une généralisation aux non-linéarités telles que, en notant ˜ ( ) := (| | 2 ) pour tout ∈ C,

• la fonction ˜ est de classe C 2 sur C vue comme fonction différentiable R 2 → R.

• la différentielle seconde de ˜ est contrôlée de la manière suivante :

• si ≥ 2, appartient à 0 +1,∞ (C) où 0 := 2 + 1 ;

• les opérateurs linéarisés autour des solitons vérifient la propriété de coercivité (1.5) appropriée à la situation de stabilité.

Notons qu'il semble nécessaire de se restreindre aux dimensions ≤ 3 ; nous utilisons en effet que 4 -1 > 0 pour assurer l'intégrabilité d'une certaine quantité et nous avons besoin de l'injection de Sobolev 1 (R ) ↩→ 6 (R ) pour contrôler les normes 6 de quantités (non-bornées) par les normes 1 correspondantes.

En dimension ≥ 4, nous pouvons aussi donner un énoncé similaire qui s'applique aux nonlinéarités générales, mais alors au prix de considérer une classe de solutions plus petite, pour laquelle un certain contrôle de

∫ +∞ ( ) -( ) ∞ est réalisé.
Il s'avère que le résultat de régularité obtenu (Théorème 1.9) est utile dans notre preuve d'unicité en dimension ≥ 2. Pour ≥ 2, nous utilisons le multi-soliton construit à valeurs dans 0 (R ).

L'idée de la preuve du théorème d'unicité est de montrer que la différence entre une solution satisfaisant les hypothèses et le multi-soliton régulier construit dans le théorème 1.9 est nulle. Pour cela, nous considérons une certaine fonctionnelle d'énergie de type Weinstein en une nouvelle variable ˜ , obtenue à partir de par modulation, comme dans [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] (l'intérêt de la modulation est d'assurer des conditions d'orthogonalité requises pour la coercivité de sur laquelle nous fondons notre preuve). Cette fonctionnelle tire son inspiration de [START_REF] Martel | Stability in 1 of the sum of solitary waves for some nonlinear Schrödinger equations[END_REF] et s'écrit :

( ) := =1 ∫ R |∇ ˜ | 2 - | | 2 | ˜ | 2 + 2Re( ˜ ) 2 | | 2 + + | | 2 4 | ˜ | 2 -• Im ∇ ˜ ˜ ( , ) . (1.11)
Le travail sur la variable ˜ semble aussi mieux adapté au cadre particulier de (NLS) considéré ici. Soulignons que, si un contrôle en O - 2 1 permet d'obtenir l'unicité dans le cadre le plus général pour (gKdV), le contrôle de la dérivée de par O 1 ˜ 2 1 explique que l'on doive se restreindre à la classe vérifiant (1.10). En raison du manque d'estimées exponentiellement décroissantes en temps (qui sont intéressantes dans la mesure où l'on ne perd rien lorsqu'on intègre en temps long), les différentes inégalités démontrées et utiles pour conclure dans ce cadre sont moins fortes que celles de [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] pour (gKdV) et demandent par conséquent une analyse plus fine. L'inégalité typique obtenue en rassemblant toutes les majorations intermédiaires et qui permet de conclure la preuve prend la forme

˜ ( ) 1 ≤ ∫ +∞ ˜ ( ) 1 .
Il ne semble pas clair de pouvoir améliorer cette inégalité avec la méthode que nous employons, fondée sur l'analyse de la dérivée de .

(proximité du soliton) ∀ ∈ R, ( , • + ( )) -0 1 ≤ , (1.12
)

(non-dispersion) ∀ > 0, ∃ > 0, ∀ ∈ R, ∫ | |> 2 ( , + ( )) ≤ , (1.13) alors il existe 1 > 0, 1 ∈ R tels que ∀ , ∈ R, ( , ) = 1 ( -1 -1 ).
Autrement dit, les solutions globales de (gKdV) qui sont suffisamment proches pour tout temps d'un état fondamental, mais qui ne sont pas des solitons, doivent nécessairement disperser au sens où la propriété de 2 -compacité (1.13) n'est pas satisfaite. Pour de telles solutions, la masse n'est plus concentrée en tout temps autour de ( ), et ceci aussi loin que l'on veut de ( ). Le théorème précédent suggère que les solitons sont des objets très rigides. À chaque fois, la stratégie mise en place pour prouver les théorèmes de Liouville précédents est la suivante : se ramener au cadre linéaire et y montrer une propriété de rigidité analogue qui concerne l'équation linéarisée autour d'un soliton, en utilisant toutefois des outils d'analyse nonlinéaire (arguments du Viriel, de monotonie) qui consistent à étudier la dérivée de quantités du type ∫ R 2 ( , ) ( ) pour des choix judicieux de fonctions et . S'il est possible de travailler avec ( ) = , les preuves, à l'instar de [START_REF] Martel | Linear problems related to asymptotic stability of solitons of the generalized kdv equations[END_REF] et [START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF], ont progressivement gagné en efficacité en considérant ( ) = -0 ( ) 0 ( ) et une variable , « duale » de , construite à l'aide de l'opérateur linéarisé autour de 0 et obtenue après modulation. Dans [START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF], Martel et Merle travaillent ainsi avec la variable Notons que les propriétés de stabilité asymptotique 1 des solitons et des multi-solitons ont été développées en parallèle des résultats de rigidité précédents pour la raison que ces derniers permettent de récupérer la stabilité asymptotique des solitons [START_REF] Martel | Asymptotic stability of solitons for subcritical gKdV equations[END_REF][START_REF] Martel | Asymptotic stability of solitons of the subcritical gKdV equations revisited[END_REF][START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF] et des multi-solitons [START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF][START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of solitons for subcritical gKdV equations[END_REF]. À notre connaissance, très peu d'articles démontrent des résultats de stabilité asymptotique pour (gKdV) sans l'appui d'une propriété de type Liouville (l'article [START_REF] Pego | Asymptotic stability of solitary waves[END_REF] est en cela un des rares exemples).

:= 0 -( 0 + ) -( 0 ) - ( 0 ) ,
Notre propos est de fournir une propriété de Liouville analogue au théorème 1.11, adaptée aux multi-solitons. Nous considérons des solutions de (gKdV) qui sont non-dispersives au sens de la définition suivante. Définition 1.12. On dit qu'un élément

∈ C ( [ , +∞), 2 (R)) est non-dispersif s'il existe > 0 tel que ∀ > 0, ∃ > 0, ∀ ≥ , ∫ < - 2 ( , ) ≤ . (1.14)
Intuitivement, une solution de (gKdV) non-dispersive au sens de la définition précédente est une solution qui ne possède pas de masse à gauche en espace et pour laquelle toute la masse se déplace à droite en espace lorsque augmente : observons en effet que la masse est essentiellement localisée à droite de .

Exemple de fonction « non-dispersive » pour > 0 au sens de (1.14) 2 Exemple de fonction qui « disperse » pour > 0 au sens de (1.14)

: ( , ) ↦ → sin( - √ ) - √ √ A titre d'exemple, une fonction ∈ C ( [ , +∞), 2 (R)) satisfaisant ∀ ≥ , ∀ ≤ , | ( , )| ≤ -| -| (1.15)
pour certaines constantes et , est non-dispersive. Nous démontrons réciproquement que toute solution ∈ C ( [ , +∞), 1 (R)) de (gKdV) nondispersive et uniformément bornée dans 1 (R) vérifie (1.15). Plus encore, il s'agit d'un élément de C ∞ ([ , +∞) × R) pour lequel toutes les dérivées partielles vérifient une inégalité de la forme (1.15). C'est une observation fondamentale, dont la preuve s'appuie sur une inégalité de monotonie forte, inspirée de Martel et Merle [START_REF] Martel | Asymptotic stability of solitons for subcritical gKdV equations[END_REF] et de Laurent et Martel [START_REF] Laurent | Smoothness and exponential decay of 2 -compact solutions of the generalized KdV equations[END_REF].

Nous prouvons au chapitre 3 que de telles solutions qui sont uniformément proches d'une somme de solitons découplés sont des multi-solitons. Théorème 1.13 (F.). Soit une solution de (gKdV) qui appartient à C ( [0, +∞), 1 (R)). Supposons l'existence de > 0 tel que

∀ > 0, ∃ > 0, ∀ ≥ 0, ∫ < - 2 ( , ) ≤ . (1.16) Soit ≥ 1 un entier et soient réels positifs 0 < 1 < • • • < . Il existe = ( 1 , . . . , , ) > 0 tel que, s'il existe fonctions 1 , . . . , : R + → R de classe C 1 qui satisfont ∀ ≥ 0, ( ) - =1 (• -( )) 1 ≤ , (1.17) et ∀ ≥ 0, ∀ ∈ {1, . . . , -1}, +1 ( ) -( ) ≥ | ln |, (1.18)
alors est un multi-soliton (en +∞). Autrement dit, il existe > 0, 0

< + 1 < • • • < + , + 1
, . . . , + ∈ R et des constantes positives telles que pour tout ≥ 0, pour tout ≥ 0,

( ) - =1 + (• -+ -+ ) ≤ -.
Le théorème précédent fournit ainsi une caractérisation dynamique des multi-solitons. Observons que les hypothèses sont faites ici pour des temps positifs ≥ 0 et non pas pour tous les temps ∈ R. De la sorte, ce théorème étend mais raffine également le théorème 1.11. Bien sûr, dans le cas 2 -sous-critique 1 < < 5, les hypothèses (1.17) et (1.18) peuvent être allégées et ne tenir en somme qu'au temps = 0 dans la mesure où les sommes de solitons découplés sont stables [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of solitons for subcritical gKdV equations[END_REF].

Dans la preuve du théorème 1.13 que nous exposons, deux ingrédients majeurs sont utilisés. D'une part, la propriété (3.6) de non-dispersion permet d'obtenir un contrôle de et de sa dérivée sur la région ≤ pour un certain > 0 suffisamment petit ; cela implique que

( ) 1 ( ≤ ) → 0 lorsque → +∞.
D'autre part, la propriété de stabilité asymptotique dans l'espace d'énergie, dont nous disposons grâce aux hypothèses complémentaires (1.17) et (1.18), permet d'avoir un contrôle, lorsque → +∞, d'une quantité de la forme ( )

-=1 + (• -( )) 1 ( ≥ ) où 0 < + 1 < • • • < + et où pour tout , les fonctions ↦ → ( ) ∈ R de classe C 1 proviennent d'un argument de modulation.
Lorsque les deux propriétés intermédiaires sont réunies, nous obtenons que

( ) - =1 + (• -( )) 1 → 0, lorsque → +∞, où +1 ( ) -( ) ≥ et ( ) ≥ pour un certain > 0.
Il reste alors à affiner ce résultat pour obtenir la convergence suivante, pour certains + 1 , . . . , + ∈ R :

( ) - =1 + (• -+ -+ ) 1 → 0, lorsque → +∞.
Pour ce faire, nous suivons le schéma de la preuve de Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]Proposition 4] et montrons en fait que la dernière convergence vers 0 est réalisée à vitesse exponentielle.

Il convient de mentionner que le contexte particulier de (KdV) qui correspond à = 2 offre une caractérisation simplifiée des multi-solitons ; aussi, nous obtenons, à l'aide des résultats de résolution en solitons d'Eckhaus et Schuur [START_REF] Eckhaus | The emergence of solutions for the generalized Korteweg-de Vries equation from arbitrary initial conditions[END_REF] et de Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF], le théorème suivant. Théorème 1.14 (F.). Soit 0 ∈ S (R) \ {0}. On suppose que la solution de (KdV) qui correspond à la donnée initiale 0 , qui est globale en temps, est non-dispersive en temps positif, c'est-à-dire satisfait (1.16). Alors est un multi-soliton.

Ajoutons que nous pouvons également caractériser de façon analogue les solutions non-dispersives de l'équation (mKdV) : ce sont génériquement des multi-breathers. L'équation (mKdV) admet des solutions particulières appelées breathers qui interviennent également dans la réponse à la question de la conjecture de résolution en solitons. Pour tout ( , Alors est un multi-breather en +∞ avec des vitesses positives : il existe 1 , 2 ∈ N avec

) ∈ R * + × R * + , et tous 1 , 2 ∈ R,
1 + 2 ≥ 1, il existe 0 < 1 < • • • < 1 , il existe , ∈ R * + pour = 1, . . . , 2 avec 0 < 2 1 -3 2 1 < • • • < 2 2 -3 2 2
et il existe > 0, des constantes positives , des signes = ±1 et des réels 0, , 1, , 2, tels que pour tout ≥ 0, appartienne à C ( [0, +∞), (R)) et

∀ ≥ 0, ( ) - 1 =1 2 , 0, ( ) - 2 =1 √ 2 , √ 2 , 1, , 2, ( ) ≤ -.
La preuve s'inspire de celle du théorème 1.14 : il s'agit d'écrire la décomposition en solitons et breathers disponible pour = 3 et d'utiliser la régularité, l'unicité et les estimées pour les multi-breathers démontrées par Semenov [START_REF] Semenov | On the existence and uniqueness of multi-breathers of (mKdV)[END_REF].

Comportement ponctuel des multi-solitons de (gKdV) : résultats et conjecture

Nous considérons à présent la question du comportement ponctuel des multi-solitons, à fixé. Un tel multi-soliton décroît-il comme une somme de solitons ?

A ce sujet, nous démontrons le théorème suivant.

Théorème 1.16 (F.). Soient > 1, 0 < 1 < • • • < , 1 , . . . , ∈ R et ∈ C ( [ , +∞), 1 (R)) un multi-soliton
( , ) ≤ -, | -| + =1 - √ | -| . (1.21)
Pour tout ∈ N et pour tout ∈ N, il existe , > 0 tel que pour tout ≥ , pour tout > , (décroissance polynomiale à droite de la dernière onde solitaire)

( , ) ≤ , ( -) . (1.22)
De plus, si ∈ ∞ (R, 1 (R)) (par exemple si < 5), alors il existe > 0, > 0 tel que pour tout ∈ N, il existe > 0 tel que pour tout ≥ , pour tout ≥ , (décroissance exponentielle à droite du dernier soliton)

( , ) ≤ -| -| . (1.23)
La décroissance exponentielle à gauche des multi-solitons de (gKdV) (sur une région de la forme ≤ ) est liée au caractère non-dispersif des multi-solitons sur une telle région et découle ainsi d'un argument de monotonie similaire à celui que nous mettons en oeuvre pour démontrer le théorème 1.13.

Sur un intervalle de la forme [ , ], l'inégalité obtenue est conséquence de la décroissance exponentielle en temps des quantités ( ) -=1 , ( ) , ∈ N. Enfin à droite, sur une région de la forme > , il semble en revanche plus délicat de montrer que les multi-solitons décroissent à vitesse exponentielle en espace. Ici, l'argument de monotonie ne s'applique plus si le multi-soliton n'est pas supposé global. A partir des estimées de convergence exponentielle en temps pour toutes les normes , nous obtenons une décroissance plus rapide que tout polynôme pour chacune des dérivées du multi-soliton. En particulier, nous obtenons que les multi-solitons appartiennent à l'espace de Schwartz. La décroissance rapide obtenue implique en particulier de la non-dispersion à droite pour le multi-soliton , ce qui permet d'obtenir (1.23) via de la monotonie dans le cas où est défini pour tout temps ∈ R ; c'est notamment le cas dans le cas 2 -sous-critique.

Notons qu'un concept essentiel dans la théorie développée pour (gKdV) réside dans « l'effet régularisant de Kato » (ou Kato smoothing effect) [START_REF] Kato | On the Cauchy problem for the (generalized) Korteweg-de Vries equation[END_REF], et si dans les cas 2 -critique et 2 -surcritique, les multi-solitons devaient également décroître à vitesse exponentielle à droite de la dernière onde solitaire (ce qui est actuellement au stade de conjecture), il est possible que le gain de dérivées en espace traduit par cet effet régularisant soit utile à la preuve correspondante.

Famille de multi-solitons pour les équations de Klein-Gordon

Nous nous intéressons ici à une autre équation pour laquelle il est possible de considérer des multisolitons : l'équation de Klein-Gordon non-linéaire qui apparaît en physique quantique des champs

2 = Δ -+ | | -1 , (NLKG)
où est une fonction de ( , ) ∈ R × R à valeurs réelles et > 1.

Cette équation de nature hyperbolique présente des caractéristiques différentes des équations (gKdV) et (NLS) introduites dans les parties précédentes. Par exemple, la norme 2 de ( ) n'est pas conservée au cours du temps ; en outre, (NLKG) ne dispose guère de propriété d'invariance par scaling.

Néanmoins, les deux quantités remarquables suivantes sont conservées par une solution de (NLKG) à valeurs dans 1 (R ) × 2 (R ) :

• l'énergie 1 2 ∫ R ( ) 2 + |∇ | 2 + 2 -2 +1 | | +1 ( , )
• le moment ∫ R { ∇ } ( , ) . On observe aussi que l'ensemble des solutions de (NLKG) est laissé invariant sous l'action de « boosts » lorentziens, c'est-à-dire sous l'action de matrices de la forme

Λ := - - + -1 | | 2 où ∈ R est de norme euclidienne | | < 1 et := 1 √ 1-| | 2 .
Aussi, l'équation (NLKG) admet des solitons, obtenus comme « ground states » boostés à partir d'une solution positive ∈ 1 (R ) de l'équation fondamentale

Δ -+ = 0,
de la façon suivante :

, 0 : ( , ) ↦ → • Λ ( , -0 ) , pour un choix de paramètres ∈ R, | | < 1 et 0 ∈ R , où := 1 √ 1-| | 2 et est la projection
canonique R 1+ → R sur les dernières coordonnées. Dans le cas unidimensionnel en particulier, le soliton précédent se réécrit :

, 0 : ( , ) ↦ → ( --0 ) . Ces solitons sont des objets orbitalement instables, et cela indépendamment de la valeur de [START_REF] Derrick | Comments on nonlinear wave equations as models for elementary particles[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] ; cela constitue une différence notable avec les équations de Korteweg-de Vries ou de Schrödinger. Comme il est commode de réécrire (NLKG) comme un système différentiel d'ordre 1 et de travailler avec des vecteurs à deux composantes, on notera dans la suite

, 0 ( , ) := , 0 ( , ) , 0 ( , )
.

La première construction d'un multi-soliton dans le contexte de (NLKG) est due à Côte et Muñoz [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF] ; elle repose sur des méthodes usuelles de compacité et d'énergie qui peuvent être mises en pratique grâce à la théorie spectrale des opérateurs linéarisés autour des solitons développée dans cet article. Notons que cette construction a été étendue par Côte et Martel [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF] en démontrant l'existence de « multi-bound states » qui sont l'analogue des multi-solitons, mais définis à l'aide d'états excités et non plus simplement à partir de « ground states ».

Focalisés plus particulièrement sur la dynamique en temps long de solutions de l'équation (NLKG) au voisinage d'une somme de solitons, nous démontrons l'existence d'une famille à paramètres de multi-solitons. Précisément, nous avons Théorème 1.17 (F.). Soit > 2. Considérons un entier naturel ≥ 2 et 2 paramètres

1 , . . . , ∈ R et 1 , . . . , ∈ R tels que 0 < | | < • • • < | 1 | < 1. Alors il existe > 0, 0 < 1 < • • • < , +, ∈ C (R, 1 (R ) × 2 (R )) pour = 1, . . . , et une famille à paramètres ( 1 ,..., ) ( 1 ,..., ) ∈R de solutions de (NLKG) telle que pour tout ( 1 , . . . , ) ∈ R , il existe 0 ∈ R et > 0 tels que ∀ ≥ 0 , Φ 1 ,..., ( ) - =1 , ( ) - =1 - +, ( ) 1 × 2 ≤ -( + ) , (1.24) où Φ 1 ,..., := 1 ,..., 1 ,..., . De plus, si ( 1 , . . . , ) ≠ ( 1 , . . . , ), alors 1 ,..., ≠ 1 ,...,
.

La famille de multi-solitons décrite par le théorème précédent est construite en adaptant à (NLKG) la construction effectuée par Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF][START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF] dans le cadre des équations (gKdV) et (NLS). Cette construction s'appuie sur un argument topologique établi originellement dans [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] et est possible grâce à la bonne connaissance des opérateurs linéarisés autour des « ground states ». Nous utilisons en effet les notations introduites par Côte et Muñoz [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF], réemployées par Côte et Martel [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF], ainsi que les propriétés spectrales démontrées dans leur article.

En ce qui concerne la question de la classification, nous montrons réciproquement le théorème suivant.

Théorème 1.18 (F.). Sous les mêmes hypothèses que le théorème précédent, si est une solution de (NLKG) telle que

( ) - =1 , ( ) 1 × 2 = O 1 lorsque → +∞, (1.25
)

où = et > 3, alors il existe 1 , . . . , ∈ R et 0 ∈ R tels que pour tout ≥ 0 , ( ) = Φ 1 ,..., ( ).
L'unicité de cette famille de multi-solitons en restriction à la classe satisfaisant (1.25) est obtenue grâce à la presque-monotonie d'une fonctionnelle F bien choisie, inspirée de [START_REF] Martel | Construction of multi-Solitons for the energy-critical wave equation in dimension 5[END_REF]. L'inégalité essentielle qui permet d'améliorer la vitesse de convergence de

( ) := ( ) - =1 , ( ) vers 0 dans 1 (R ) × 2 (R ) est de la forme -F ( ) ≤ F ( ) + ℎ ( , ( ) 1 × 2 ) ,
pour un certain paramètre ∈ (1, -1) et une fonction ℎ telle que ↦ → ℎ ( , ( ) 1 × 2 ) est intégrable au voisinage de +∞ et dont le comportement en temps long est meilleur que simplement quadratique en ( )

1 × 2 , essentiellement en sup ≥ ( ) 2 1 × 2 .
Lorsqu'un seul soliton est considéré, nous obtenons une caractérisation complète des solutions qui convergent vers ce soliton lorsque → +∞.

Théorème 1.19 (F.). Soit ∈ R avec | | < 1 et soit > 2. Il existe > 0, +, ∈ C (R, 1 (R )× 2 (R )) et une famille ( ) ∈R à un paramètre de solutions de (NLKG) telle que pour tout ∈ R, il existe 0 = 0 ( ) ∈ R tel que pour tout ≥ 0 ( ) -( ) -- +, ( ) 1 × 2 ≤ -2 , (1.26
)

où := . De plus, si ≠ , alors ≠ .
Par ailleurs, si est une solution de (NLKG) telle que

( ) -( ) 1 × 2 → 0, lorsque → +∞, (1.27) où = , alors il existe ∈ R et 0 ∈ R tels que pour tout ≥ 0 , ( ) = ( ).
L'existence d'une infinité de solutions est générée par la direction d'instabilité due à l'existence de la valeur propre négative (liée à ) de l'opérateur linéarisé autour du soliton . Ce phénomène est naturel pour l'étude de solitons instables.

Si l'énoncé précédent se construit sur le modèle de [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF] attaché à (gKdV) et s'inspire de plusieurs résultats de classification obtenus auparavant, comme pour (NLS) cubique tridimensionnelle [START_REF] Duyckaerts | Threshold solutions for the focusing 3D cubic Schrödinger equation[END_REF], (NLS) 1 -critique [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical NLS[END_REF] ou encore l'équation des ondes dans le cas 1 -critique [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF], notre démonstration relève d'une approche différente, dans la mesure où celle-ci repose pleinement sur la théorie spectrale des opérateurs linéarisés au voisinage des solitons.

La classification établie dans le théorème 1.19 laisse supposer que pour ≥ 1 quelconque, les -solitons construits dans le théorème 1.17 forment exactement l'ensemble des multi-solitons tels que

- =1 , ( ) 1 × 2 → →+∞ 0 où =
. Cependant, cela demeure au stade de conjecture et l'application d'un argument usuel de coercivité ne permet apparemment pas de décrire et d'améliorer de façon évidente la vitesse de décroissance de -=1 , ( ) 1 × 2 vers 0. En outre, il existe une différence profonde entre le cas où un seul soliton est considéré ( = 1) et le cas où ≥ 2 : le recours à un argument de type point fixe, possible dans le cas = 1 et initié par Duyckaerts et Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF][START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical NLS[END_REF] ne paraît pas adapté au cadre des multi-solitons.

Malgré tout, notre argument permet de récupérer tous les multi-solitons d'une classe raisonnable, plus large que la classe à décroissance exponentielle en temps. De plus, la démonstration du théorème 1.18 donne l'espoir d'une réponse à la question de l'unicité ou de la classification générale des multi-solitons pour d'autres équations aux dérivées partielles, au moins dans la classe des solutions qui convergent polynomialement en temps vers une somme de solitons, c'est-à-dire au sens de (1.25).

Organisation des chapitres

Les chapitres qui suivent cette introduction générale sont consacrés aux développements des résultats obtenus dans le cadre de cette thèse.

Aussi, le deuxième chapitre étudie les propriétés de régularité et d'unicité des multi-solitons des équations de Schrödinger non-linéaires (NLS) et s'attache tout particulièrement aux théorèmes 1.9 et 1.10.

Le troisième volet ouvre sur la propriété de non-dispersion qui s'avère essentielle pour caractériser les multi-solitons des équations de Korteweg-de Vries généralisées (gKdV). Ce chapitre a donc vocation à contenir les démonstrations des théorèmes 1.13, 1.14 et 1.15.

La quatrième partie fait en outre état du comportement des multi-solitons de (gKdV) en espace (théorème 1.16).

Enfin, la cinquième partie fait l'objet de la construction de multi-solitons pour les équations de Klein-Gordon non-linéaires (NLKG) avec le théorème 1.17 ; elle aborde aussi le problème d'unicité pour la classe de solutions qui convergent vers un soliton (théorème 1.19) ou vers une somme de solitons (théorème 1.17).

Introduction

Generalities on the non-linear Schrödinger equations

We consider non-linear Schrödinger equations in R which admit traveling solitary waves (solitons). More precisely, we focus on

= Δ + (| | 2 ) , (NLS)
where

: × R → C, ⊂ R is a time interval, and : [0, +∞) → R is an 1 -subcritical non-linearity.
For ≤ 3 and for particular functions , equation (NLS) arises in the mathematical description of many physical phenomena; it is used mainly to model non-linear wave dynamics. For instance, it is fundamental in the description of the dynamic of particles moving in electromagnetic fields [START_REF] Malomed | Nonlinear Schrödinger Equations[END_REF] and quantum systems like Bose-Einstein condensates [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF]. With particular non-linearities obtained by linear combinations of quadratic, cubic, and quintic terms it appears also when one tries to describe the propagation of laser beams in some mediums [START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF] or of more general ultrashort optical pulses (see for example [START_REF] Kumar De | Few-cycle optical solitary waves in cascaded-quadratic-cubic-quintic nonlinear media[END_REF] for the study of some solutions of these equations), with applications in medical imaging, material processing and optical communications (we refer to [START_REF] Felice | A study of a nonlinear Schrödinger equation for optical fibers[END_REF] for further Ce chapitre fait l'objet d'un article rédigé en collaboration avec Raphaël Côte et accepté pour publication dans Communications in Partial Differential Equations [START_REF] Côte | On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations[END_REF].

CHAPTER 2. ON SMOOTHNESS AND UNIQUENESS OF NLS MULTI-SOLITONS

details concerning the applications of (NLS) to fiber optics for example).

For the purpose of the exposition, we focus in this paragraph on pure power non-linearities

( ) = -1 2 , 1 < < 1 + 4 ( -2) + , ≥ 0. (2.1) (If = 1 or 2, the condition is > 1 and if ≥ 3, the condition is 1 < < 1 + 4 -2
). We will give results on general non-linearities in paragraph 2.1.3.

Ginibre and Velo [START_REF] Ginibre | On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case[END_REF] proved that (NLS) is locally well-posed in 1 (R ): for all 0 ∈ 1 (R ), there exist > 0 and a unique maximal solution ∈ C ( [0, ), 1 (R )) of (NLS) such that (0) = 0 . For any such 1 solution, the following quantities are conserved for all ∈ [0, ):

• the 2 mass ∫ R | ( , )| 2 . • the energy ∫ R 1 2 |∇ ( , )| 2 - 1 + 1 | ( , )| +1 . • the momentum Im ∫ R ∇ ( , ) ( , ) .
Furthermore, for all ∈ N \ {0, 1}, if

: ↦ → (| | 2 ) = | | -1
is C on C as an R-differentiable function (that is if > or is an odd integer), and in case where < 2 , if in addition < 1 + 4 -2 , then (NLS) is locally well-posed in (R ) according to Kato [START_REF] Kato | On nonlinear Schrödinger equations. II, solutions and unconditional wellposedness[END_REF]Theorem 4.1]. Also (NLS) is invariant under properties of space-time translation, phase, and galilean invariances: if 0 ∈ R, ∈ R , 0 ∈ R , ∈ R, and is a solution to (NLS), then

( , ) ↦ → ( -0 , -0 -) 1 2 • -| | 2 4 + (2.2)
is also a solution to (NLS). What is more, (NLS) with a pure power non-linearity (2.1) is scaling invariant: if > 0 and is a solution to (NLS), then

( , ) ↦ → 1 1/( -1) , 1/2 (2.3)
is still a solution to (NLS).

Let us introduce now some particular solutions of (NLS) which are essential in the theory and on which our paper is based. Given > 0, Berestycki and Lions [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF] proved the existence of a (non-vanishing) positive radial solution ∈ 1 (R ) to the following elliptic problem

Δ + ( 2 ) = , > 0 (2.4)
(By scaling, it suffices to prove the existence for = 1). A solution to (2.4) is called a ground state (and if one relaxes the sign condition, we speak of bound state). Using a Pohozaev identity [START_REF] Stanislas | Eigenfunctions of the equation Δ + ( ) = 0[END_REF], one can show that (2.4) has no solution in 1 (R ) for ≥ 1 + 4 ( -2) + . Moreover, for ∈ N * and if is C on [0, +∞), then is C +2 on R and one has exponential decay (see [2, proof of Lemma 1]): there exists > 0 such that for each multi-index

∈ N with | | ≤ + 2, ∀ ∈ R , ( ) ≤ - √ 2 | | . (2.5)
Then the function

( , ) ↦ → ( ) (2.6)
is a solution to (NLS). Using the invariances (2.2) of the equation, one obtains a whole family of solutions of (NLS) known as solitons.

Dynamical properties of solitons have been extensively studied. One important result is related to their orbital stability: solitons are orbitally stable if < 1 + 4 and unstable if ≥ 1 + 4 . Recall that the case = 1 + 4 corresponds to the 2 -critical exponent: in this particular case, the 2 norm of a solution is invariant by scaling (2.3).

In this article, we are interested in qualitative properties of multi-solitons, that is solutions of (NLS) which behave as a sum of decoupled solitary waves for large times.

Let us begin with the definition of some further notations. Fix ∈ N \ {0, 1} and for all = 1, . . . , , let > 0, ∈ R, 0 ∈ R , and ∈ R such that for all ≠ , ≠ .

For all = 1, . . . , , we consider

( , ) = ( -0 - ) 1 2 • + - | | 2 4 + ,
which is a soliton of (NLS) moving on the line = 0 + . We denote also

:= =1 .
In general, is obviously not a solution to (NLS) because of the non-linearity. A multi-soliton is a solution of (NLS) defined on [ 0 , +∞) for some 0 ∈ R and such that

lim →+∞ ( ) -( ) 1 = 0. (2.7)
Multi-solitons were explicitly constructed in the integrable case, that is with ( ) = and = 1, using the inverse scattering method (see Zakharov and Shabat [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]).

The first construction in a non-integrable context is due to Merle [START_REF] Merle | Construction of solutions with exactly blow-up points for the Schrödinger equation with critical nonlinearity[END_REF], in the critical case = 1 + 4 . Later, following closely the ideas of Martel in [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] for the construction of multi-solitons for the Korteweg-de Vries equations, Martel and Merle [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF] constructed multi-solitons of (NLS), in the 2 -subcritical case 1 < < 1 + 4 . This result was extended to 2 -supercritical exponent by Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]. Let us recall the results. Theorem 2.1 (Merle [85], Martel and Merle [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF], Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]). There exist > 0 (depending on , for 1 ≤ ≤ ), 0 ≥ 0, and a solution

∈ C ( [ 0 , +∞), 1 (R )) of (NLS) such that ∀ ≥ 0 , ( ) -( ) 1 ≤ -2 . (2.8)
Let us also mention the works by Le Coz and Tsai [START_REF] Le | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] and Le Coz, Li and Tsai [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF] where infinite trains of solitons are constructed, in the context of (NLS). The construction of multi-solitons in 1 was done for many other non-linear dispersive models (besides the generalized Korteweg-de Vries equations) such as the non-linear Klein-Gordon equation [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF], the Hartree equation [START_REF] Krieger | Two-soliton solutions to the threedimensional gravitational Hartree equation[END_REF], the water-waves system [START_REF] Ming | Multi-solitons and related solutions for the water-waves system[END_REF], and in both stable and unstable contexts, which means assuming that all are stable or not. Even though solutions of (NLS) behaving as a sum of decoupled general bound states (that is, solutions to (2.4) which change sign) have been studied in the last years (see for example [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF] on (NLS) or [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF] on non-linear Klein-Gordon equation), in the present paper we concentrate only on multi-solitons based on ground states. Our goal here is to study uniqueness and smoothness issues.

To our knowledge, the only work where multi-solitons are shown to be more regular than 1 is for the generalized Korteweg-de Vries equation (which is one-dimensional), with monomial nonlinearity, by Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF], where the exponential convergence (2.8) is shown to hold in (R) for all ∈ N (with a constant depending on in front of the exponential term and a convergence rate independent of ): see Proposition 5 and its proof for the 2 -subcritical and critical cases; the 2 -supercritical case can be treated likewise, as it is mentioned in [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]Remark 1]).

A natural question is thus to understand for (NLS) whether the multi-soliton in Theorem 2.1 is smoother than 1 : for example, does it belong to C ( [ 0 , +∞), (R )) for > 1 and does it hold ( ) -( ) → 0 as → +∞? Another natural question is the uniqueness or the classification of multi-solitons. Again, to our knowledge, the only complete study of the question was done for the generalized Korteweg-de Vries equations: multi-solitons were proved to be unique in the 2 -subcritical and critical cases by Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF], and were classified in the 2 -supercritical case by Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF] (there is a -parameter family of -solitons, each instability direction yielding a free parameter). Actually, smoothness of the multi-solitons constructed in Theorem 2.1 is an important ingredient in the proof of uniqueness (or classification) in dimension ≥ 2.

Main results

Our first result concerns the construction of a multi-soliton in (R ), where the regularity index > 1 depends on the regularity of the function . We prove in particular that the convergence occurs with an exponential rate in (R ). The result is stated here for pure power non-linearities, and we will discuss general non-linearities in the next paragraph. Theorem 2.2 (Smoothness of multi-solitons). Assume that ≥ 3. Let > 0 be defined as in Theorem 2.1 and 0 = -1 ≥ 2, or 0 = +∞ if is an odd integer. There exist 1 > 0 and ∈ C ([ 1 , +∞), 0 (R )) a solution of (NLS) with pure power nonlinearity (2.1) such that for all non-negative integer ≤ 0 , there exists ≥ 1 such that

∀ ≥ 1 , ( ) -( ) ≤ -2 +1 . (2.9)
Moreover, if is an odd integer, then for all integer ≥ 0,

∀ ≥ 1 , ( ) -( ) ≤ -.
(2.10)

Remark 2.1. Theorem 2.2 completes Theorem 2.1 by showing the existence of smooth multisolitons. Notice that its applications are limited to dimensions ≤ 3, since we consider the pure power case and due to the 1 -subcritical assumption < 1+ 4 -2 which is required for the existence of solitons.

In particular, in dimension = 1 and = 2, multi-solitons belong to ∞ (R ) when is an odd integer, and in dimension = 3, multi-solitons are ∞ (R 3 ) when = 3 (which corresponds to the most physically relevant case).

The exponential decay rate 2 +1 is optimal for ≥ 2 (obviously, the estimate (2.9) obtained for = 1 is worse than (2.8) but this is not of interest in the context of Theorem 2.2). We underline that this exponential decay rate does depend on (vanishing when is large); this is due to some loss, passing from the proof of the -estimate to that of the +1 -estimate. This dependence could be a problem for some applications. Observe however that if one is willing to consider only regularity indices ≤ 0 2 (say), then a straightforward interpolation argument between the 1 and 0 bounds gives the convergence with uniform exponential decay rate :

∀ ≤ 0 2 , ∀ ≥ 1 , ( ) -( ) ≤ -.
Our second goal is to obtain some kind of uniqueness result for (NLS). We derive one for 2subcritical and critical (NLS), in the class of multi-solitons such that ( ) -( ) 1 decreases faster than a high power of 1 for large values of . More precisely, we state the following:

Theorem 2.3 (Conditional uniqueness). Let ≤ 2 and 3 ≤ ≤ 1 + 4 . There exists ∈ N large such that there is a unique ∈ C ( [ 1 , +∞), 1 (R )) solution to (NLS) such that ( ) -( ) 1 = O 1 , as → +∞. (2.11)
In particular, the multi-solitons of Theorems 2.1 and 2.2 coincide (and one can take 0 = 1 ).

Remark 2.2. The crucial point in Theorem 2.3 is obviously the uniqueness part. For pure power non-linearities, Theorem 2.3 provides conditional uniqueness in the sense of (2.11), in the 2subcritical and critical cases with ≥ 3 in dimension 1, and in the 2 -critical case = 3 in dimension 2.

The requirement that the non-linearity be 2 -subcritical or 2 -critical is to be expected as no uniqueness holds in the 2 -supercritical case; see Côte and Le Coz [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF] for example.

Of course, one would expect an unconditional uniqueness result, that is uniqueness in the class of solutions defined for large enough times and convergent to the profile without decay rate:

( ) -( ) 1 → 0 as → +∞.
This seems out of reach with our method, but we would like to point out that Theorem 2.3 already allows to break the class of exponential convergence, in which multi-solitons naturally lie, and as it was done in [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Le | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] for example.

General non-linearities

In order to consider general non-linearities, one must make a number of assumptions which we discuss in this paragraph.

Well-posedness in 1 (R ) is classically done under the hypothesis that : C → C is C 1 and satisfies (H1) (0) = 0 and there exists ∈ 1, 1

+ 4 -2 such that ( ), ( ) = O(| | -1 ) as | | → +∞.
In order that the Cauchy problem for (NLS) be well-posed in for ∈ N \ {0, 1}, Kato [48, Theorem 4.1] requires furthermore that is C , and if ≤ 2 , one also needs:

(H2) if is a polynomial in and , its degree is deg < 1 + 4 -2 if is not a polynomial, there exists ∈ , 1 + 4 -2 such that -( ) = O(| | -) as | | → +∞, for all = 0, . . . , , with
the smallest integer greater or equal to .

The existence of solitons with frequency > 0 is not as immediate as in the pure power case. Under the assumption that

∃ ∈ R * + , ( ) > where
( ) := ∫ 0 ( ) (2.12)
for ≥ 0, Berestycki and Lions [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF] showed the existence of a positive radial ground state ∈ 1 (R ) to (2.4). Note that if there exist ˜ > 0, > 0 and 0 > 0 such that ∀ ≥ 0 , ( ) ≥ ˜ , then (2.12) holds for all > 0. If = 1, a necessary and sufficient condition for the existence of a positive solution (2.4) is that is such that

0 := inf > 0 ( ) = (2.13)
exists and ( 0 ) > (see [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF]). Let us denote by O a subset of (0, +∞) such that for all ∈ O, there exists a solution to (2.4).

(2.14)

Recall that when it exists, a positive solution of (2.4) is radial (see [2, section 3] and also Gidas, Ni and Nirenberg [33, Theorem 1'] for non-linearities such that ↦ → ( 2) is increasing). We underline that it is not unique in general. Indeed, Dàvila, Pino and Guerra [START_REF] Dávila | Non-uniqueness of positive ground states of non-linear Schrödinger equations[END_REF] showed the existence of at least three positive 1 solutions of

Δ + + 2 =
for some > 0 and ∈ (1, 5) in dimension = 3. See [START_REF] Dávila | Non-uniqueness of positive ground states of non-linear Schrödinger equations[END_REF] for other counterexamples in dimension 3.

On the other side, Kwong [START_REF] Kwong | Uniqueness of positive solutions of Δ -+ = 0 in R[END_REF] showed uniqueness of a positive radial ground state in the pure power case, and one can extend this to more general non-linearities; we refer to Mc Leod and Serrin [START_REF] Mc | Uniqueness of positive radial solutions of Δ + ( ) = 0 in R[END_REF],

Serrin and Tang [START_REF] Serrin | Uniqueness of ground states for quasilinear elliptic equations[END_REF] and Jang [START_REF] Jang | Uniqueness of positive radial solutions of Δ + ( ) = 0 in R , ≥ 2[END_REF] for full details. One of the most important statements may be found in Serrin and Tang [START_REF] Serrin | Uniqueness of ground states for quasilinear elliptic equations[END_REF]: a sufficient condition for uniqueness when ≥ 3 is the existence of > 0 such that

       ∀ ∈ (0, ], ( ) ≤ 1, and ∀ ∈ ( , +∞), ( ) > 1 ↦ → ( ) ( ) -1
is not increasing on ( , +∞).

In [START_REF] Jang | Uniqueness of positive radial solutions of Δ + ( ) = 0 in R , ≥ 2[END_REF], a slightly more general condition (inspired by [START_REF] Serrin | Uniqueness of ground states for quasilinear elliptic equations[END_REF]) yields uniqueness for (2.4) in any dimension ≥ 2.

Let us point out that conditions for existence and uniquenes of a ground state have been discussed for specific non-linearities in the litterature. For example, Berestycki and Lions condition concerning existence and Serrin and Tang condition concerning uniqueness of a ground state apply to the (important) cubic-quintic non-linearity (corresponding to

( ) = | | 2 -| | 4 or ( ) = -2
). Killip, Oh, Pocovnicu and Visan studied more precisely the properties of ground states associated with this nonlinearity and showed in particular that existence and uniqueness of a positive radially symmetric solution to

Δ + 3 -5 =
hold if and only if ∈ O := 0, 3 16 ; see [51, Lemma 2.1 and Theorem 2.2].

Pursuing with general non-linearities, we will also need a number of assumptions on the linearized operators around solitons. Fix ∈ O, and let

L : 1 (R , C) → 1 (R , C) = 1 + 2 ↦ → -Δ + -( ( 2 ) + 2 2 ( 2 ) 1 )
so that the linearized equation of (NLS) around ( + ) is = L . We also define the linearized energy around , for any

= 1 + 2 ∈ 1 (R , C) ( ) : = ∫ R |∇ | 2 + | | 2 -( 2 )| | 2 + 2 2 ( 2 ) 2 1 = Re ∫ R L = ∫ R +, 1 1 + ∫ R -, 2 2
, where +,

1 := -Δ 1 + 1 -( 2 ) + 2 2 ( 2 ) 1 -, 2 := -Δ 2 + 2 -( 2 ) 2 .
We do two (mutually incompatible) coercivity assumptions, depending on whether is stable or not. They write as follows:

(H3) (Stable case) There exists + > 0 such that for all

= 1 + 2 ∈ 1 (R , C) ( ) ≥ + 2 1 - 1 + ∫ R 1 2 - 1 + =1 ∫ R 1 2 - 1 + ∫ R 2 2
.

(2.15) (H4) (Unstable case) There exists an eigenfunction

= 1 + 2 ∈ 1 (R , C) of L (with eigenvalue 0 > 0) and + > 0 such that for all = 1 + 2 ∈ 1 (R , C), ( ) ≥ + 2 1 - 1 + ∫ R 1 2 2 - 1 + =1 ∫ R 1 2 - 1 + ∫ R 2 1 2 + ∫ R 2
Ker( +, 0 ) = Span 0 , = 1, . . . , , (2.17)
we have the following dichotomy:

• If | = 0 ∫ R ( ) 2 > 0, then (2.15
) holds, and as a consequence, 0 is orbitally stable in 1 (R ).

• If | = 0 ∫ R ( ) 2 < 0, then 0 is orbitally unstable in 1 (R ).
We also refer to Cazenave and Lions [5, Theorem II.2 and Remark II.3] for another approach to 1 orbital stability of the solitons based on 0 . For the pure power case,

∫ R ( ) 2 = 2 -1 -2 -+3 -1 -2 ∫ R 1 ( ) 2
so that it is positive when 1 < < 1 + 4 , that is in the 2 -subcritical case (and in particular (H3) holds in that case) and it is negative when 1 + 4 < < +2 -2 , that is in the 2 -supercritical case.

Regarding the unstable case, following the ideas of Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF], Duyckaerts and Roudenko [START_REF] Duyckaerts | Threshold solutions for the focusing 3D cubic Schrödinger equation[END_REF], and Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF], the coercivity result below holds. Proposition 2.5 ((3.6) in [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]). Let ∈ O such that L admits a non zero eigenfunction ∈ 1 (R ). Then (2.16) holds.

An important step is therefore the construction of an eigenfunction : this can be done in the 2 -supercritical pure power case ( > 1 + 4 ) for all > 0, and so (H4) holds in that case.

We are now in a position to state our results for general non-linearities. For smoothness, it reads as follows.

Theorem 2.2 . Let 0 > 2 . Assume that satisfies (H1) and belongs to 0 +1,∞ (C). Assume moreover that for all = 1, . . . , , belongs to O and satisfies either (H3) or (H4). Then the conclusions of Theorem 2.2 hold.

And below is about uniqueness. Theorem 2.3 . Let ≤ 3 and ˜ : ↦ → (| | 2 ) be of class C 2 on C (as an R-differentiable function), such that its second differential satisfies

2 ˜ = O | | 4 -2 , as | | → +∞. (2.18)
If is not the pure power non-linearity, assume that for all = 1, . . . , , ∈ O and satisfies (H3), and in the case where ≥ 2, assume moreover that belongs to 0 +1,∞ (C), where

0 := 2 + 1.
Then the conclusion of Theorem 2.3 holds. Remark 2.3. Theorems 2.3 and 2.3 are restricted to dimensions ≤ 6. For ≥ 7, a similar uniqueness result can be proved (using the same method as that we develop in section 2.3), provided a smaller class of multi-solitons is considered, and for which a bound on ( ) -( ) ∞ is furthermore assumed. This is the purpose of the next proposition. Proposition 2.6. Let ≥ 4, 0 := 2 + 1, and ˜ : ↦ → (| | 2 ) be of class C 2 on C (as an R-differentiable function), such that its second differential satisfies

2 ˜ = O | | 4 -2 , as | | → +∞. (2.19)
Assume that belongs to 0 +1,∞ (C). Assume moreover that for all = 1, . . . , , ∈ O and satisfies (H3). Then for any > 0, there exists ∈ N * such that there exists a unique

∈ C ( [ 1 , +∞), 1 (R ) ∩ ∞ (R )) solution to (NLS) such that ( ) -( ) 1 = O 1 and ∫ +∞ ( ) -( ) ∞ = O 1 , as → +∞.

Outline of the paper and the proofs

The main content

We will prove Theorems 2.2 and 2.3 which generalize Theorems 2.2 and 2.3 respectively when applied to pure power non-linearities. Section 2 is devoted to the proof of our regularity result, that is Theorem 2.2 . We start from a well-chosen sequence ( ) of solutions satisfying uniform 1 estimates and which were constructed in [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] and [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF] (we emphasize that we do not work with the already built multi-soliton in 1 (R ) given in Theorem 2.1). Taking some inspiration from Martel [63, section 3] in the context of the generalized Korteweg-de Vries equations, we prove uniform estimates for ( ) via an induction on the index of regularity. We can combine both stable and unstable cases since we start from the same uniform exponential 1 estimates obtained in [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] and [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF]. From these estimates we deduce (by a usual compactness argument) the existence of a multi-soliton satisfying the conclusions of Theorem 2.2 .

The induction argument relies on the study of a functional related to 2 , suitably modified so as to cancel ill-behaved terms; this functional takes the same form in all dimensions (see (2.30) in subsection 2.2.2). In [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF], for gKdV, from = 3, all quantities of the type ( ) -( ) introduced are shown to decrease exponentially in large time with the same rate. Our proof is more technical, insofar as the algebra is not as favorable. In the context of (NLS), the terms involving real and imaginary parts can not be treated in the same way at once, and in dimension ≥ 2, derivative can fall on terms in many various ways. As nonlinearities are not necessarily smooth (as it is the case in [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]), we need to count carefully the number of times one can perform an integration by parts. This explains why in the case of (NLS), the rate of the exponential decay on ( ) -( ) is halved when passing from to + 1 (see (2.25) and Proposition 2.8). We then obtain the decay rate of (2.9) by a simple interpolation argument.

Regarding the regularity assumption on the nonlinearity, recall that the 1 estimate in Theorem 2.1 holds when

: ↦ → (| | 2 ) is of class C 1 .
As fas as 0 regularity is concerned, Kato's wellposedness result [START_REF] Kato | On nonlinear Schrödinger equations. II, solutions and unconditional wellposedness[END_REF] in 0 (R ) assumes of class C 0 . In Theorem 2.2 , we require a bit more regularity for to prove an 0 estimate for 0 ≥ 2 for the multi-soliton. From a technical point of view, many estimates rely indeed on the local boundedness of the derivatives (in the sense of distributions) of the functions -, where = 0, . . . , 0 and = 0, . . . , . The preceding property is typically used at two levels. First, we need the local Lipschitz condition which is satisfied by functions in 1,∞ : this is for example the case for (2.42) in subsection 2.2.2. In order to obtain the desired 0 estimate, we need also to integrate by parts a particular term (at least one time) which contains derivatives with respect to the space variable of maximal order 0 of both and in order thatappears with a derivative of order 0 -1, thus can be controlled (see in particular (2.49) in subsection 2.2.2). For this, one shall ensure that the distributional derivative of ↦ → 0 0 -( ( )) belong to some Lebesgue space ; this is in fact the case if the derivative of ↦ → 0 0 -is bounded on a certain disk centered at the origin. Therefore, we assume that is an element of 0 +1,∞ (C). Notice that this condition is met when is the pure power nonlinearity (2.1) with 0 = -1 (and also in the particular case when is an even integer).

Besides, we emphasize that assumption 0 > 2 in Theorems 2.2 and 2.2 (which is automatically satisfied for ∈ {1, 2, 3}) seems to be needed to obtain the desired estimates, judging from (2.43). In order to relax this, one should work out an argument involving Strichartz type estimates. But to be effective, the dispersive estimates are to be done on the linearized equation around a sum of solitons, that is a sum of potentials which are decoupled and smooth, but large and not decaying in time. Such estimates would actually be very useful for other purposes, for example the stability of multi-solitons. To our knowledge, they are however not (yet) available. Section 3 is devoted to the proof of the uniqueness result, which combines some ideas of [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] and of [START_REF] Martel | Stability in 1 of the sum of solitary waves for some nonlinear Schrödinger equations[END_REF]. We will consider a solution satisfying (2.11) and (2. [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]) and show that it is in fact the multi-soliton constructed in the proof of Theorem 2.2 : we therefore study the difference of these two solutions and show that it is 0. One main tool for this is a Weinstein type functional, which is coercive provided we assume some adequate orthogonality properties. Depending on the stable or 2 -critical case considered, these orthogonality conditions differ. The coercivity result available in the latter case (where : ↦ →

2 ) is the object of Proposition 2.25. The fact that we do the difference with an already constructed multi-soliton which is sufficiently regular is crucial, at least up to dimension 2. In fact, what we truly need is the 2 (R ) decay for ≥ 2, and also at several times, that the constructed multi-soliton takes values in ∞ (R ).

Note also that, finding like us his inspiration in [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF], Combet [START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF], in the one-dimensional pure power unstable case, has already obtained estimates similar to those we develop in section 2.3 for general in the stable case. Last, the lack of (backward in time) monotonicity properties of (NLS) explains somehow the difficulty to obtain unconditional uniqueness, that is to prove uniqueness in the whole class of multi-solitons in the sense of (4.2) (without decay rate); see Remark 2.7 for more details.

Some notations and writing practices used through the text

Solutions of (NLS) or functions constructed with such solutions take values in C. As usual, the modulus of a complex number will be denoted by |.|. Our computations are generally done in all dimensions . To that extent,

• for any vector ∈ R , we denote by Re( ) (respectively Im( )) the vector of R which components are the real parts (respectively the imaginary parts) of the components of .

• • denotes the euclidean scalar product in R and |.| denotes also the euclidean norm from which it derives.

• we use the usual notation for multi-indices.

As usual, it is also convenient to denote by some positive constant which can change from one line to the next but which is always independent of the index of any sequence considered.

The main functional spaces we will work with are the Sobolev spaces (R ) for ∈ N * endowed with the usual norms defined by:

∀ ∈ (R ), := | | ≤ 2 2 1 2
.

We consider also ∞ (R ) := ∈N * (R ) and the Sobolev spaces ,∞ (C) (identified with

,∞ (R 2 )) for ∈ N * .
Furthermore, many computations are presented formally for ease of reading but can be justified by standard regularization arguments which often involve the local well-posedness of (NLS) in with continuous dependence on compact sets of time (see [START_REF] Dai | Continuous dependence of Cauchy problem for nonlinear Schrödinger equation in[END_REF]Theorem 1.6]).

Existence of smooth multi-solitons of (NLS)

In this section, let us concentrate on the proof of Theorem 2.2 . Let 0 > 2 be an integer and assume that : ↦ → (| | 2 ) is in 0 +1,∞ (C) and satisfies (H1).

Step 1: Uniform 1 -estimate for a sequence of solutions

In order to prove Theorem 2.2, we start from the following proposition, which applies to both stable and unstable cases, and which has already been established in preceding papers. This proposition gives rise to some control in the 1 norm on a constructed sequence of solutions of (NLS) which turns out to be relevant to achieve our goal. Proposition 2.7 (Martel and Merle [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF], Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]). There exist an increasing sequence ( ) ∈N of times such that → +∞, a sequence ( ) ∈ 0 (R ) N , and constants > 0, 1 > 0, 0 > 0 with 0 > 0 such that for all ∈ N:

• 0 ≤ 1 -2
• the maximal solution of (NLS) such that

( ) = ( ) + belongs to C ( [ 0 , ], 0 (R )) and satisfies ∀ ∈ [ 0 , ], ( ) -( ) 1 ≤ 1 -2 .
Remark 2.4. Note that the sequence ( ) can be chosen in the following form.

• For the stable case, we take = 0 for all (see [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF]).

• For the unstable case, we take = ∈ {1,..., },± ± , ± ( ) for all with ± defined by

± : ( , ) ↦ → ± - -0 1 2 • + - | | 2 4 + (2.20)
and with = ± , ∈ {1,..., },± ∈ R 2 well chosen (see [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] for full details) so that

∀ ∈ N, | | ≤ -2 .
(2.21)

Some particular estimates will be useful throughout the proof. Firstly we retain

( ) -( ) 0 ≤ 1 -2 (2.22)
(since (2.21) holds and the quantities ± ( ) are independent of ). We emphasize also that

∀ ∈ N, ∀ ∈ [ 0 , ], ( ) -( ) 1 ≤ 1 -2 . (2.23)
In addition, the exponential decay property (2.5) of the ground states and their derivatives lead to the following assertion, which is also crucial to establish many estimates:

∀ ∈ [ 0 , +∞), ∀ ≠ , ∀ | 1 |, | 2 | ∈ {0, . . . , 0 + 2}, ∫ R 1 . 2 ( ) ≤ -2 .
(2.24)

2.2.2

Step 2: Proof of uniform -estimates for -, = 1, . . . , 0

From now on, let := -.

Step 2.1: Performance of preliminary uniform -estimates Define 0 := 2 , 1 := , and for all ≥ 2, := min

-1 2 , 2 + 1 , (2.25)
so that = 2 -2 ( +1) for all ≥ 2. We prove the following statement, which is the core of our main existence result.

Proposition 2.8.

There exists 1 ≥ 0 such that for all ∈ {1, . . . , 0 }, there exists ≥ 0 such that for all ∈ N, if

≥ 1 then ∀ ∈ [ 1 , ], ( ) ≤ - . (2.26)
To prove Proposition 2.8, we resort to a "bootstrap" argument. Recall that for all ∈ N * , there exists

≥ 0 such that ∀ ∈ R, ± ( ) ≤ . (2.27)
For all , set

* := inf ≥ 0 | ∀ ∈ [ , ], ( ) 0 ≤ 0 , for some constant 0 > 2 0 . Note that * indeed exists since ( ) = =1 ± , ± ( ).
Hence for all ∈ N, we have ∀ ∈ ( * , ], ( ) 0 ≤ 0 . Due to the continuity of : [ 0 , ] → 0 (R ) in * , we also have for all ∈ N:

∀ ∈ [ * , ], ( ) 0 ≤ 0 .
(2.28)

We will show that * can be chosen independently of and improve the preceding estimate by showing first: Proposition 2.9. For all ∈ N, for all ∈ {1, . . . , 0 }, for all ∈ [ * , ],

( ) ≤ -.

Proof. We argue by induction. The existence of 1 ≥ 0 such that for all ∈ N,

∀ ∈ [ * , ], ( ) 1 ≤ 1 -
is already known. Assume that for some ∈ {2, . . . , 0 }, for all ∈ {1, . . . , -1}, there exists ≥ 0 such that for all ∈ N,

∀ ∈ [ * , ], ( ) ≤ - . (2.29)
We aim at showing that the same estimate is valid for = . For this purpose, let us consider for all ∈ N the functional

, : ↦ → ∫ R        | |= | | 2 - | |= -1 -1 Re 2 2 (| | 2 )        ( ) . (2.30)
More precisely we prove, in what follows, how to obtain the following statement, which is essential in the proof of estimate (2.29) corresponding to = .

Lemma 2.10. For all ∈ N, and for all ∈ [ * , ], we have

| , ( ) -, ( )| ≤ -min{ -1 , 4 +1 } , (2.31)
for some constant independent of , and 0 .

Remark 2.5. The fundamental reason why it is worth introducing the functional , is that no quadratic term involving for | | = appears in its first derivative and no term with | | > appears either. Thus we manage to control , ( ). Nevertheless, we do not claim that the functional , is the only one that can be used to prove (2.29).

Proof of Lemma 2.10. We will work on the derivative of , and show in fact that

, ( ) ≤ -min{ -1 , 4 +1 } .
(2.32)

The computations and estimates are established rather in terms of the function instead of ; by this means, they are considerably less burdensome. Besides, in accordance with Remark 2.6 below, the calculations indicate that (2.32) would still be true for more generalized functions which satisfy Im = Re .

Let us introduce also some further notations. For ease of reading, we will write instead of or ( ) and instead of , . In addition, we denote by 1 the real part of and by 2 its imaginary part. Moreover, for all ∈ {1, . . . , }, we specify as the -tuple (0, . . . , 1, . . . , 0) for which all components except the -th one are zero.

We divide the proof of Lemma 2.10 into three steps.

Step 1: Computation of the derivative of .

First of all, observe that for all ( ,

) ∈ R 2 ( , ) = 2 ( + ) ( 2 + 2 ) + ( 2 + 2 ) ( , ) = 2 ( + ) ( 2 + 2 ) + ( 2 + 2 ), (2.33)
so that + ( ) = 2 2 (| | 2 ).
In particular, can be rewritten in terms of as follows:

( ) = ∫ R        | |= | | 2 - 1 2 | |= -1 -1 Re ( ) + ( ) 2        ( ) . Let = ( 1 , . . . , ) ∈ N be such that | | = .
There exists ( ) ∈ {1, . . . , } such that ( ) ≥ 1. Then, using the fact that satisfies (NLS), the following holds true:

∫ R | | 2 = -2 Im ∫ R =1 2 + ( ( )) = -2 Im ∫ R ( ( )) = -2 Im ∫ R ( )Re ( ) + ( )Im ( ) + = -2 ∫ R Im 2 ( ) 2 (| | 2 ) + , (2.34)
where

= -2 Im ∫ R -( ) ( ( )) ( ) 1 + -( ) ( ) ( ) 2
.

Due to Faà di Bruno formula, is also a linear combination of the following terms:

, , , ˜ 1 ,..., ˜ ( ) := ∫ R ˜ 1 1 . . . ˜ Im -( ) , (2.35)
where ∈ {2, . . . , }, ∈ {0, . . . , }, =1 | ˜ | = , and for all ∈ {1, . . . , }, ˜ ≥ 1 and ∈ {1, 2}.

Similarly, we have for each multi-index such that | | = -1:

∫ R 1 2 Re ( ) + ( ) 2 = Im ∫ R ( ) + ( ) =1 +2 + ( ( )) + 1, = - =1 Im ∫ R ( ) + ( ) + 2 + 1, + 2, + 3, = -2 =1 ∫ R Im 2 ( + ) 2 (| | 2 ) + 1, + 2, + 3, , (2.36)
where we denote

1, = 1 2 ∫ R Re 2 2 ( ) 1 + 2 ( ) 2 + 2 ( ) 1 + 2 2 ( ) 2 2 2, = =1 Im ∫ R ( ) + ( ) + 3, =Im ∫ R ( ) + ( ) ( ( )) .
We observe that

| |= ( ) 2 - | |= -1 -1 =1 + 2 = 0.
(2.37) Thus,

( ) = | |= - | |= -1 -1 1, + 2, + 3, . (2
| | = , 2 Im ( )Re ( ) + ( )Im ( ) = Im ( ) + ( ) ( ) 2 .
Step 2: Control of the derivative of .

Take ∈ N such that | | = . Let , , and ˜ 1 , . . . , ˜ be as in (2.35), and denote by , , , ˜ 1 ,..., ˜ ( ) the integral defined exactly as in (2.35) by replacing by the soliton , for all = 1, . . . , . Then we have

, , , ˜ 1 ,..., ˜ ( ) - =1 , , , ˜ 1 ,..., ˜ ( ) ≤ -min{ -1 , 4 +1 } , (2.39)
for some constant independent of . In order to prove (2.39), one proceeds by decomposition of , , , ˜ 1 ,..., ˜ ( ) as follows. The basic idea is to make terms in (which provide the expected exponential term at the right-hand side of (2.39)) appear. Let us explicit this decomposition:

, , , ˜ 1 ,..., ˜ ( ) = ∫ R ˜ 1 1 . . . ˜ Im -( ) - -( ) ( ,1 ) + ∫ R ˜ 1 1 ˜ 2 2 . . . ˜ Im -( ) ( ,2 ) + ∫ R ˜ 1 1 ˜ 2 2 ˜ 3 3 . . . ˜ Im -( ) ( ,3 ) + . . . ( ,... ) + ∫ R ˜ 1 1 ˜ 2 2 . . . ˜ -1 -1 ˜ Im -( ) ( , +1 ) + ∫ R ˜ 1 1 . . . ˜ Im -( ) ( , +2 ) + ∫ R ˜ 1 1 . . . ˜ Im -( ) ( , +3 )
Now, we control each preceding term ,1 , . . . , , +3 occurring in the preceding decomposition by means of the induction assumption and some classical tools in functional analysis, namely Hölder inequality, Sobolev embeddings, and Gagliardo-Nirenberg inequalities.

Let us notice that sup ∈R ( ) ∞ < +∞.

(2.40)

Considering that 0 > 2 , we deduce then from (2.28) and the Sobolev embedding 2 +1 (R ) ↩→ ∞ (R ) that there exists ≥ 0 such that for all ,

∀ ∈ [ * , ], ( ) ∞ ≤ . (2.41) Since -is 1,∞ on C (
| ,1 | ≤ ∫ R | || || ˜ 1 | . . . | ˜ | . (2.42)
We now estimate the integral

∫ R | || || ˜ 1 | . . . | ˜ |
by means of Hölder inequality. For this, we have to be careful concerning the choice of the involved Lebesgue spaces (or in other words the Hölder exponents) considering that ˜ ∈ 0 -| ˜ | . We define

I := ∈ {1, . . . , } | 0 -| ˜ | < 2 , J := ∈ {1, . . . , } | 0 -| ˜ | > 2 , and :=        2 -2( 0 -| ˜ |) if ∈ I ∞ if ∈ J .
For ∈ {1, . . . , } \ (I ∪ J ), we take ∈ (0, +∞) large enough so that

=1 1 < 1 2
, which is possible since

1 2 - ∈I 1 + ∈ J 1 = 1 2 - ∈I 1 = 1 2 - ∈I -2( 0 -| ˜ |) 2 ≥ 1 2 - 1 2 - 0 -≥ (1 -) 1 2 - 0 > 0.
(2.43) due to our assumption on 0 and the fact that > 1. Then, we observe that for all = 1, . . . , ,

˜ ∈ 0 -| ˜ | (R ) ↩→ (R )
by the classical Sobolev embedding theorem. Using Hölder inequality, we obtain

| ,1 | ≤ 2 =2 ˜ ≤ . (2.44)
where ≥ 2, and 1 = 1 2 -=1 1 ≥ ( -1) 0 -1 2 > 0 by definition of the , = 1, . . . , . The following Gagliardo-Nirenberg inequality

≤ 0 1- 2 ,
(2.45)

with 0 := 2 + 1 ≤ +1 2 and := 0 1 2 -1 (which implies 1 -≥ 1 0 , since 2 0 -≥ 1) leads finally to | ,1 | ≤ 1- 2 ≤ -2 0 ≤ -4 +1 .
(2.46)

To estimate ,2 , . . . , , +1 , one proceeds as before. For instance, let us explain how to deal with ,2 ; the same would be done for the other integrals. We choose

1 such that 1 -1-| ˜ 1 | ↩→ 1 (R ) and 1 2 -1 1 -=2 1 > 0.
Then, again due to Hölder inequality, we have:

| ,2 | ≤ 2 ˜ 1 1 =1 ˜ -( ) ∞ ≤ -1 ≤ --1 .
(2.47)

Similarly, we check that

∀ ∈ {3, . . . , + 1}, | , | ≤ --1 . (2.48)
Now, let us deal with , +2 . By (2.24) and the fact that -∈ 1,∞ (C), we have:

, +2 = =1 ∫ R ˜ 1 , 1 . . . ˜ , Im -( ) + -2 .
Again by assumption, each partial derivative of -( ) is bounded, thus the integral

∫ R - ˜ 1 , 1 . . . ˜ , -( )
makes sense and one can integrate once by parts to obtain

∫ R ˜ 1 , 1 . . . ˜ , Im -( ) = Im ∫ R - ˜ 1 , 1 . . . ˜ , -( ) ≤ ( ) -1 ≤ --1 .
(2.49)

Thus, | , +2 | ≤ --1 . (2.50)
Finally, by (2.24) and using once more that -is in The expressions 1, , 2, , 3, (given before) consist of terms that can be controlled in a similar manner. Let us denote by , ( ) the same integral as , where replaces for all = 1, 2, 3 and for all = 1, . . . , . One can check that

1, - =1 1, ( ) ≤ -min{ -1 , 4 +1 } ;
(2.52)

2, - =1 2, ( ) ≤ -min{ -1 , 4 +1 } ; (2.53) 3, - =1 3, ( ) ≤ --1 .
(2.54)

Step 3: Related functional involving .

Let ∈ {1, . . . , }. An immediate induction argument shows that for all multi-index ∈ N such that | | = , for all multi-index , there exists ∈ C such that

( , ) = ( -0 - ) 1 2
• + -

| | 2 4 + . Therefore ∫ R | | 2 ( ) = ∫ R ( -0 - ) 2 = ∫ R ( ) 2 , so that ∫ R | | 2 = 0. (2.55)
Furthermore for all multi-index such that

| | = -1, 2 2 ( , ) = 2 ( -0 - ) ( -0 - ) 2 ,
from which we infer also

∫ R Re 2 2 (| | 2 ) = 0. (2.56)
Hence, gathering (2.55) and (2.56), Integrating the preceding inequality between and yields directly Lemma 2.10.

0 = ∫ R        | |= | | 2 - | |= -1 -1 Re 2 2 (| | 2 )        . ( 2
Let us now conclude the proof of Proposition 2.9.

We observe that, for all ∈ [ * , ],

| |= ∫ R | ( )| 2 = , ( ) -, ( ) + 2 | |= Re ∫ R -( ) ( ) + ( ) ( ) + | |= ∫ R | ( )| 2 - ∫ R | ( )| 2 + | |= -1 -1 ∫ R Re 2 ( ) 2 (| | 2 ) ( ) - ∫ R Re 2 ( ) 2 (| | 2 ) ( ) + | |= -1 -1 ∫ R Re 2 ( ) 2 (| | 2 ) ( ) - ∫ R Re 2 ( ) 2 (| | 2 ) ( ) + | |= ∫ R | ( )| 2 -2 | |= Re ∫ R -( ) ( ) + ( ) ( ) .
(2.59) Then, by means of (2.22), (2.55), and (2.56), we infer

| |= ∫ R | ( )| 2 ≤ , ( ) -, ( ) + -1 + -2 + ∫ R Re 2 ( ) 2 (| | 2 ) ( ) - ∫ R Re 2 ( ) 2 (| | 2 ) ( ) . (2.60)
Now, from (2.24), (2.31), (2.57), and from the inequality

∫ R Re 2 ( ) 2 (| | 2 ) ( ) - ∫ R Re 2 ( ) 2 (| | 2 ) ( ) ≤ ( ) -1 , (2.61
+ ( ) = 2 (| | 2 ), we deduce the existence of ≥ 0 such that for all ∈ N, ∀ ∈ [ * , ], ( ) 2 ≤ 2 -min{ -1 , 4 +1 } . (2.62)
This is rewritten as follows:

∀ ∈ [ * , ], ( ) ≤ - , (2.63)
which is exactly the expected result. Thus the induction argument implies that for all ∈ N,

∀ ∈ [ * , ], ( ) 0 ≤ 0 -0 . (2.64)
This puts an end to the proof of Proposition 2.9. Now we explain how to deduce from Proposition 2.9 that * can be chosen independently of , and by this means, we finish the proof of Proposition 2.8. We pick up 1 ≥ 0 such that 0 -0 1 < 0 . Let ∈ N be such that ≥ 1 , and assume by contradiction that * > 1 . Then by continuity of in * (and by definition of * as infimum), we have ( * ) 0 = 0 . On the other hand,

( * ) 0 ≤ 0 -0 * ≤ 0 -0 1 < 0 , (2.65)
which yields a contradiction. Thus * ≤ 1 . Hence, for all ∈ N such that ≥ 1 , we have

∀ ∈ [ 1 , ], ( ) 0 ≤ 0 -0 .
If necessary we drop the first terms of the sequence ( ) and re-index it in order to obtain:

∀ ∈ N, ∀ ∈ [ 1 , ], ( ) 0 ≤ 0 -0 . (2.66)
Hence, Proposition 2.8 is established.

Step 2.2: Independence of 1 with respect to Now, we justify that 1 can be chosen independent of > 2 , which is useful to obtain

∀ ∈ N * , ∃ ≥ 0, ∀ ∈ [ 1 , ], ( ) ≤ - , (2.67)
in the case where 0 = ∞.

If is C ∞ on C as an R-differentiable function, it is in particular of class C 2 +2
, so that we can apply the previous result: there exists 1 ≥ 0 such that for all ∈ N,

∀ ∈ [ 1 , ],
( )

2 +1 ≤ 2 +1 - 2 +1 .
(2.68)

Let ≥ 2 + 2 and assume that for all ∈ { 2 + 2, . . . , },

∀ ∈ [ 1 , ], ( ) -1 ≤ -1 --1 .
Then define *

, := inf{ ≥ 1 | ∀ ∈ [ , ], ( ) ≤ },
for some constant > max{2 , 1} to be determined. We show exactly as before (that is considering the functionals , ) the existence of ˜ > 0 independent of , , and such that

∀ ∈ [ * , , ], ( ) 2 ≤ ˜ 2 -2 , (2.69)
or also

∀ ∈ [ * , , ], ( ) ≤ ˜ 1 2 - . (2.70)
Indeed, the constant in (2.41) does not depend on and ≥ 1 so that we have for example as in (2.44) and then (2.46):

| ,1 | ≤ -4 +1 ,
with independent of . Choosing > ˜ 2 -2 1 and arguing as in (2.65), we conclude that * , = 1 . Hence, 1 is uniform with respect to .

Step 2.3: Looking for optimal exponential decay rates in the uniform -estimates

The next result uses and improves that of Proposition 2.8. Proposition 2.11. For all ∈ {1, . . . , 0 }, there exists ˜ ≥ 0 such that for all ∈ [ 1 , ],

( ) ≤ ˜ -2 +1 .
(2.71)

Proof. Let ∈ {1, . . . , }. By (2.23), (2.26), and the following interpolation inequality

( ) ≤ ( ) 2 ( ) 1-
, with = -, we have for all ∈ [ 1 , ],

( ) ≤ -2 - . Now, set * := inf{ ≥ 1 | ∀ ∈ [ , ], ( ) ≤ ˜ -},
for some ∈ (0, 2 ) and for some ˜ ≥ 1 to be determined later.

Let belong to [ * , ]. Then by the proof set up before,

( ) 2 ≤ ( ) -1 + -2 .
In addition, we obtain once again by interpolation

( ) -1 ≤ ( ) 1 2 ( ) -1 ≤ ˜ 1-1 -2 --1 .
Since ≤ 2 , we have 2 + ( -1) ≤ 2 , and so there exists ˜ ≥ 0 (independent of ˜ ) such that

( ) 2 ≤ ˜ ˜ 1-1 -2 --1 . (2.72)
Now, choose

:= 2 + 1 and ˜ > ˜ +1 .
By a similar argument as that set up to prove Proposition 2.8, we see that * = 1 . Indeed, if we had * > 1 , then by the definition of * and by continuity of in * , we would obtain

˜ 2 -2 * = ( * ) 2 ≤ ˜ ˜ 1-1 -2 * --1 * ,
thus, by the choice of ,

˜ 2 ≤ ˜ ˜ 1-1 ,
which is a contradiction. Consequently, estimate (2.71) does indeed hold.

Step 3: Conclusion of the proof of Theorem 2.2

We construct now the multi-soliton using the same arguments as those of Martel [63, paragraph 2,

Step 2] and Martel and Merle [71, Paragraph 2]. The crucial point is the following lemma, obtained by a compactness argument.

Lemma 2.12. There exist ∈ 0 (R ) and a subsequence ( ( 1 )) of ( ( 1)) such that

( 1 ) - 0 -→ →+∞ 0.
Note that the main ingredients to show this lemma are:

• the uniform 0 -estimate obtained in Step 2.

• the following 2 -compactness assertion: for all > 0, there exists K a compact subset of

R such that ∀ ∈ N, ∫ K | ( 1 , )| 2 ≤ .
Then by local well-posedness of (NLS) in 0 (R ) with continuous dependence on compact sets of time [21, Theorem 1.6], the solution of (NLS) such that ( 1 ) = is defined in 0 (R ) and for all ≥ 1 , ( ) -( ) 0 → 0 as → +∞. Thus turns out to be the desired multisoliton. Besides, the quantities ( ) -( ) decrease exponentially; this result is obtained by passing to the limit as tends to +∞ in the -uniform estimates given by Proposition 2.11, that is for all = 1, . . . , 0 , for large enough:

( ) -( ) ≤ -2 +1 .
This yields precisely (2.9).

Note that in the case where is C ∞ (for example when we consider the pure power nonlinearity with an odd integer), we obtain (2.10) as a consequence of (2.9), by interpolating the corresponding -estimates, and by the independence of 1 with respect to proved in Step 2.2.

Conditional uniqueness for multi-solitons of (NLS)

In this section, we prove the uniqueness result stated in Theorem 2.3 , that is for ≤ 3. The strategy developed here would also work to prove Proposition 2.6 under the corresponding stronger assumptions.

Our uniqueness result holds due to the coercivity properties of the linearized operators around ground states, namely assumption (H3) when is not the 2 -critical non-linearity and (2.142) in Proposition 2.25 in the 2 -critical pure power non-linearity case. The proof follows essentially the same lines in these two cases; the differences are only rooted in the use of the appropriate coercivity result.

We first develop the proof in the stable case, assuming ˜ : [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]). This covers in particular the 2 -subcritical assumption with 3 ≤ < 1 + 4 and = 1 in Theorem 2.3. In subsection 2.3.4, we explain how to modify the calculations in order to perform the proof in the 2 -critical case, that is assuming = 1 + 4 and : ↦ → -1 2 , which will extend the uniqueness result as stated in Theorems 2.3 and 2.3 .

↦ → (| | 2 ) of class C 2 satisfying (2.
Let us denote the multi-soliton of (NLS) constructed in Theorem 2.1 for = 1 and in Theorem 2.2 for ≥ 2 (which is possible to consider by hypothesis). Set := 2 3 , where is defined in Theorem 2.1 and let 1 > 0 such that belongs to C ( [ 1 , +∞), 1 (R)) and

∀ ≥ 1 , ( ) -( ) 1 ≤ - (2.73)
for = 1, and such that belongs to C ( [ 1 , +∞), 0 (R )) and

∀ ≥ 1 , ( ) -( ) 0 ≤ - (2.74)
for ≥ 2 (where 0 = 2 + 1.) In particular, due to the Sobolev embedding

2 +1 (R ) ↩→ ∞ (R ) for ( ≥ 1), we emphasize that, for all ≥ 1, ∈ C ( [ 1 , +∞[, ∞ (R )) and ∀ ≥ 1 , ( ) -( ) ∞ ≤ -. (2.75)
Now, let us take in the class of multi-solitons satisfying (2.11) and define := -the difference of the two multi-solitons so that

= Δ + | + | 2 + | + | 2 - | | 2 , (2.76)
and

( ) 1 = O 1 , as → +∞, (2.77)
for some integer ≥ 1 to be determined later.

We will show that = 0. The idea is to practice some kind of modulation of the variable in order to ensure some orthogonality relations, needed to make use of the coercivity properties mentioned before. In other words, we obtain a new function (denoted by ˜ ) which seems to be adapted to the proof; this is the aim of subsection 2.3.1. Then, the control of the modulation parameters allows us to obtain an estimate of ( ) 1 in terms of ˜ ( ) 1 ; this combined with the estimate of the derivative of some kind of Weinstein functional ˜ (that we introduce in paragraph 2.3.2) enables us finally to see that ˜ = 0.

Change of function to ensure a coercivity property in the stable case

Introduction of a new variable

We introduce a new function ˜ on [ , +∞) × R for sufficiently large by

∀ ( , ) ∈ [ , +∞) × R , ˜ ( , ) := ( , ) + =1 ( ( ) ( , ) + ( ) • ∇ ( , )) , (2.78)
where ( ) ∈ R and ( ) ∈ R are chosen so that ∀ ∈ {1, . . . , }, ∀ ∈ {1, . . . , },

           Im ∫ R ˜ = 0 Re ∫ R ˜ = 0.
(2.79)

Existence of ( ) and ( ) for large enough follows from: (2.81)

Proof of Lemma 2.13. Let us introduce the symmetric block matrix

( ) :=          0,0 ( ) 1,1 ( ) • • • 1, ( ) 1,1 ( ) 1,1 ( ) • • • 1, ( ) . . . . . . . . . . . . 1, ( ) ,1 ( ) • • • , ( )         
, where , ( ) and 1, ( ) are × -matrices with real entries defined by

0,0 = Re ∫ R ( , ) , ∀ ( , ) ∈ {1, . . . , } 2 , , = Re ∫ R ( , ) , ∀ ∈ {1, . . . , }, 1, = Im ∫ R ( , )
.

Set also Then relations (2.79) rewrite clearly

( ) = 1 , . . . , , 1
( ) ( ) = ( ).
Consequently, we have to show that det ( ) ≠ 0 for large enough to ensure existence and uniqueness of ( ) and ( ) for those values of . To do this, observe that

Re ∫ R ( ) ( ) =        ∫ R 2 if = O - if ≠ , Re ∫ R ( ) ( ) =        ∫ R + , , 4 2 if = O - if ≠ , Im ∫ R ( ) ( ) =        , 2 ∫ R 2 if = O - if ≠ .
Let us now compute det( ( )). For all = 1, . . . , , let denote the -th line of the block matrix

0,0 ( ) 1,1 ( ) • • • 1, ( ) .
For all = 1, . . . , , and for all = 1, . . . , , replacing the -th line , of the block matrix

1, ( ) ,1 ( ) • • • , ( ) by , -,

2

, we obtain det( ( )) = det( ( )) where

( ) :=          0,0 ( ) 1,1 ( ) • • • 1, ( ) 1 ( ) 1,1 ( ) • • • 1, ( ) . . . . . . . . . . . . ( ) ,1 ( ) • • • , ( )          and
( ) has entries zero on the diagonal and ( -) everywhere else and , has entries ∫ R on the diagonal and ( -) everywhere else. Thus

det( ( )) = =1 ∫ R 2 ( ) det( ( )) + O( -), (2.82)
where ( ) is the sub-matrix of ( ) with block matrices , ( ).

We observe that ( ) admits a limit as → +∞ which we denote by (∞) and which corresponds to the block matrix

       1,1 (∞) • • • 1, (∞) . . . . . . . . . ,1 (∞) • • • , (∞)        where , (∞) is a diagonal matrix with entries ∫ R . Due to the continuity of the determinant, det( ( )) → det (∞) as → +∞. Thus, det ( ) → =1 ∫ R 2 ( ) det( (∞)), as → +∞. ( 2

.83)

Moreover for all = ( , ) ∈ R different from 0, we have

(∞) = , =1 =1 ∫ R , , = =1 ∫ R =1 , 2
which is a positive quantity for large values of since for all , the functions , = 1, . . . , are linearly independent (this can be seen using that is radial but it is in fact also related to a more general result corresponding to Proposition 2.30 in Appendix). Hence, det( ∞)) > 0 and also det( ( )) > 0 for large values of by (2.83). In particular, ( ) is invertible for large values of . Applying Cramer's formula, we obtain an explicit expression of ( ) and ( ) in terms of ( ), from which we derive the content of Lemma 2.13. Let us justify it. The entries of ( ) are bounded functions of so that the transpose of the comatrix of ( ) is bounded too (with respect to ). In addition, we have proved the existence of > 0 such that for large, det ( ) > . Hence, there exists 0 > 0 such that for all sufficiently large,

| ( )| ≤ 0 | ( )|.
This immediately implies (2.80). We moreover observe that the entries of ( ) are C 1 functions of (by (2.5) and Lebesgue's dominated convergence theorem); in particular ↦ → det ( ) is C 1 . Then, the differentiability of ( ) and ( ) and estimate ( ) -( ) 1 → 0, as → +∞.

(2.84)

Now, by Fubini theorem and the differentiability of ↦ → ( , ) ( , ) for all ∈ R , we obtain

∫ 1 0 ∫ R ( , ) ( , ) ( ) = ∫ R ∫ 1 0 ( , ) ( , ) ( ) = - ∫ R ∫ 1 0 ((Δ + ( + ) -( )) + ) ( , ) ( ) = - ∫ 1 0 ( ) ∫ R ((Δ + ( + ) -( )) + ) ( , ) = - ∫ 1 0 ( ) ∫ R (∇ • + ( ( + ) -( )) + ) ( , ) .
(We recall that ( ) = (| | 2 ) for ∈ C.) Passing to the limit as → +∞ by using (2.84) leads to

∫ 1 0 ∫ R ( , ) ( , ) ( ) = - ∫ 1 0 ( ) ∫ R (∇ • + ( + ) -( ) + ) ( , ) . (2.85) Thus ↦ → ∫ R ( , )
is differentiable in the sense of distributions; its differential is

↦ → ∫ R (∇ • + ( ( + ) -( )) + ) ( , )
and is thus bounded by ( ) 1 . This finishes proving the lemma.

Even if it means taking a larger 1 , we can suppose that the preceding lemma holds on [ 1 , +∞). Then it results also immediately that

∀ ≥ 1 , ( ) 1 ≤ ˜ ( ) 1 + =1 | ( )| + | ( )| . (2.86) and ∀ ≥ 1 , ˜ ( ) 1 ≤ ( ) 1 .
(2.87)

The statement of a coercivity property in terms of the new variable

In this paragraph, we come to some crucial inequality, on which the proof is essentially based. First of all, let us define some notations, and particularly well-chosen cut-off functions. By a classical argument given in [71, Claim 1], we can assume (without loss of generality) that

1,1 < 2,1 < • • • < ,1 . (2.88)
Now let 0 ∈ 0, 1 2 min ∈ {2,..., } { ,1 --1,1 } and define

: R → R ↦ →                  1 if < -0 ∫ 0 -0 - 2 0 2 0 -2 -1 ∫ 0 - 2 0 2 0 -2 if -0 ≤ ≤ 0 0 if > 0 ,
which is obviously a smooth bounded non-increasing function. For all ∈ {1, . . . , -1}, let

:= 1 2 ,1 + +1,1 and := 1 2 0 ,1 + 0 +1,1 . Then define on R × R 0 : ( , ) ↦ → 0 : ( , ) ↦ → 1 -- for ∈ {1, . . . , -1} : ( , ) ↦ → 1
Proposition 2.15. There exists > 0 such that

∀ ≥ 1 , ˜ ( ) 2 1 - 1 =1 Re ∫ R ˜ ( ) ( ) 2 ≤ ( ).
(2.94)

Proof. This result follows from our assumption (H3), from (2.79), and an immediate adaptation to all dimensions of the proof given for the one-dimensional case in [81, appendix B] which consists in localizing in some sense each version of (H3) for all = 1, . . . , .

Proof of some needed estimates

This subsection, which is probably the most technical one, precises the tools and estimates which will allow us to make use of Proposition 2.15 and actually to conclude the proof of uniqueness in subsection 2.3.3. It consists in giving some controls of ( ), of the scalar products

Re ∫ R ˜ ( ) ( ) ,
and also of the modulation parameters ( ) and ( ).

Control of

We typically improve the a priori control of by O ˜ 2 1 by differentiation of the functional. Actually, for the sake of simplification, we will compute the derivative of the following related functional ˜ : .12).) The next proposition, which compares and ˜ , justifies that it suffices to control ˜ in order to obtain a similar estimate for . Proposition 2.16. We have

[ 1 , +∞) → R defined by ∀ ≥ 1 , ˜ ( ) = =1 ∫ R |∇ ˜ | 2 -(| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) + + | | 2 4 | ˜ | 2 -• Im ∇ ˜ ˜ ( , ) . (2.95) (Recall that ( ) = ∫ 0 ( ) (2
( ) = ˜ ( ) + O ˜ ( ) 3 1 + -˜ ( ) 2 1 .
(2.96)

Proof. Let us first observe that ˜ : ↦ → (| | 2 ) is C 3 on C. Indeed, since ˜ is C 2 on C, the function is C 2 on (0, +∞) and thus, ˜ is C 3 on C \ {0}.
Moreover, for all = (Re( ), Im( )) = ( , ) ∈ C \ {0}, we obtain by differentiation of ˜ and ˜ :

˜ ( ) = 2 (| | 2 ) = 2 ˜ ( ) ˜ ( ) = 2 (| | 2 ) = 2 ˜ ( ) ˜ ( ) = 2 (| | 2 ) + 4 2 (| | 2 ) = 2 ˜ ( ) + 2 ˜ ( ) ˜ ( ) = 4 (| | 2 ) = 2 ˜ ( ) ˜ ( ) = 2 (| | 2 ) + 4 2 (| | 2 ) = 2 ˜ ( ) + 2 ˜ ( ) ˜ ( ) = 4 ˜ ( ) + 2 ˜ ( ) ˜ ( ) = 2 ˜ ( ) ˜ ( ) = 2 ˜ ( ) ˜ ( ) = 4 ˜ ( ) + 2 ˜ ( ).
Since ˜ is C 2 , the partial differentials of ˜ up to order 3 admit limits as ( , ) → (0, 0) in R 2 , from which we deduce that ˜ is C 3 . Then we have the following Taylor expansion: for ≥ 1 and ∈ R ,

˜ ( ˜ + ) -˜ ( ) -Re( ˜ ) ˜ ( ) -Im( ˜ ) ˜ ( ) ( , ) = 1 2 (Re ˜ ) 2 ˜ + 2Re ˜ Im ˜ ˜ + (Im ˜ ) 2 ˜ ( , ) + ( , ), (2.97)
where

| ( , )| ≤ 1 6 | ˜ | 3 sup ∈ [ ( , ), ( ˜ + ) ( , ) ] 3 ˜ ≤ | ˜ | 3 sup ∈ [ ( , ), ( ˜ + ) ( , ) ] ˜ + | ˜ | 2 ˜ ≤ | ˜ | 3 1 + ˜ | 4 -1
by assumption (2.19). We then note that the preceding Taylor expansion rewrites

(| ˜ + | 2 )-(| | 2 )-2Re( ˜ ) (| | 2 ) = | ˜ | 2 (| | 2 )+2Re( ˜ ) 2 (| | 2 )+O(| ˜ | 3 +| ˜ | 4 +2 ), (2.98)
uniformly with respect to both variables and . Let us underline that, for ≥ 2, one can not claim whether ( ) or ˜ ( ) belong to ∞ (R ) and even less whether or

˜ belong to ∞ [ 1 , +∞), ∞ (R ) , which prevents us from simplifying O | ˜ | 3 + | ˜ | 4 +2 by O | ˜ | 3 .
Moreover, we have noticed that

| ˜ | 2 (| | 2 ) + 2Re( ˜ ) 2 (| | 2 ) = 1 2 2 ˜ ( ˜ , ˜ ) so that =1 ∫ R | ˜ | 2 | | 2 + 2Re( ˜ ) 2 | | 2 = =1 ∫ R 2 ˜ ( ˜ , ˜ ) .
We now observe that for all = 1, . . . , ,

∫ R 2 ˜ ( ˜ , ˜ ) -2 ˜ ( ˜ , ˜ ) ≤ ∫ R |( -) || ˜ | 2 sup ∈ [ , ] 3 ˜ ≤ ∫ R |( -) || ˜ | 2 sup ∈ [ , ] 3 ˜ + ≠ ∫ R | || ˜ | 2 sup ∈ [ , ]
3 ˜ .

We note that sup ∈ [ , ] 3 ˜ is bounded by a constant independent of and because , belong to ∞ ( [ 1 , +∞), ∞ (R )) and ↦ → 3 ˜ is continuous on C. Then (2.74) and (2.89) lead to

=1 ∫ R | ˜ | 2 | | 2 + 2Re( ˜ ) 2 | | 2 = =1 ∫ R | ˜ | 2 | | 2 + 2Re( ˜ ) 2 | | 2 + O -˜ 2 1 . (2.99)
We finally obtain (2.96) as a direct consequence of (2.98), (2.99), the Sobolev embeddings 1 (R ) ↩→ 3 (R ) (indeed available for ≤ 3) and 1 (R ) ↩→ 4 +2 (R ), and the fact that 4 ≥ 1.

Now, we state and prove the crucial

Proposition 2.17. The derivative of ˜ is given by

˜ ( ) = Main( ) + O -˜ ( ) 1 ( ) 1 + ˜ ( ) 1 ( ) 2 1 , as → +∞. (2.100)
where Main(

) := =1 + | | 2 4 ∫ R | ˜ | 2 + 2 ∫ R Im( 1 ˜ ˜ ) 1 - =1 ,1 2 ∫ R |∇ ˜ | 2 1 - 1 2 ∫ R | ˜ | 2 3 1 - =1 • ∫ R Im ∇ ˜ ˜ = O 1 ˜ ( ) 2 1 .
Remark 2.7. The bound O 1 ˜ ( ) 2 1 in above is the one which constrains us to prove uniqueness in the class satisfying (2.11). In order to prove unconditionnal uniqueness, one would need to improve this bound to O ( ) ˜ ( ) 2 1 , where ( ) is integrable in time (in the KdV context, [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] proves it with ( ) = -for some > 0).

Let us begin with some preliminaries (Lemma 2.18 and Lemma 2.19 below), which are needed to obtain Proposition 2.17.

Lemma 2.18.

There exists > 0 such that:

(| + | 2 ) -(| | 2 ) ≤ | | 4 + | | (| + | 2 ) -(| | 2 ) -2Re( ) (| | 2 ) ≤ | | 2 + | | 4 .
Proof. By the mean value theorem applied to ˜ ,

˜ ( + ) -˜ ( ) ( , ) ≤ | ( , )| sup ∈ [ ( , ), ( + ) ( , ) ]
˜ By (2. [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]) and the boundedness of with respect to and , it results

˜ ( + ) -˜ ( ) ( , ) ≤ | ( , )| + | ( , )| 4 .
Similarly, the second estimate stated in Lemma 2.18 is obtained by direct application of Taylor formula for ˜ at order 2 and by using sup

∈ [ ( , ), ( + ) ( , ) ] 2 ˜ ≤ (1 + | ˜ | 4 -2 ).
Lemma 2.19 (Expression of ˜ ). We have

˜ = Δ ˜ + (| + | 2 ) + (| + | 2 ) -(| | 2 ) + =1 (| | 2 ) + • ∇ (| | 2 ) ) + =1 + • ∇ = Δ ˜ + (| | 2 ) ˜ + (| ˜ + | 2 ) -(| | 2 ) + =1 + • ∇ + , (2.101)
where is a function of and such that

∫ R | | | ˜ | + |∇ ˜ | + ∫ R Δ ˜ ≤ - 1 ˜ 1 + 2 1 ˜ 1 . (2.102)
Proof. The first equality concerning ˜ is quite immediate. Let us precise how to obtain the second equality. Decomposing

(| + | 2 ) = (| | 2 ) + (| + | 2 ) -(| | 2 ) ,
and using the expression of in terms of ˜ and the and , = 1, . . . , given by (2.78), we have that

(| + | 2 ) + =1 (| | 2 ) + • ∇ (| | 2 ) ) = (| | 2 ) ˜ + =1 (| | 2 ) -(| | 2 ) + • ∇ (| | 2 ) -(| | 2 ) + =1 • ∇( (| | 2 )) + (| + | 2 ) -(| | 2 ) (2.103)
Moreover, using the second estimate obtained in Lemma 2.18,

(| + | 2 ) -(| | 2 ) = (| ˜ + | 2 ) -(| | 2 ) -2Re ( ˜ -) (| | 2 ) + O(| | 2 + | | 4 ) = (| ˜ + | 2 ) -(| | 2 ) -2 =1 Re • ∇ (| | 2 ) + ℎ + O(| | 2 + | | 4 ) = (| ˜ + | 2 ) -(| | 2 ) - =1 • ∇ (| | 2 + h, (2.104
) where ℎ and h satisfy the same property (2.102) as due to (2.74) and (2.89). Lemma 2.19 is now a consequence of (2.103), (2.104), and the fact that 4 + 1 ≥ 2 (for ≤ 3).

We are now in a position to prove (2.100).

Proof of Proposition 2.17. The proof decomposes essentially into two parts. We first differentiate successively each term constituting ˜ by means of Lemma 2.19. For this, integrations by parts are sometimes necessary in order not to keep terms carrying second spatial derivatives for . Then we put together suitable terms in the expression of ˜ in order to get better estimates than the a priori control by O ˜ ( ) 2 1 . Besides, we put annotations for the different terms we have to work on for ease of reading; terms associated with the same letter A, B, or C are to be gathered.

Step 1: Differentiation of ˜ • Using Lemma 2.19, one computes

∫ R |∇ ˜ | 2 = 2 Re ∫ R ∇ ˜ • ∇ ˜ = 2 Im ∫ R (| | 2 ) ˜ Δ ˜ + 2 Im ∫ R (| ˜ + | 2 ) -(| | 2 ) Δ ˜ -2 Re ∫ R =1 + • ∇ Δ ˜ + O ( -+ 1 ) 1 ˜ 1 .
Similarly one obtains directly

∫ R (| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) = 2 Re ∫ R (| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) + 2 Re ∫ R ˜ ˜ (| ˜ + | 2 ) + 2 Re ∫ R ( ˜ + ˜ ) (| ˜ + | 2 ) -(| | 2 ) = -2 Im ∫ R Δ ˜ ˜ (| ˜ + | 2 ) -2 Im ∫ R Δ (| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) -2 Im ∫ R (| ˜ + | 2 ) -(| | 2 ) ˜ ( (| ˜ + | 2 ) -2 Im ∫ R Δ ˜ (| ˜ + | 2 ) -(| | 2 ) -2 Im ∫ R Δ + (| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) -2 Im ∫ R (| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) + 2 Re ∫ R =1 + • ∇ ˜ (| ˜ + | 2 ) + (| ˜ + | 2 ) -(| | 2 ) + O ( -+ 1 ) 1 ˜ 1 .
Thus, we have at this point

∫ R |∇ ˜ | 2 -(| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) = 2 Im ∫ R Δ (| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) ( 1 ) + 2 Im ∫ R Δ + (| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) ( 1 ) -2 Re ∫ R =1 + • ∇ Δ ˜ + ˜ (| ˜ + | 2 ) ( 1 ) -2 Re =1 ∫ R • ∇ (| ˜ + | 2 ) -(| | 2 ) ( 2 ) + O -+ 1 1 ˜ 1 .
• We differentiate then the next term appearing in the expression of ˜ . For all = 1, . . . , ,

+ | | 2 4 ∫ R | ˜ | 2 = + | | 2 4 ∫ R | ˜ | 2 + 2 + | | 2 4 ∫ R Im 1 ˜ ˜ 1 ( 1, ) -2 + | | 2 4 ∫ R Im( ˜ ) ( ( ˜ + | 2 ) -(| | 2 )) ( 2, ) + 2 + | | 2 4 Re ∫ R + • ∇ ˜ ( 3, ) + O ( -+ 1 ) 1 ˜ 1 .
• To finish with, using integrations by parts and (2.92), we obtain for all = 1, . . . , ,

∫ R Im( • ∇ ˜ ˜ ) = -2 • Im ∫ R ˜ ∇ ˜ -• Im ∫ R ˜ ˜ ∇ + • Im ∫ R ∇ ˜ ˜ = 2 ,1 ∫ R |∇ ˜ | 2 1 - ,1 2 ∫ R | ˜ | 2 3 1 + • ∫ R Im ∇ ˜ ˜ ( 2, ) + 2 • ∫ R Re ∇ ˜ ) (| ˜ + | 2 ) -(| | 2 ) ( 3, ) + • ∫ R ∇( (| | 2 ))| ˜ | 2 ( 2, ) -2 • Re ∫ R ∇ ˜ (| ˜ + | 2 ) -(| | 2 ) ( 3, ) -2 • Im ∫ R + • ∇ ∇ ˜ ( 4, ) + O ( -+ 1 ) 1 ˜ 1 .
We now continue the proof by showing how the corresponding terms put together can yield estimation (2.100).

Step 2: Estimate concerning ˜ We first deal with the terms 1 , 2, , and 3, ( = 1, . . . , ). We see that

1 -2 =1 Im ∫ R Δ + (| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) ≤ 2 ∫ R Δ( -) ˜ (| ˜ + | 2 ) -(| | 2 ) + ∫ R (| | 2 ) -(| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) + ∫ R (| | 2 ) ( -) ˜ (| ˜ + | 2 ) -(| | 2 ) + =1 ∫ R (| | 2 ) -(| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) .
(2.105)

As for the proof of Lemma 2.18, by the mean value theorem, we observe also that

           (| | 2 ) -(| | 2 ) ≤ | -| 4 + | -| (| | 2 ) -(| | 2 ) ≤ ≠ | |.
(2.106) Moreover, we deduce from Lemma 2.18 that

∫ R Δ( -) ˜ (| ˜ + | 2 ) -(| | 2 ) ≤ - ˜ 4 +1 1 + ˜ 2 1 ≤ - ˜ 2 1 .
(2.107)

Let us establish the preceding inequality in each dimension = 1, 2, 3.

• For = 1, we firstly make use of one integration by parts:

∫ R Δ( -) ˜ (| ˜ + | 2 ) -(| | 2 ) = - ∫ R ∇( -) • ∇ ˜ (| ˜ + | 2 ) -(| | 2 ) - ∫ R ∇( -) • ∇ (| ˜ + | 2 ) -(| | 2 ) ˜ .
We thus obtain by Lemma 2.18:

∫ R Δ( -) ˜ (| ˜ + | 2 ) -(| | 2 ) ≤ ∫ R |∇( -)||∇ ˜ |(| | + | | 4 ) + ∫ R |∇( -)||∇ |(| | + | | 4 ) . (2.108)
Then we note that for all ∈ 1 (R ), (by the embedding

1 (R) ↩→ ∞ (R)) ∫ R |∇ ||∇ ˜ || ˜ | ≤ ∇ ˜ 2 ∇ 2 ˜ ∞ ≤ ˜ 2 1 1 , ∫ R |∇ ˜ ||∇ || ˜ | 4 ≤ ˜ 4 1 ∇ ˜ 2 ∇ 2 ≤ ˜ 4 +1 1 1 , ∫ R |∇ ||∇ || ˜ | 4 ≤ ˜ 4 1 ∇ 2 ∇ 2 ≤ ˜ 4 1 1
1 .

• For = 2: for all ∈ 2 (R ), (by the embeddings

1 (R 2 ) ↩→ (R 2 ) for each ∈ [2, +∞)) ∫ R |∇ ||∇ ˜ || ˜ | ≤ ∇ ˜ 2 ∇ 4 ˜ 4 ≤ ˜ 2 1 2 , ∫ R |∇ ||∇ ˜ || ˜ | 4 ≤ ∇ ˜ 2 ˜ 4 16 ∇ 4 ≤ ˜ 4 +1 1 2 ∫ R |∇ ||∇ || ˜ | 4 ≤ ˜ 4 8 ∇ 4 ∇ 4 ≤ ˜ 4 1 2
2 .

• For = 3, | | + | | 4 ≤ | | + | | 4 3 ≤ 2(| | + | | 2 ).
We have for all ∈ 2 (R ), (by the embedding 1

(R 3 ) ↩→ 6 (R 3 )) ∫ R |∇ ||∇ ˜ || ˜ | ≤ ∇ ˜ 2 ∇ 4 ˜ 4 ≤ ˜ 2 1 2 , ∫ R |∇ ||∇ ˜ || ˜ | 2 ≤ ∇ ˜ 2 ˜ 2 6 ∇ 6 ≤ ˜ 3 1 2 ∫ R |∇ ||∇ || ˜ | 2 ≤ ∇ 2 ˜ 3 6 ∇ 6 ≤ ˜ 2 1 2 2 .
Hence, gathering (2.105), (2.106), and(2.107), using (2.74) and the fact that 4 + 1 ≥ 2, it results

1 -2 =1 Im ∫ R Δ + (| | 2 ) ˜ (| ˜ + | 2 ) -(| | 2 ) ≤ - ˜ 2 1 .
(2.109) In a similar way, for all = 1, . . . , ,

2, + 2 + | | 2 4 ∫ R Im( ˜ ) (| ˜ + | 2 ) -(| | 2 ) ≤ 2 + | | 2 4 ∫ R ( -) ˜ (| ˜ + | 2 ) -(| | 2 ) + 2 + | | 2 4 ≠ ∫ R ˜ (| ˜ + | 2 ) -(| | 2 ) ≤ ∫ R - | ˜ | 4 +1 + | ˜ | 2 + ≠ ∫ R | ˜ | 4 +1 + | ˜ | 2 ≤ - ˜ 2 1 .
(2.110)

We have then for all = 1, . . . , ,

3, -2 • ∫ R Re(∇ ˜ ) (| ˜ + | 2 ) -(| | 2 ) ≤ ∫ R |∇( -)| | ˜ | + | ˜ | 2 + ≠ ∫ R ∇ | ˜ | 4 +1 + | ˜ | 2 ≤ - ˜ 2 1 .
(

         = -• ∇ + + | | 2 4 = Δ + (| | 2 ) , (2.112)
we notice that

Im (Δ + (| | 2 ) ) ˜ - + | | 2 4 Im ˜ -•Re ∇ ˜ = -Re (1 -) ˜ .
As a consequence of (2.90), we obtain a control of

∫ R 2Im (Δ + (| | 2 ) ) ˜ -2 + | | 2 4 Im ˜ -2 • Re ∇ ˜ (| ˜ + | 2 ) -(| | 2 ) by -˜ 2 1 . Finally, 1 + =1 2, -3, ≤ - ˜ 2 1 . (2.113)
Let us focus now on the terms identified by the letter . We observe that

2, = -• ∫ R (| ˜ + | 2 )∇(| ˜ | 2 ) + O ( -+ 1 ) 1 ˜ 1 . (2.114)
Then, we obtain

2, + 3, = -• Re ∫ R ∇ | ˜ + | 2 (| ˜ + | 2 ) + • Re ∫ R ∇ | | 2 (| ˜ + | 2 ) + 2 • Re ∫ R ∇( ˜ ) (| | 2 ) + O ( -+ 1 ) 1 ˜ 1 . (2.115) Notice next that • Re ∇ = Im Δ ,
which allows us to rewrite

1 -2 =1 • Re ∫ R ∇ (| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 )
as a sum of quantities in which the differences between and or the products ∇ for ≠ appear. Use moreover 1. on the one hand, the second estimate proven in Lemma 2.18 2. on the other, the following inequalities:

• for = 1: for all ∈ 1 (R ), ∫ R |∇ ||∇ ˜ || ˜ | ≤ ∇ ˜ 2 ∇ 2 ˜ ∞ ≤ ˜ 2 1 1 , ∫ R |∇ | 2 | ˜ | 2 ≤ ˜ 2 1 ∇ 2 2 ≤ ˜ 2 1 2 1 , ∫ R |∇ ||∇ || ˜ | 4 ≤ ˜ 4 1 ∇ 2 ∇ 2 ≤ ˜ 4 1 1 1 ,
• for ∈ {2, 3}: for all ∈ 2 (R ),

∫ R |∇ ||∇ ˜ || ˜ | ≤ ∇ ˜ 2 ∇ 4 ˜ 4 ≤ ˜ 2 1 2 ,

CHAPTER 2. ON SMOOTHNESS AND UNIQUENESS OF NLS MULTI-SOLITONS

∫ R |∇ | 2 | ˜ | 2 ≤ ˜ 2 4 ∇ 2 4 ≤ ˜ 2 1 2 2 , ∫ R |∇ ||∇ || ˜ | 4 ≤ ˜ 4 6 ∇ 6 ∇ 6 ≤ ˜ 4 1 2 2 .
Remark 2.8. Regarding the higher dimensions in order to prove Proposition 2.6, one would make use of the following inequality, valid for ≥ 3: for all ∈ 2 +1 (R ), for all ˜ ∈ 1 (R ),

∫ R |∇ ||∇ ˜ || ˜ | ≤ ˜ 2 -2 ∇ ˜ 2 ∇ ≤ ˜ 1 ˜ 2 2 +1 .
Then we conclude that

1 -2 =1 • Re ∫ R ∇ (| ˜ + | 2 ) -(| | 2 ) -2Re( ˜ ) (| | 2 ) ≤ - ˜ 4 +1 1 + ˜ 2 1 ≤ - ˜ 2 1 .
(2.116) Thus, from (2.115) and (2.116), we deduce that:

1 - =1 2, + 3, (2.117)
≤ - ˜ 2 1 + ( -+ 1 ) 1 ˜ 1 + =1 • Re ∫ R ∇ | | 2 -| | 2 (| ˜ + | 2 ) + =1 -• ∫ R ∇ | | 2 (| | 2 ) + • Re ∫ R ∇(| ˜ + | 2 ) (| ˜ + | 2 ) ≤ -+ 1 1 ˜ 1 .
(2.118)

Notice that we have used (2.87),

∫ R ∇ | | 2 (| | 2 ) = 0, and ∫ R ∇ | ˜ + | 2 (| ˜ + | 2 ) = 0.
To finish with, we have to obtain estimates for the terms with the letter involving and . Due to (2.81), (2.90), and (2.112), we compute

-2 =1 ∫ R Re Δ ˜ + Re ˜ - + | | 2 4 Re ˜ -• Im ∇ ˜ = 2 =1 Im ∫ R ˜ Δ + | | 2 - + | | 2 4 - • ∇ + O -˜ 1 1 = O -˜ 1 1 . (2.119)
On the other hand,

-2 =1 • ∫ R Re(∇ Δ ˜ ) + Re(∇ ˜ (| ˜ + | 2 )) + Re ∇ (| ˜ + | 2 ) -(| | 2 ) - + | | 2 4 Re ∇ ˜ -Im • ∇ ˜ = 2 =1 • Re ∫ R ∇ ˜ Δ + | | 2 - + | | 2 4 - • ∇ + 2 =1 • ∫ R Re ˜ ∇ | | 2 -Re ∇ | ˜ + | 2 - | | 2 + O ( -+ ˜ 1 ) ˜ 1 1 = O ( -+ ˜ 1 ) ˜ 1 1 ,
(2.120) again due to (2.81), (2.90), and (2.112). Consequently, (2.119) and (2.120) lead to

1 + 2 + =1 { 3, -4, } = O -+ ˜ 1 ˜ 1 1 . ( 2

Control of the directions

We have the following estimate which expresses that the variation in time of the real scalar products Re ∫ R ˜ ( ) ( ) (which appear in (2.94)) is essentially of order two in ( ).

Lemma 2.20. For all

≥ 1 , Re ∫ R ˜ ( ) ( ) ≤ - ( ) 1 + ( ) 2 1 .
(2.122)

Proof. We notice that

Re ∫ R ˜ = Re ∫ R + =1 Re ∫ R + =1 Re ∫ R • ∇ = Re ∫ R + Re ∫ R | | 2 + • Re ∫ R ∇ + ( ) = Re ∫ R + ( ), (2.123)
where is a complex-valued function defined on a neighborhood of +∞, differentiable in the sense of distributions, and such that ( ) = O ( - ( ) 1 ). Moreover,

Re ∫ R = Re ∫ R Δ + | + | 2 + | + | 2 - | | 2 + Re ∫ R -Δ + | | 2 = Re ∫ R | | 2 - | | 2 + Re ∫ R | + | 2 - | | 2 + O ( ) 2 1 , (2.124)
where we have used ≥ 2 and

∫ R Δ = ∫ R Δ and (| + | 2 ) -(| | 2 ) ≤ (| | + | | 4 ).

By means of

(| | 2 ) -(| | 2 ) ≤ | -| 2 + | -| + ≠ | |
(which is consequence of the application of the mean value theorem, as for the proof of Lemma 2.18), and by means of (2.24) and (2.74), we see that

Re ∫ R (| | 2 ) -(| | 2 ) ≤ - ( ) 2 .
(2.125)

Similarly by Lemma 2.18, and noting in addition that Re = Re (( -) + ( -)) ,

we have also

Re ∫ R (| + | 2 ) -(| | 2 ) ≤ - ( ) 1 . ( 2
                       Δ ˜ = Δ + =1 Δ + • ∇(Δ ) ˜ = + =1 + • ∇ + =1 + • ∇ = Δ + (| + | 2 ) + (| + | 2 ) -(| | 2 ) ,
differentiation with respect to of equality Im ∫ R ˜ = 0 (2.79) implies:

0 = Im ∫ R ˜ + Im ∫ R ˜ = Re ∫ R Δ + (| + | 2 ) + (| + | 2 ) -(| | 2 ) + =1 Im ∫ R + • ∫ R ∇ + ∫ R + • ∫ R ∇ -Re ∫ R + =1 + • ∇ Δ + (| | 2 ) ,
or equivalently

0 = O ( ) 2 1 + Re ∫ R (| | 2 ) -(| | 2 ) + Re ∫ R (| + | 2 ) -(| | 2 ) + =1 Re ∫ R + • Im ∫ R ∇ + =1 -Im ∫ R Δ + (| | 2 ) + • Re ∫ R Δ + (| | 2 ) ∇ - =1 -Im ∫ R Δ + (| | 2 ) + • Re ∫ R ∇ Δ + (| | 2 ) = ∫ R (| + | 2 ) -(| | 2 ) | | 2 + ∫ R 2 + 1 2 • ∫ R 2 + O ( ) 2 1 + - ( ) 1 .
(2.129) Similarly, exploiting the -dimensional equality Re ∫ R ˜ ∇ = 0 (2.79), we see that

0 = Im ∫ R Δ ˜ - =1 Δ - =1 • ∇Δ ∇ + O ( ) 2 -Im ∫ R (| | 2 ) ˜ - =1 - =1 • ∇ ∇ -Im ∫ R (| + | 2 ) -(| | 2 ) ∇ + =1 Im ∫ R ∇ + • Re ∫ R ∇ ∇ - =1 Re ∫ R Δ ∇ - =1 Re ∫ R (| | 2 ) ∇ - =1 • Im ∫ R ∇(Δ )∇ - =1 • Im ∫ R ∇ (| | 2 ) ∇ + Im ∫ R ˜ ∇(Δ ) + ∇ (| | 2 ) ,
or equivalently, using

∫ R ∇ ( 2 ) 2 = - =1 ∫ R ( 2 ) ( 2 ) = 0, 0 = O ( ) 2 1 + - ( ) 1 + ∫ R (| + | 2 ) -(| | 2 ) Im ∇ + Im ∫ R ˜ ∇( (| | 2 )) + 2 ∫ R 2 + Re ∫ R ∇ ∇ × = O ( ) 2 1 + - ( ) 1 + ∫ R (| + | 2 ) -(| | 2 ) Im ∇ + Im ∫ R ˜ ∇ (| | 2 ) + 2 ∫ R 2 + ∫ R ∇ ∇ + 1 4 2 × , (2.130)
Using Lemma 2.18 and (2.24), we obtain

∫ R (| + |) 2 -(| | 2 ) Im ∇ = 2 ∫ R Re( ) (| | 2 )Im ∇ + O 2 1 + 4 1 = 2 ∫ R Re( ) (| | 2 )Im ∇ + O - 1 + 2 1 + 4 1 .
Recalling the definition of ˜ (2.78), this reads also as follows:

∫ R (| + |) 2 -(| | 2 ) Im ∇ = 2 ∫ R Re ˜ (| | 2 )Im ∇ + O - 1 + 2 1 = O ˜ 2 + - 1 + 2 1 + 4 1 .
Considering that Im ∇ = | | 2 , we then deduce from (3.98) and (3.99) that

∫ R ∇ ∇ × = O ˜ 2 + - 1 + 2 1 + 4 1 2 . Since ∫ R ∇ ∇
is a positive definite symmetric matrix (by Proposition 2.30), it is invertible, so that inequality (2.128) holds for | ( )|. Then we conclude that the same inequality is true for | ( )| by (3.98).

End of the proof

We now conclude the proof of our uniqueness result, that is Theorem 2.3 by gathering (2.94) and the different controls obtained in subsection 2.3.2.

Let us begin with a control of 1 in terms of ˜ 1 which relies on the integrability of ↦ → ( ) 1 + ( ) Proposition 2.23. For large enough,

( ) 1 ≤ sup ≥ ˜ ( ) 1 + ∫ +∞ ˜ ( ) 1 . ( 2

.131)

Proof. Recall that we have already seen (2.86):

1 ≤ ˜ 1 + =1 | | + | |
Therefore, using the control of the modulation parameters obtained before, that is (2.128),

( ) 1 ≤ ˜ ( ) 1 + ∫ +∞ ˜ ( ) 1 + ∫ +∞ ( ) 2 1 + ( ) 4 1 + ∫ +∞ - ( ) 1 .
(2.132)

Since ↦ → ( ) 1 + ( ) 4 -1 1
is integrable in the neighborhood of +∞, we have for large enough:

∫ +∞ ( ) 2 1 + ( ) 4 1 ≤ ∫ +∞ ( ) 1 + ( ) 4 -1 1 sup ≥ ( ) 1 .
(

˜ ( ) 1 + ∫ +∞ ˜ ( ) 1 + ∫ +∞ ( ) 1 + ( ) 4 -1 1 + -sup ≥ ( ) 1 . (2.135)
Hence, for large enough,

sup ≥ ( ) 1 ≤ sup ≥ ˜ ( ) 1 + ∫ +∞ ˜ ( ) 1 , ( 2
˜ ( ) 2 1 ≤ ∫ +∞ 1 ˜ ( ) 2 1 + - ( ) 1 ˜ ( ) 1 + ( ) 2 1 ˜ ( ) 1 ≤ ∫ +∞ 1 ˜ ( ) 1 + - ( ) 1 + ( ) 2 1 sup ≥ ˜ ( ) 1 .
(2.137)

We deduce from the preceding line that for large enough

˜ ( ) 1 ≤ ∫ +∞ 1 ˜ ( ) 1 + - ( ) 1 + ( ) 2 1 .
(2.138) Using (2.133), (2.134), and (2.136), this leads to the fact that for large enough

˜ ( ) 1 ≤ ∫ +∞ 1 ˜ ( ) 1 + -+ ∫ +∞ ( ) 1 sup ≥ ˜ ( ) 1 + ∫ +∞ ˜ ( ) 1 .
Thus, for large values of ,

˜ ( ) 1 ≤ ∫ +∞ 1 ˜ ( ) 1 + -+ ∫ +∞ ( ) 1 ∫ +∞ ˜ ( ) 1 . (2.139)
Since by assumption ( ) 1 = O 1 with > 2 and since -≤ 1 -1 for large enough, there exist ˜ ≥ 0 and ≥ 1 such that for all ≥ ,

˜ ( ) 1 ≤ ˜ ∫ +∞ 1 ˜ ( ) 1 + 1 -1 ∫ +∞ ˜ ( ) 1 . ( 2
∃ ≥ 0, ∀ ≥ , ˜ ( ) 1 ≤ . Then := sup ≥ { ˜ ( ) 1 } is well defined. Let us pick up ˜ ≥ such that ˜ ˜ ( ˜ ) 1 ≥ 2 .
Now, replacing by ˜ in (2.140), we obtain

2 ˜ ≤ ˜ 1 ˜ + 1 ( -1) ˜ + -2 ≤ 2 ˜ ( -1) ˜ .
(2.141) Supposing ≠ 0 would lead to a contradiction because of the choice of . Consequently ∀ ≥ , ˜ ( ) 1 = 0.

We deduce from Proposition 2.23 and Lemma 2.24 that ∀ ≥ , ( ) 1 = 0.

The local well-posedness in 1 (R ) of (NLS) implies then = . Hence Theorem 2.3 is proved.

Uniqueness result for the critical pure-power case

In this paragraph, let ≥ 1 and : ↦ → 2 . Our proof of uniqueness in the class of multi-solitons such that ( ) -( ) 1 = →+∞ O 1 (for some ∈ N * sufficiently large to be determined later) and in the 2 -critical case consists in exploiting the same ideas as for the subcritical case. Nevertheless it is this time based on Proposition 2.25, stated below and proved in Appendix.

Proposition 2.25. Assume that ( ) = 2 and let > 0. There exists > 0 such that for all

= 1 + 2 ∈ 1 (R, C), ( ) ≥ 2 1 - 1 ∫ R 1 2 - 1 =1 ∫ R 1 2 + ∫ R 1 ( • ∇ ) 2 2 + ∫ R 2 2
.

(2.142)

In order not to be too redundant, we only explicit the main modifications of the proof given for the stable case.

Change of variable

We still consider , , and as defined at the beginning of Section 2.3. In order to apply Proposition 2.25 (which states a coercivity property available in the critical case), one has to take into account a third family of directions indexed by = 1, . . . , . More precisely, let us introduce ( , ) := --0 ∈ R , for all = 1, . . . , , and

˜ ( , ) := ( , ) + =1 { ( ) ( , ) + ( ) • ∇ ( , )} + =1 ( ) 2 + • ∇ - 2 • ( , ),
where , , and are well defined on [ 1 , +∞) (even if it means taking a larger 1 ) with values respectively in R, R , and R such that

∀ ≥ 1 ,                      Im ∫ R ˜ ( ) ( ) = 0 Re ∫ R ˜ ( )∇ ( ) = 0 Re ∫ R ˜ ( ) 2 + • ∇ - 2 • ( ) = 0.
(2.143)

As for the stable case, we can prove that ( ), ( ), and ( ), = 1, . . . , , are uniquely determined by the preceding orthogonality conditions. This time, we have to show indeed that the following block matrix is invertible:

˜ ( ) :=            0,0 ( ) 1,1 ( ) • • • 1, ( ) 0 ( ) 1,1 ( ) 1,1 ( ) • • • 1, ( ) 1 ( ) . . . . . . . . . . . . . . . 1, ( ) ,1 ( ) • • • , ( ) ( ) 0 ( ) 1 ( ) • • • ( ) ( )           
, where 0,0 , , , 1, are defined in paragraph 2.3.1, ( ) ( = 0, . . . , ) has entries zero on its diagonal and O( -) elsewhere, and ( ) possesses the coefficients

∫ R • ∇ 2 , = 1
, . . . , on its diagonal and O( -) elsewhere. For the sake of completeness, let us justify how to determine the coefficients of ( ) ( = 1, . . . , ) which are the less obvious ones to compute. By the orthogonality condition Re ∫ R ˜ ( )∇ ( ) = 0, the coefficient ( , ) of located at line and column is equal to

( , ) = 2 Re ∫ R + Re ∫ R • ∇ + 1 2 • Im ∫ R .
Thus for ≠ , we have ( , ) = O( -) by (2.24), and for = , we obtain:

( , ) = Re ∫ R • ∇ - 2 = ∫ R • ∇ = =1 ∫ R -1 ∫ R 1 . . . ˆ . . . = 0,
since for all ∈ {1, . . . , }, ↦ → ( 1 , . . . , ) ( 1 , . . . , ) is an odd integrable function on R in view of the fact that is radial.

Hence, we obtain that

det ˜ ( ) = det ( ) =1 ∫ R • ∇ 2 + O( -)
is strictly positive for large enough (see (2.167) in Appendix).

Moreover, for all ≥ 1 , (2.145)

| ( )|, | ( )|, | ( )| ≤ ( ) 2 , (2.144)
Remark 2.10. The consideration of ˜ turns out to be appropriate judging by the properties stated in Lemma 2.27. Besides let us note that the particular non-linearity satisfies the ODE ( ) = 2 ( ) in the 2 -critical case; this will be truly useful to control the third family of directions associated with the coefficients . First of all, let us begin with the useful computation of the derivative of ˜ with respect to the time variable. Lemma 2.26. We have

˜ = Δ ˜ + (| | 2 ) ˜ + (| ˜ + | 2 ) -(| | 2 ) + =1 + • ∇ -2 =1 + =1 2 + • ∇ - 2 • ( , ) + 1 ,
where 1 is a function of and such that

∫ R | 1 | | ˜ | + |∇ ˜ | + |Δ ˜ | ≤ - Proof. Note that ˜ decomposes like ˜ = + =1 (• • • ) + (• • • ) + (• • • ) + (• • • ) + =1 2 + • ∇ - 2 • + =1 2 (Δ + (| | 2 )) ) -• ∇ + 2 | | 2 + • ∇ Δ + (| | 2 ) + 1 2 • Δ + (| | 2 ) . ( 2
Δ • ∇ = • ∇(Δ ) + 2Δ , (2.147) Δ - 2 • = - 2 • Δ - • ∇ , (2.148)
(| + | 2 ) = (| + | 2 ) -(| | 2 ) + (| | 2 ) ˜ -(| | 2 ) (• • • ) + (• • • ) + 2 + • ∇ - 2 • ( , ) , (2.149)
and

(| + | 2 ) -(| | 2 ) = (| ˜ + | 2 ) -(| | 2 ) + 2Re ( -˜ ) (| | 2 ) + 1 = (| ˜ + | 2 ) -(| | 2 ) -2 =1 Re (• • • ) + 2 + • ∇ (| | 2 ) + 1 .
(2.150) Inserting each equality (2.147), (2.148), (2.149), and (2.150) in (2.146) leads to

˜ = Δ ˜ + (| | 2 ) ˜ + (| ˜ + | 2 ) -(| | 2 ) + =1 + • ∇ + =1 2 + • ∇ - 2 • ( , ) + =1 -2 • ∇ + 2 | | 2 -2 Δ -| | 2 (| | 2 ) + 1 .
Finally, we conclude by means of

| | 2 (| | 2 ) = 2 (| | 2 ) (2.151)
(which indeed holds in the 2 -critical case as mentioned in Remark 2.10) and the two possibilities given in (2.112) to write .

Control of ˜ and of the modulation parameters

Take again ˜ as defined at the end of paragraph 2.3.1 and consider still ( ) as in Proposition 2.17. Then we can state Lemma 2.27 (Control of the derivative of the Weinstein functional). The following assertion holds true:

˜ ( ) = ( ) -4 =1 ( ) - | | 2 4 Re ∫ R ˜ ( ) + O -+ ( ) 1 ˜ ( ) 1 ( ) 1 .
Proof. Take again the proof of Proposition 2.17. Concerning the expression of the derivative of ˜ , observe that everything is kept unchanged in the present context except that we have to take care of the additional terms involving the parameters ( ) and ( ), for all = 1, . . . , .

Let us define the C-linear endomorphism L of 1 (R ) by

L ( ) := -Δ -(| | 2 ) + + | | 2 4 + • ∇ .
Observe that L ( ) = 0 and for all , ∈ 1 (R ),

Re ∫ R L ( ) = Re ∫ R L ( ) -2Re ∫ R ∇ • ∇ + Re ∫ R • ∇ , ( 2
.152) Using (2.90), (2.92), and (2.152), we deduce that for all ∈ 1 (R ):

Re ∫ R L ( ) = Re ∫ R L ( ) + O - 1 = Re ∫ R L ( ) + O - 1 = O - 1 .
(2.153)

The term associated with ( ) in the expression of ˜ writes

2 ( )Re ∫ R (-2 )L ( ˜ ) .

By (2.153), it is thus bounded by

-˜ 1 1 .
It remains us to obtain the term associated with in ˜ . This term corresponds to I 1, -I 2, , where

I 1, = 2 Re ∫ R 2 + • ∇ - 2 • ( , ) L ( ˜ )
and

I 2, = 2 Re ∫ R 2 + • ∇ (| ˜ + | 2 ) -(| | 2 ) .
Let us concentrate first on I 1, . By (2.152),

I 1, = 2 Re ∫ R L 2 + • ∇ - 2 • ( , ) ˜ + O -˜ 1 .
Moreover,

Re ∫ R ˜ L 2 = 0; Re ∫ R ˜ L ( • ∇ ) = Re ∫ R ˜ -• ∇(Δ ) -2Δ -Re ∫ R ˜ (| | 2 ) • ∇ + Re ∫ R ˜ • ∇( • ∇ ) + Re ∫ R ˜ + | | 2 4 • ∇ = Re ∫ R ∇ ˜ • + ˜ Δ -2 Re ∫ R Δ + Re ∫ R ∇ ˜ • + ˜ (| | 2 ) + Re ∫ R ˜ • ∇( (| | 2 )) -Re ∫ R ∇ ˜ • • ∇ -( -1)Re ∫ R ˜ • ∇ -Re ∫ R ∇ ˜ • + ˜ + | | 2 4 + O -˜ 2 = -2 Re ∫ R ˜ Δ + Re ∫ R ˜ • ∇ + 2 Re ∫ R ˜ (| | 2 ) • Re ∇ + O( -˜ 2 ); Re ∫ R ˜ L 2 • = Re ∫ R ˜ - 2 ( • )Δ - • ∇ + Re ∫ R ˜ + | | 2 4 2 • - 1 2 Re ∫ R ˜ • ∇ ( • ) -Re ∫ R 2 ˜ (| | 2 ) • + O -˜ 2 = -Re ∫ R ˜ • ∇ - | | 2 2 Re ∫ R ˜ + O -˜ 2 .
Note that to establish the three preceding equalities, we have used once again L ( ) = 0. Thus, gathering the preceding calculations, we infer

I 1, = 2 Re ∫ R ˜ • ∇ (| | 2 ) -4 Re ∫ R ˜ Δ + • ∇ - | | 2 4 + O -˜ 2 1 .
(2.154) Now, let us focus on the second integral I 2, . On the one hand, by means of a Taylor expansion, by (2.24), (2.75), and (2.151), we obtain

∫ R | | 2 (| ˜ + | 2 ) -(| | 2 ) = ∫ R | | 2 × 2Re( ˜ ) (| | 2 ) + O -˜ 2 + ˜ 2 2 = 4 ∫ R (| | 2 )Re( ˜ ) + O -˜ 2 + ˜ 2 2 .
(2.155)

On the other, we observe that

2 Re ∫ R • ∇ (| ˜ + | 2 ) -(| | 2 ) = 4 Re ∫ R • ∇ Re ˜ (| | 2 ) + O -˜ 2 + ˜ 2 2 .
(2.156) Thus, we deduce from (2.155), (2.156), and (??) that

I 2, = Re ∫ R ˜ Δ + • ∇ - | | 2 4 + 2 Re ∫ R ˜ • ∇ (| | 2 ) + O -˜ 2 + ˜ 2 2 .
(2.157) From (2.154) and (2.157), we conclude that the term associated with in ˜ is equal to

-4 Re ∫ R ˜ Δ + • ∇ + (| | 2 ) - | | 2 4 + O( -˜ 2 1 ),
that is to

4 - | | 2 4 Re ∫ R ˜ + O( -˜ 2 1 ).
This finishes the proof of Lemma 2.27.

Lemma 2.28 (Control of the modulation parameters).

We have for all = 1, . . . , :

∀ ≥ 1 , | ( )| + | ( )| + | ( )| ≤ - 1 + 2 1 + ˜ 2 .
Proof. This lemma follows from the preliminary computations

Δ ˜ =Δ + =1 Δ + • ∇(Δ ) + 2 + 2 Δ + • ∇(Δ ) - =1 2
• Δ + • ∇ ˜ = + =1 + • ∇ + 2 + • ∇ - 2 • + =1 + • ∇( ) + =1 2 + • ∇( ) - 2 • -• ∇ + 2 | | 2 ,
from (3.98), (3.99) (which still take the same form in the present context with the consideration of a third direction), and from the derivation with respect to of the third family of orthogonal conditions

Re ∫ R ˜ 2 + • ∇ - 2 • = 0, which yields 0 = ∫ R • ∇ 2 - 2 4 ∫ R 2 + O 2 1 + - 1 + ˜ 1 . (2.158)
We finish the proof of Lemma 2.28 with (2.167) in Appendix. Now, to exploit Lemma 2.27 in order to perform an estimate of ˜ ( ), we have to control the scalar products Re ∫ R ˜ , = 1, . . . , . In fact, we state and prove next the analogue of Lemma 2.20.

Lemma 2.29 (Control of the directions). We have

Re ∫ R ˜ ≤ - 1 + 2 1 .
Proof. Note that (2.123) is still guaranteed here by observing moreover that

=1 Re ∫ R 2 + • ∇ - 2 = ∫ R 2 | | 2 + Re ∫ R • ∇ + 2 ( ) = 2 ( ),
where 2 ( ) = O ( - ( ) 1 ). Now, the rest of the proof of Lemma 2.20 is kept unchanged and thus we obtain Lemma 2.29.

We deduce from Lemma 2.28 and from Lemma 2.29 that

Re ∫ R ˜ ≤ - 1 + 2 1 + ˜ 1 ∫ +∞ - 1 + 2 1 . (2.159)
Next, it follows from (2.159) and Lemma 2.27 that

˜ ( ) ≤ ∫ +∞ 1 ˜ ( ) 2 1 + ( -+ ( ) 1 ) ˜ ( ) 1 ( ) 1 + ∫ +∞ - ( ) 1 + ( ) 2 1 + ˜ ( ) 1 ∫ +∞ - ( ) 1 + ( ) 2 1 . (2.160)

Conclusion of the proof of uniqueness in the critical case

Proposition 2.25 allows us to obtain the coercivity estimate in Proposition 2.15) so that by (2.127) (which follows also from Lemma 2.29) and (2.160), we obtain

˜ ( ) 2 1 ≤ ∫ +∞ 1 ˜ ( ) 2 1 + ( -+ ( ) 1 ) ˜ ( ) 1 ( ) 1 + ∫ +∞ - ( ) 1 + ( ) 2 1 + ˜ ( ) 1 ∫ +∞ - ( ) 1 + ( ) 2 1 .
On the other hand, Proposition 2.23 and in fact (2.133), (2.134), and (2.136) are still available here in the critical case; this is guaranteed by Lemma 2.28. Thus, adapting the proof of Lemma 2.24, we deduce the following estimate in which ˜ is the only variable that appears: for large enough, and for some > 4,

˜ ( ) 1 ≤ ∫ +∞ 1 ˜ ( ) 1 + 1 -2 ∫ +∞ ∫ +∞ ˜ ( ) 1 (2.161)
(with independent of for the same reasons as those mentioned in Remark 2.9). It results then ˜ ( ) = 0 in the neighborhood of +∞. Note that one requires here ( ) 1 = O 1 , with > 4, to hold. Consequently, uniqueness of a multi-soliton associated with the , = 1, . . . , in the sense of (2.11) is proved also in the 2 -critical case.

Appendix

Linear independence of ∈{1,..., } Proposition 2.30. Let ∈ 1 (R ) be such that there exists ∈ R \ {0} such that

∀ ∈ R , • ∇ ( ) = 0. Then = 0.
Proof. Even if it means completing | | in an orthonormal basis of R and considering the passage matrix between the canonical basis and this new basis, we can always assume that is the last vector of the canonical basis of R . In that case, our assumption in Proposition 2.30 reads:

∀ ( 1 , . . . , ) ∈ R , ( 1 , . . . , ) = 0,
or, in other words, for all 1 , . . . , -1 ∈ R, the application

↦ → ( 1 , . . . , -1 , ) is constant, equal to ( 1 , . . . , -1 , 0). Since ∈ 2 (R ), for all 1 , . . . , -1 ∈ R, one must have that ∫ R | ( 1 , . . . , -1 , )| 2 = ∫ R | ( 1 , . . . , -1 , 0)| 2
is a finite quantity by Fubini theorem. This is the case if and only if ( 1 , . . . , -1 , 0) = 0. Thus = 0.

If 1 ≥ + ( -0 ) , then 1 > ,1 and one obtains again

| ( , )| ≤ - - √ 4 | -| .
Hence, using in addition (2.91), we deduce from what precedes that

| ( , ) 1 ( , )| ≤ - - √ 4 | -| .
In this manner, we obtain (2.92).

Proof of Proposition 2.25: coercivity property in the 2 -critical case

In the 2 -critical (pure power) case, we consider : ↦ → 2 so that the linearized operators around rewrite +, ( ) = -Δ + -1 + 4 , = 0 so that we can set := inf ∈ +, , , where is the set of all ∈ 1 (R ) such that , = 0, for all = 1, . . . , , , = 0, • ∇ , = 0, and 1 = 1. We have obviously ≥ 0; we aim to show that > 0.

Assume by contradiction that = 0. Then for all ∈ N, there exists ∈ such that

+, , ≤ 1 +1 . This implies 0 < min{ , 1} 2 1 ≤ ∫ R |∇ | 2 + ∫ R | | 2 ≤ 1 + 4 ∫ R 4 2 + 1 + 1 . (2.

162) In addition, (

1 ) is uniformly bounded so that, up to extraction, ( ) converges in 1 (R ) for the weak topology, say to * ∈ 1 (R ). And so, we have , . Even if it means considering * * 1 , we will assume moreover * 1 = 1. We are thus led to the following Lagrange multiplier condition:

, * = 0, ∀ ∈ {1, . . . , }, , * = 0, • ∇ , * = 0. ( 2
+, * = * + + =1 + • ∇ , (2.166)
for some reals , , , and . Since 0 = +, * , * , (2.163) implies that = 0. Then, for all ∈ {1, . . . , },

0 = +, , * = +, * , = =1 , .
Given that 1 , . . . , are linearly independent in 2 (R ), the × -matrix with entries , is invertible. Consequently, for all ∈ {1, . . . , }, = 0. Now, using

0 = -2 , * , +, 2 + • ∇ = -2
(which is specific to the critical case), and the symmetry of the bilinear form +, •, • , we deduce that

0 = +, * , 2 + • ∇ = 2 2 2 - 2 4 - 2 2 2 + ∫ R ( • ∇ ) 2 = ∫ R ( • ∇ ) 2 - 2 4 ∫ R 2 . But we have ∫ R • ∇ 2 - 2 4 ∫ R 2 > 0, (2.167)
considering that this quantity is nothing but the square of the 2 norm of 2 + • ∇ (and 2 + • ∇ is obviously not zero). Hence = 0, and finally (2.166) reduces to +, * = . We claim now that ≠ 0: otherwise (using the well-known non-degeneracy condition (2.17) of +, in the present case) * would be a linear combination of the , = 1, . . . , , and then it would result * = 0 (since for all , * , = 0), which is not the case.

Thus * = -2 2 + • ∇ and 0 = * , • ∇ = - 2 2 + • ∇ , • ∇ = - 2 ∫ R ( • ∇ ) 2 - 2 4 ∫ R 2 ,
which contradicts (2.167). So we come to the conclusion that is positive; hence Proposition 2.25 is established.

Introduction

Setting of the problem and known results

We consider the generalized Korteweg-de Vries equations

+ ( 2 + ) = 0 (0) = 0 ∈ 1 (R) (gKdV)
where ( , ) are elements of R × R and > 1 is an integer.

Recall that the Cauchy problem for (gKdV) is locally well-posed in 1 (R) from a standard result by Kenig, Ponce and Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] and that the two following quantities are conserved for each solution of (gKdV) for all :

• (the 2 -mass) ∫ R 2 ( , ) • (the energy) ∫ R 1 2 2 - 1 + 1 +1 ( , ) .
Ce chapitre fait l'objet d'un article publié dans les Annales de l'Institut Henri Poincaré C, Analyse non linéaire [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF].

In addition, the set of solutions of (gKdV) is conserved under scaling transformation

↦ -→ ( , ) ↦ → 2 -1 3
, , for all > 0, and the ( ) -norm is invariant under this transformation, where ( ) := 1 2 -2 -1 . Let us recall that the global dynamics of the solutions depends on the sign of ( ). The case ( ) < 0 that is 1 < < 5, is called 2 -subcritical, and all 1 -solutions of (gKdV) are then global (in time) and 1 -uniformly bounded. If ( ) = 0, that is = 5, we are in 2 -critical case and solutions might blow up in finite time [START_REF] Martel | Blow up in finite time and dynamics of blow up solutions for the 2 -critical generalized KdV equation[END_REF][START_REF] Martel | Nonexistence of blow-up solution with minimal 2 -mass for the critical gKdV equation[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. II: Minimal mass dynamics[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. III: Exotic regimes[END_REF]. In the 2 -supercritical case, corresponding to ( ) > 0 (or > 5), much less is known but finite time blow up is expected: existence of * > 5 and of blow-up solutions for all ∈ (5, * ) are proven in [START_REF] Lan | Blow-up solutions for 2 -supercritical gKdV equations with exactly k blow-up points[END_REF].

Moreover it is well-known that (gKdV) admits a family of explicit traveling wave solutions indexed by R * + × R. Let be the unique (up to translation) positive solution in 1 (R) (known also as ground state) to the following stationary elliptic problem associated with (gKdV)

+ = ,
given by the explicit formula

( ) = + 1 2ch 2 -1 2 1 -1
.

Then for all 0 > 0 (velocity parameter) and 0 ∈ R (translation parameter),

0 , 0 ( , ) = 0 ( -0 -0 ) (3.1)
is a global traveling wave solution of (gKdV) classically named soliton solution, where 0 ( )

= 1 -1 0 ( √ 0 )
. It is orbitally stable if and only if < 5 ( 2 -subcritical case) (see Weinstein [START_REF] Michael | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF], Bona, Souganidis and Strauss [START_REF] Bona | Stability and instability of solitary waves of Korteweg-de Vries type[END_REF], Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], and Martel and Merle [START_REF] Martel | Instability of solitons for the critical generalized Korteweg-de Vries equation[END_REF]).

Solitons are special objects which enjoy very specific properties. Let us recall the following rigidity result, which roughly asserts that non dispersive solutions to (gKdV) which are close to solitons are actually exactly solitons. Theorem 3.1 (Liouville property near a soliton; Martel and Merle [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF][START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF]). Let 0 > 0. There exists > 0 such that if ∈ C (R, 1 (R)) is a solution of (gKdV) satisfying, for some C 1 function : R → R,

(closeness to a soliton) ∀ ∈ R, ( , • + ( )) -0 1 ≤ , (3.2) (non-dispersion) ∀ > 0, ∃ > 0, ∀ ∈ R, ∫ | |> 2 ( , + ( )) ≤ , (3.3) then there exist 1 > 0, 1 ∈ R such that ∀ , ∈ R, ( , ) = 1 ( -1 -1 ).
This striking result has its own interest of course, but we emphasize that it is also a key ingredient to prove asymptotic stability of (gKdV) solitons (we refer to [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF][START_REF] Martel | Asymptotic stability of solitons for subcritical gKdV equations[END_REF][START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF]). We highlight the fact that this result applies in each mass subcritical, critical, and supercritical case by requiring the solution to remain close to a soliton (up to translation) for all times (3.2). In the 2 -subcritical case where solitons are known to be stable, (3.2) can be relaxed to hold only at = 0.

Finally let us note that solitons play a fundamental role in the study and the understanding of the (gKdV) flow; the important soliton resolution conjecture asserts that any solution with generic initial condition behaves as a sum of solitons plus a radiative-dispersive term as time goes to infinity. In this spirit, built upon solitons, we are interested in other solutions to our problem, namely multisoliton solutions, defined as follows.

Definition 3.2. Let ≥ 1 and consider solitons

, as in (3.1) with speeds 0

< 1 < • • • < . A multi-soliton in +∞ (resp. in -∞) associated with the
, is an 1 -solution of (gKdV) defined in a neighborhood of +∞ (resp. -∞) and such that

( ) - =1 , ( ) 1 → 0, as → +∞ (resp. as → -∞). (3.4)
Multi-solitons are known to exist for all > 1; they are even explicit for = 2 (KdV) [89, section 16] and for = 3 (mKdV) [101, Chapter 5, formula (5.5)]. What is more, the classification of the multi-solitons of (gKdV) is complete. Let us gather the main results. Theorem 3.3 (Martel [63]; Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]; Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]). Let > 1 be an integer and let ≥ 1, 0 < + 1 < • • • < + , and + 1 , . . . , + ∈ R. If ≤ 5, there exists 0 ≥ 0 and a unique multi-soliton ∈ C ( [ 0 , +∞), 1 (R)) associated with the + , + , ∈ {1, . . . , }. If > 5, there exists a one-to-one map Φ from R to the set of all 1 -solutions of (gKdV) defined in a neighborhood of +∞ such that is a multi-soliton in +∞ associated with the + , + if and only if there exist ∈ R and 0 ≥ 0 such that | [ 0 ,+∞) = Φ( ) | [ 0 ,+∞) . Moreover, in each case, belongs to C ( [ 0 , +∞), (R)) for all ≥ 0, and there exist > 0 and positive constants such that for all ≥ 0, for all ≥ 0 , ( ) -

=1 + , + ( ) ≤ -. (3.5)
In the 2 -subcritical case (like solitons), sums of decoupled and ordered solitons are stable in 1 (R), even asymptotically stable (Martel, Merle and Tsai [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of solitons for subcritical gKdV equations[END_REF] and Martel and Merle [START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF]), and so are multi-solitons.

Main results

Several properties available for solitons have been adapted or even extended to multi-solitons. This article precisely takes this step since it aims at providing an analogue of the rigidity property of Theorem 3.1 in the multi-soliton case. We consider solutions of (gKdV) that are non dispersive in some sense and uniformly close to the sum of solitons, and show that they are exact multisolitons.

Theorem 3.4 (Liouville property near a multi-soliton). Let be a solution of (gKdV) which belongs to C ([0, +∞), 1 (R)). Assume the existence of > 0 such that

∀ > 0, ∃ > 0, ∀ ≥ 0, ∫ < - 2 ( , ) ≤ . (3.6)
Let ≥ 1 be an integer and consider positive real numbers 0 < 1 < • • • < . There exists = ( 1 , . . . , , ) > 0 such that the following holds: if there exist functions 1 , . . . , :

R + → R of class C 1 satisfying ∀ ≥ 0, ( ) - =1 (• -( )) 1 ≤ , (3.7
)

and ∀ ≥ 0, ∀ ∈ {1, . . . , -1}, +1 ( ) -( ) ≥ | ln |, (3.8)
then is a multi-soliton (in +∞). In other words, there exist > 0, 0

< + 1 < • • • < + , + 1
, . . . , + ∈ R and positive constants such that for all ≥ 0, (3.5) is granted.

Remark 3.5. Assumption (3.8) is done so that solitons are sufficiently decoupled, and thus do not collide. In the 2 -subcritical case 1 < < 5, as sum of decoupled solitons are stable, assumptions (3.7) and (3.8) can be relaxed to hold only at time = 0.

This result is a natural extension of Theorem 3.1 to multi-solitons in +∞, which are the only solutions which are non dispersive in the sense (3.6) (and remain close to a sum of solitons): this is a nice dynamical characterization of multi-solitons among solutions to (gKdV). By contraposition, it means that if a solution remains in large time sufficiently close to a multi-soliton but is not a multi-soliton, then it disperses insofar as (3.6) fails.

We emphasize that, in contrast with the original statement for one soliton (Theorem 3.1), the non dispersion assumption (3.6) requires the mass to be located essentially for ≥ for some small positive speed > 0 (and almost touches = 0); it allows (seemingly) for much more room than in the condition (3.3), which requires that the mass be essentially concentrated in a moving ball of fixed size .

Furthermore, the assumptions in Theorem 3.4 are done only for positive times ≥ 0 (and not for all times ∈ R). As it applies of course to the case of a single soliton, Theorem 3.4 actually extends and refines Theorem 3.1.

We must underline that this improvement to focus on the behavior for positive times only is actually very meaningful. Indeed, in view of the above result, a solution which would be non dispersive at times +∞ and -∞ would be a multi-soliton at both ends: but such a behavior is not to be expected, except in the integrable cases of = 2 (KdV) and = 3 (mKdV) and the Gardner nonlinearity 2 -3 . To support this, let us refer to the work by Martel and Merle [START_REF] Martel | Description of two soliton collision for the quartic gKdV equation[END_REF][START_REF] Martel | Inelastic interaction of nearly equal solitons for the quartic gKdV equation[END_REF][START_REF] Martel | On the nonexistence of pure multi-solitons for the quartic gKdV equation[END_REF] (see also Muñoz [90]) on the description of 2-solitons: starting with a 2-soliton solution at -∞ (for the quartic = 4 (gKdV)), the collision is almost but not elastic, and there is a non zero defect (which one can quantify), so that it is not a 2-soliton at +∞ (and so, by Theorem 3.4, it must be dispersive in the sense of (3.6)).

In principle, the computations in the articles above could extend to -solitons for ≥ 3, but it has not been performed yet, and one could still wonder if there is always a defect. If one is willing to assume non dispersion for all time ∈ R, our conclusion is that the solution under consideration is a multi-soliton in +∞ for which all derivatives decay exponentially in space for each fixed values of . More precisely, we have

∀ > 0, ∃ > 0, ∀ ∈ R, ∫ B ( , | |+ ) 2 ( , ) ≤ . (3.9)
Then the conclusion of Theorem 3.4 holds, and also the following exponential decay property at fixed time, for all ∈ N, and for some possibly larger constant :

∀ ≥ 0, ∀ ∈ R, ( , ) ≤ =1 -| -+ | . (3.10)
As it was first observed in [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF], non dispersion (for all times) actually self improves to smoothness and exponential decay in space (outside the center of mass). Of course, this is very relevant for solitons, which exhibit precisely this spatial behavior. But it has yet to be proven that multi-solitons do have spatial exponential decay (3.10) as well; even though it is a very natural conjecture, and that it is known that multi-solitons are smooth. To be able to conclude to (3.10), one has currently to make the assumption (3.9) (in fact, it would be sufficient to assume that and (-) satisfy (3.6) and to assume in addition that the analog of (3.6) with > ( + ) + holds for positive times), and for the time being, the above Corollary 3.1 is meaningful. Remark 3.6. In the 2 -subcritical case 1 < < 5, assumptions (3.7) and (3.8) can be relaxed to hold only a time = 0. If they hold for large enough times, positive and negative (or outside of the collision period), the conclusion can be strengthened to being a multi-soliton at +∞ and -∞, and satisfying (3.10) for all ∈ R.

In the context of the particular (KdV) equation (corresponding to = 2), we claim next a result which gives rise to a simplified characterization of multi-solitons among all 1 -solutions. Theorem 3.7. Let = 2 and 0 ∈ S (R) \ {0} be such that the corresponding solution of (KdV), which is defined globally in time, is non dispersive for positive times, that is, satisfies (3.6). Then is a multi-soliton (in +∞ and -∞).

The proof of this theorem relies on the soliton resolution result for (KdV), set up by Eckhaus and Schuur [START_REF] Eckhaus | The emergence of solutions for the generalized Korteweg-de Vries equation from arbitrary initial conditions[END_REF] and refined in Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF].

Remark 3.8. Requiring that the initial condition 0 belongs to the Schwartz space is not necessary in order to reach the conclusion in Theorem 3.7. Considering the non dispersion assumption made in Theorem 3.7, it would be sufficient for example that all derivatives up to order 4 of 0 decay faster than -11 when → +∞. Actually, we only need to assume that 0 is smooth enough and decays sufficiently rapidly for | | → +∞ for the whole of the inverse scattering method to work, thus for the soliton resolution result for (KdV) to hold [START_REF] Cohen | Existence and regularity for solutions of the Korteweg-de Vries equation[END_REF][START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF]. However, our goal is not to obtain the most general statement, and for clarity purposes, we will not attempt to optimize the regularity and decay assumptions on 0 .

Similarly, we can characterize non dispersive solutions of the (mKdV) equation. Recall that, in addition to solitons, (mKdV) admits other particular solutions, known as breathers, which are also important with respect to the soliton resolution conjecture. Breathers do not correspond to a superposition of solitons but are instead periodic in time and can move both in the left and right directions; for all ( , ) ∈ R * + × R * + , and for all 1 , 2 ∈ R, the breather , , 1 , 2 with envelope velocity := 2 -3 2 , phase velocity := 3 2 -2 , and translation parameters 1 , 2 ∈ R takes the following expression:

, , 1 , 2 ( , ) := 2 √ 2 arctan sin( ( --1 )) cosh( ( --2 )) . (3.11)
We refer to Alejo and Muñoz [START_REF] Alejo | Nonlinear stability of mKdV breathers[END_REF] for the introduction and the study of stability in 2 of these solutions. Note that the decomposition result in terms of solitons and breathers available for (mKdV) solutions and stated in [101, Chapter 5, Theorem 5.1] and more recently in [8, Theorem 1.10] holds under the assumption that the initial data 0 is generic in the following sense: the set of all ∈ C such that the classical Jost solutions ( ) and ( ) to the Zakharov-Shabat system

1 2 = - 0 -0 1 2
are R-linearly dependent is finite and consists in the scattering data Our result on non dispersive solutions of (mKdV) writes as follows.

{ √ 1 , . . . , √ 1 , 1 + 1 , . . . , 2 + 2 }, (3.12)
with 1 , 2 ∈ N, 1 , . . . , 1 , 1 , . . . , 2 , 1 , . . . , 2 ∈ R * + such that 1 < • • • < , 2 1 -3 2 1 < • • • < 2 2 -3 2 2 , (3.13)
Theorem 3.9. Let = 3 and 0 ∈ S (R) \ {0} be generic (in the above sense) with scattering data (3.12), and such that the corresponding global solution of (mKdV) is non dispersive for positive times (that is, satisfies (3.6)).

Then is a multi-breather with positive speeds in +∞: we have 1 + 2 ≥ 1 and for all = 1, . . . , 2 , 2 -3 2 > 0, and there exist > 0, positive constants , signs = ±1, and real parameters 0, , 1, , 2, such that for all ≥ 0, belongs to C ( [0, +∞), (R)) and

∀ ≥ 0, ( ) - 1 =1 2 , 0, ( ) - 2 =1 √ 2 , √ 2 , 1, , 2, ( ) ≤ -.
The proof is done by adapting that of Theorem 3.7 by writing the soliton/breather resolution for = 3, and then using smoothness and uniqueness of multi-breathers and the estimates in higher order Sobolev spaces proved by Semenov [START_REF] Semenov | On the existence and uniqueness of multi-breathers of (mKdV)[END_REF]. Remark 3.10. Let us notice that Remark 3.8 applies also in the context of Theorem 3.9.

Comments

The proofs of Theorem 3.4 and Corollary 3.1 are in the spirit of the original Liouville result by Martel and Merle [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF], and also the work of Laurent and Martel [START_REF] Laurent | Smoothness and exponential decay of 2 -compact solutions of the generalized KdV equations[END_REF] on smoothness and decay of non dispersive solutions. An important ingredient is the observation that a crucial monotonicity formula holds under a much relaxed non dispersion assumption than previously made, see Proposition 3.11, which has its own interest. Also we underline a subtle but key difference in the strategy of the proof: we crucially rely at some point on the asymptotic stability of multi-solitons in the energy space (from [START_REF] Martel | Asymptotic stability of solitons of the subcritical gKdV equations revisited[END_REF], stated in Theorem 3.15). But let us recall this result itself is a consequence of the rigidity result for one soliton stated in Theorem 3.1: in some sense, the roles are reversed here.

In fact, our proofs use and combine several previous results on the (gKdV) flow around solitons and multi-solitons. This sheds a new light on many results established so far and which have their own interest, which are here linked together to yield new statements. It seems to us that this phenomenon is an interesting point of this paper.

Our results lead to several open questions. We already mentioned above the first one, but repeat it here: we conjecture that, for each time where defined, multi-solitons for (gKdV) have pointwise exponential decay (along with their derivatives); this is only known in the integrable case, where explicit formulas are known. As second question is whether similar rigidity results (as well as asymptotic stability properties) hold for other dispersive models. The Liouville theorem for solitons holds for the Zakharov-Kuznetsov equation in 2D for example, it would be nice to know if an analog for multi-solitons holds as well. A very natural context is that of the non-linear Schrödinger equations, for which the understanding of non-dispersive solutions remains mostly open. This article is organized as follows. After the introduction, we present in section 3.2 a general property of exponential decay satisfied by non-dispersive solutions which is an important new observation and interesting in itself. The third section is then devoted to the proof of Theorem 3.4 and Corollary 3.1. In the forth section, we consider the integrable case and sketch the proofs of Theorems 3.7 and 3.9.

Acknowledgments

The author would like to thank his supervisor Raphaël Côte for suggesting the idea of this work and for fruitful discussions.

Smoothness and exponential decay for non dispersive solutions

The goal of this section is to show the following propositions which extend Laurent and Martel [57, Theorem 1]. Proposition 3.11. Let be a neighborhood of +∞ and ∈ C ( , 1 (R)) be a solution of (gKdV) which belongs also to ∞ ( , 1 (R)). Suppose that there exists > 0 such that

∀ > 0, ∃ > 0, ∀ ∈ , ∫ < - 2 ( , ) ≤ . (3.15)
Then belongs to C ∞ ( × R) and there exists > 0 such that for all ∈ N, there exists

> 0 such that ∀ ∈ , ∀ < , ( , ) ≤ -2 | -| . (3.16)
We state next another generalized version, which is useful in the proof of Corollary 3.1.

Proposition 3.12. Let be a neighborhood of +∞ and ∈ C ( , 1 (R)) be a solution of (gKdV) which belongs also to ∞ ( , 1 (R)). Suppose that there exist , > 0 and two

C 1 functions , : → R such that ∀ ∈ , ≤ ( ) ≤ ( ) ≤ , (3.17
)

and ∀ > 0, ∃ > 0, ∀ ∈ , ∫ < ( )- 2 ( , ) ≤ , (3.18)
where ( ) := min{ ( ), ( )}.

Then belongs to C ∞ ( × R) and there exists > 0 such that for all ∈ N, there exists

> 0 such that ∀ ∈ , ∀ < ( ), ( , ) ≤ -2 | -( ) | . (3.19)
Remark 3.13. It is to be noticed that, if in Proposition 3.12 is replaced by a neighborhood of -∞, then we conclude with an estimate at the right of ( ) := max{ ( ), ( )}, or more precisely with the existence of > 0 such that for all ∈ N, there exists > 0 such that

∀ ∈ , ∀ > ( ), ( , ) ≤ -2 ( -( )) . (3.20)
This is justified by the following symmetry property for (gKdV) and the assumption in Proposition 3.12. Denoting ˆ ( , ) := (-, -), ˆ ( ) := -(-), ˆ ( ) := -(-), ˆ ( ) := -(-), and ˆ ( ) := -(-), we observe that satisfies the assumptions of Proposition 3.11 on a neighborhood of -∞ if and only if ˆ satifies the same assumptions on -(which is a neighborhood of +∞) with , , , and replaced respectively by ˆ , ˆ , ˆ , and ˆ in Proposition 3.12. Thus once we have proved (3.19) as stated in Proposition 3.12, we have immediately the pointwise estimate on ˆ ( ) at the left of ˆ ( ) for ∈ -, which precisely provides (3.20), that is the desired pointwise estimate on ( ) at the right of ( ) for ∈ . Obviously, Propositions 3.11 and 3.12 apply in particular with = R, in which case both estimates (3.19) and (3.20) hold. Now, proceeding essentially as Laurent and Martel [57, Theorem 1], we derive the proof of Proposition 3.12.

Proof. Step 1: Estimates to be established

By the classical Sobolev embedding 1 (-∞, ( )) ↩→ ∞ (-∞, ( )), it suffices to see that

∃ > 0, ∀ ∈ , ∫ < ( ) 2 ( , ) + 2 ( , ) ( ( )-) ≤ (3.21)
holds to have the desired conclusion, that is (3.19), for = 0 and for almost every < ( ). Using that ( ) is continuous on R by 1 (R) ↩→ C (R), we deduce that (3.19) is true for = 0.

Similarly, to reach the whole conclusion, we have to show that for each ∈ N, there exists ˜ > 0 such that

∀ ∈ , ∫ < ( ) ( , ) 2 ( ( )-)
≤ ˜ .

In order to prove (3.22), it is convenient to introduce a well-chosen C 1 function defined on , denoted by ˜ , which replaces somehow in the case where is not already C 1 . By this means, we get around the difficulty of a possible point where is not differentiable. This is the purpose of the following: Claim 3.14. There exists ˜ : → R of class C 1 such that for all ∈ , ( ) ≤ ˜ ( ) ≤ ( ) + 1, and ˜ ( ) ≥ .

Proof of Claim 3.14. Define ˜ by

∀ ∈ , ˜ ( ) := 1 + ( ) + ( ) 2 -1 + ( ) -( ) 2 2
.

Then ˜ is C 1 on and given that min{

, } = + 2 -| -| 2 , one can check that ˜ satisfies ( ) ≤ ˜ ( ) ≤ ( ) + 1 by means of the well-known inequality √ + ≤ √ + √ , valid for all , ≥ 0.
Moreover, by a straightforward computation, we have

˜ ( ) ≥ ( ) + ( ) 2 - ( ) -( ) 2 ≥ ( ).
Consequently, Claim 3.14 is proved. Now, we consider ˜ as in the previous claim. Judging by the fact that for all ∈ , ( ) ≤ ˜ ( ), we can write for all ∈ N

∫ < ( ) ( , ) 2 ( ( )-) ≤ ∫ < ˜ ( ) ( , ) 2 ( ˜ ( )-) . (3.23)
Thus, to achieve our goal (3.22), it suffices to show the existence of > 0 such that

∀ ∈ , ∫ < ˜ ( ) ( , ) 2 ( ˜ ( )-) ≤ . (3.24)
Step 2: Proof of (3.24) for = 0

We will obtain (3.24) by a strong monotonicity property which is the purpose of Lemma 3.2 and Lemma 3.3 below.

Let us introduce, for some > 0 to be determined later, the function defined by

( ) = 1 2 - 1 arctan( ).
It satisfies the following properties

∃ 0 > 0, ∀ ∈ R, 0 -| | < -( ) < 1 0 -| | , (3.25)
∀ ∈ R, | (3) ( )| ≤ -2 ( ). (3.26)
∃ 1 > 0, ∀ ≥ 0, 1 - ≤ ( ). (3.27)
Moreover, let us observe that

∫ < ˜ ( ) 2 ( , ) ( ˜ ( )-) = ∫ <0 2 ( , + ˜ ( )) - , (3.28)
and that, for all 0 < 0,

∫ 0 ≤ <0 2 ( , + ˜ ( )) - ≤ -0 ∫ ≥ 0 2 ( , + ˜ ( )) -( -0 ) ≤ 1 1 -0 ∫ ≥ 0 2 ( , + ˜ ( )) ( -0 ) ≤ 1 1 -0 ∫ R 2 ( , + ˜ ( )) ( -0 ) . (3.29)
By Claim 3.14, for all ∈ , ˜ ( ) ≥ . Therefore there exists > 0 and an increasing affine function : R → R such that

∀ ∈ , -( ) + ˜ ( ) ≥ . (3.30)
Now, for fixed 0 ∈ and 0 in R, consider

( 0 , 0 ) : → R + ↦ → ∫ R 2 ( , + ˜ ( )) -0 + ( ) -( 0 ) .
We have

∀ ∈ , ( 0 , 0 ) ( ) = ∫ R 2 ( , ) -0 + ( ) -( 0 ) -˜ ( ) , (3.31)
so that by differentiation with respect to , we obtain

( 0 , 0 ) ( ) = -3 ∫ R 2 ( , ) ( ˜ ) --( ) + ˜ ( ) ∫ R 2 ( , ) ( ˜ ) + ∫ R 2 ( , ) (3) ( ˜ ) + 2 + 1 ∫ R +1 ( , ) ( ˜ ) , (3.32)
where ˜ := -0 + ( ) -( 0 ) -˜ ( ).

Set := 2 . We claim then Lemma 3.2. There exists 0 > 0 such that for all 0 ∈ R, and for all 0 , ∈ ,

( 0 , 0 ) ( ) ≥ -0 --0 + ( )-( 0 ) . (3.33)

SMOOTHNESS AND EXPONENTIAL DECAY FOR NON DISPERSIVE SOLUTIONS 99

Proof of Lemma 3.2. Due to the choice of and property (3.26) of , we have

∫ R 2 ( , ) (3) ( ˜ ) ≤ - 2 ∫ R 2 ( , ) ( ˜ ) . (3.34)
Furthermore we control the non-linear part by considering, for > 0,

1 ( ) := ∫ | ˜ |>-0 -+ ( )-( 0 ) +1 ( , ) ( ˜ ) and 2 ( ) := ∫ R +1 ( , ) ( ˜ ) -1 ( ).
On the one hand, we have due to (3.25)

1 ( ) ≤ 1 0 --0 -+ ( )-( 0 ) ∫ R | | +1 ( , ) ≤ --0 -+ ( )-( 0 ) ,
where we have used the Sobolev embedding 1 (R) ↩→ +1 (R) and the fact that belongs to ∞ ( , 1 (R)). Note that > 0 is independent of 0 , 0 , and . On the other, we observe that if | ˜ | ≤ -0 -+ ( ) -( 0 ), then ≤ ˜ ( ) -in particular, and therefore by Claim 3.14 we have also ≤ ( ) -+ 1. Thus, it follows

2 ( ) ≤ ( ) -1 ∞ ≤ ( )-+1 ∫ ≤ ( )-+1 2 ( , )| ( ˜ )| ≤ √ 2 -1 ( ) -1 2 2 ≤ ( )-+1 ( ) -1 2 2 ≤ ( )-+1 ∫ R 2 ( , )| ( ˜ )| ≤ √ 2 -1 ( ) -1 2 2 ≤ ( )-+1 sup ∈R ( ) -1 2 1 ∫ R 2 ( , )| ( ˜ )| . (3.35)
By the non-dispersion assumption (3.15), we can choose > 1 such that

√ 2 ( ) -1 2 2 ≤ ( )-+1 sup ∈R ( ) -1 2 1 ≤ + 1 4
.

Taking into account (3.35), this leads eventually to the following estimate

2 + 1 ∫ R +1 ( , ) ( ˜ ) ≤ - 2 ∫ R 2 ( , ) ( ˜ ) + 0 --0 -+ ( )-( 0 ) , (3.36)
where 0 := 2 +1 is independent of 0 , 0 , and . Gathering (3.30), (4.18), and (4.22) in (4.53) we deduce finally

( 0 , 0 ) ( ) ≥ -3 ∫ R 2 ( , ) ( ˜ ) -0 --0 -+ ( )-( 0 ) .
Thus Lemma 3.2 is established.

As a consequence of the preceding lemma,

∃ 1 > 0, ∀ 0 ∈ R, ∀ ≥ 0 , ( 0 , 0 ) ( 0 ) ≤ ( 0 , 0 ) ( ) + 1 0 , (3.37)
with 1 independent of the parameters 0 and 0 . Next, we claim the following: Lemma 3.3. For fixed 0 ∈ R and 0 ∈ , ( 0 , 0 ) ( ) → 0 as → +∞.

Proof. To show this lemma, we just repeat the arguments given by Laurent and Martel [57, paragraph 2.1, Step 2]. Let be a positive real number. By Claim 3.14 and by (3.18), there exists ˜ > 0 such that

∫ < ˜ ( )-˜ 2 ( , ) ≤ 2 .
Since 0 ≤ ≤ 1, this enables us to see that

∫ <-˜ 2 ( , + ˜ ( )) -0 + ( ) -( 0 ) ≤ ∫ < ˜ ( )-˜ 2 ( , ) ≤ 2 . (3.38)
Now, recall that is decreasing so that

∫ ≥-˜ 2 ( , + ˜ ( )) -0 + ( ) -( 0 ) ≤ -˜ -0 + ( ) -( 0 ) ( ) 2 2 ≤ -˜ -0 + ( ) -( 0 ) , (3.39) with = ( ) 2 2
for all ∈ . Moreover, since ( ) → +∞ as → +∞ and ( ) → 0 as → +∞, there exists ∈ R such that for all ≥ ,

-˜ -0 + ( ) -( 0 ) ≤ 2 .
Then, for all ≥ ,

( 0 , 0 ) ( ) ≤ 2 + 2 = .
Hence, we have finished proving Lemma 3.3.

Due to (4.23) and Lemma 3.3, we obtain

∀ 0 ∈ , ∀ 0 ∈ R, ( 0 , 0 ) ( 0 ) ≤ 1 0 . (3.40)
Thus, (4.14) leads to: for all ∈ ,

∫ 0 ≤ <0 2 ( , + ˜ ( )) - ≤ 1 1
.

Thus letting 0 tend to -∞, we deduce from (4.13) that

∫ < ˜ ( ) 2 ( , ) ( ˜ ( )-) ≤ 1 1
.

Step 3: Proof of (3.22) for ≥ 1

Starting from the fact that for all ≥ 0 and for all 0 < 0

( 0 , 0 ) ( 0 ) -( 0 , 0 ) ( ) ≤ 1 1 0 + 3 ∫ 0 ∫ R 2 ( , + ˜ ( )) ( -0 + ( ) -( 0 ))
and arguing like Laurent and Martel [57, paragraph 2.1

Step 3], one can show

∫ 0 +1 0 ∫ R 2 ( , + ˜ ( )) ≤ ,
where is independent of 0 and 0 . Now, one proves by induction on ∈ N the existence of ≥ 0 such that for all ∈ , ( ) ≤ and

∫ R 2 ( , + ˜ ( )) + ∫ +1 ∫ R 2 ( , + ˜ ( )) ≤ . (3.41)
In particular, estimates (3.22) are then performed. Moreover, we deduce from the equation satisfied by that the partial derivatives with respect to and of all order exist and are continuous, thus

∈ C ∞ ( × R).
For simplification purposes, we will not explicit the proof of (3.41) and refer instead to [57, paragraph 2.3 and paragraph 2. [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF] Step 2]; the induction argument works since we assume ( ) ≤ in (3.17), which implies that ˜ is bounded on .

Proof of Theorem 3.4 and Corollary 3.1

We split the proof into four steps. The first three steps are common to both theorems and are valid under the hypotheses of Theorem 3.4, whereas the last one is specific to the proof of Corollary 3.1 where exponential decay properties are established and for which the stronger non dispersion assumption (3.9) is required. Consider which satisfies the assumptions of Theorem 3.4.

Step 1: Asymptotic stability in the energy space

The following asymptotic stability result in the energy space is to be considered as a crucial tool for the proof. In particular, we point out the importance of assumption (3.8) in Theorem 3.4. Theorem 3.15 (Martel,Merle and Tsai [80]; Martel and Merle [START_REF] Martel | Asymptotic stability of solitons of the gKdV equations with general nonlinearity[END_REF]). Fix 0 < 0 1 < • • • < 0 . For all > 0, there exist 0 > 0 and 0 = 0 ( )

> 0 such that if ∈ C ( [0, +∞), 1 (R)) is a solution of (gKdV) satisfying ∀ ≥ 0, inf ∈ R +1 -> 0 ( ) - =1 0 (• -) 1 < 0 + - (3.42)
for some positive constants and , then the following holds.

(Asymptotic stability in the energy space) There exist

C 1 functions ↦ → ( ) ∈ R * + , ↦ → ( ) ∈ R for ∈ {1, . . . , } such that lim →+∞ ( ) - =1 ( ) (• -( )) 1 > = 0.
(3.43)

(Convergence of the scaling parameter) There exists

+ ∈ R * + such that lim →+∞ ( ) = + .
Set := 1 2 min 1 , min ∈ {1,..., -1} { +1 -} . Using Theorem 3.15 and adapting the classical modulation argument set up in the proof by Martel and Merle [72, section 5], we have the following. For all ∈ {1, . . . , }, there exist : [0, +∞) →

R * + , : [0, +∞) → R of class C 1 such that, defining : ( , ) ↦ → ( , ) - =1 ( ) ( -( )), (3.44)
and for small enough in Theorems 3.4 and 3.1, we have 1. the conclusion of Theorem 3.15, that is

lim →+∞ ( ) 1 > 1 = 0, (3.45)
with > 3 such that

1 < -, (3.46)
(take = 0 for Theorem 3.4) and

∀ ∈ {1, . . . , }, ∃ + ∈ R * + , lim →+∞ ( ) = + ; (3.47)
2. control on the modulation parameters [80, proof of Lemma 1]: more precisely, there exists > 0 such that for all large enough,

∀ ∈ {1, . . . , -1}, +1 ( ) -( ) ≥ , (3.48)
and for all ∈ {1, . . . , },

| ( ) -| + ( ) 1 ≤ + 1 , (3.49)
| ( ) -( )| ≤ ∫ R 2 ( , ) -√ 1 | -( ) | 1 2 . (3.50)
Remark 3.16. The preceding choices of , , and of the functions ↦ → ( ) as defined before are possible, provided is sufficiently small. Note also that estimate (3.49) and assertion (3.47) guarantee that 0

< + 1 < • • • < + (due to the choice of ).
Step 2: Convergence of ( ) -=1 + (• -( )) as → +∞ Lemma 3.4. We have

( ) - =1 + (• -( )) 1 → 0, as → +∞.
This lemma follows immediately from Claim 3.17 and Claim 3.18 below. We begin with this first observation. Claim 3.17. We have

lim →+∞ ( ) - =1 + (• -( )) 1 > 1 = 0. (3.51)
Let us justify this fact. Using the triangular inequality and taking into account (3.45), it suffices in fact to see that for all ∈ {1, . . . , }, lim

→+∞ ( ) (• -( )) -+ (• -( )) 1 > 1 = 0. (3.52)
But the quantity

( ) (• -( )) -+ (• -( ))
1 > 1 is bounded by ( ) -+ 1 which tends to 0 as tends to +∞. We recall indeed that the map

R * + → 1 (R), ↦ →
is continuous by application of Lebesgue's dominated convergence theorem. Hence (3.52) holds and Claim 3.17 is proved.

Due to the assumption of non-dispersion made in Theorem 3.4, we claim moreover: Claim 3.18. We have

lim →+∞ ( ) - =1 + (• -( )) 1 ≤ 1 = 0. (3.53)
In what follows, we prove actually that each quantity ( ) 1 ≤ 1 and

+ (• -( )) 1 ≤ 1 for ∈ {1, .
. . , } tends to 0 as tends to +∞.

1. Proof of ( ) 1 ≤ 1 -→ →+∞ 0. Let > 0. By (3.6) or (3.9), there exists > 0 such that for all ≥ ,

∀ ≥ 0, ∫ < ( -) - 2 ( , ) ≤ 2
. Now, by means of Proposition 3.11, there exist > 0 and 1 > 0 such that for all ≥ 0,

∀ ≤ ( -) , | ( , )| ≤ 1 -| -( -) | . Pick ≥ such that 2 1 - ∫ R -| | ≤ 2 .
For large enough, 1 < ( -) -due to (3.46), and therefore

∫ ≤ 1 2 ( , ) ≤ 2 , and ∫
≤ 1 2 ( , ) ≤ ∫ ≤ ( -) - 2 ( , ) ≤ 1 ∫ ≤ ( -) - -| ( , )| ≤ 2 1 - ∫ ≤ ( -) - -| -( -) | ≤ 2 1 - ∫ R -| -( -) | ≤ 2 .
As a consequence, for large enough, ( )

2 1 ≤ 1 ≤ . 2. Proof of + (• -( )) 1 ≤ 1 -→ →+∞ 0.
Notice first that, recalling (3.49) and (3.50), we have for large enough

| ( ) -| ≤ | ( ) -( )| + | ( ) -| ≤ ( ) 2 + | ( ) -| ≤ ( + 1) ( ) 2 + | ( ) -| ≤ 1 2 .
In particular, for large enough, (

By integration of the preceding inequality, we deduce that for large values of , ( ) ≥ 1 3 . Thus, for these values,

( ) - 1 ≥ 1 1 3 - 1 , (3.55)
with 1 3 -1 > 0. Due to the exponential decay property of the integrable functions + and + , we deduce then from (3.55) that

+ (• -( )) 1 ≤ 1 -→ →+∞ 0.
Now, it follows from Claim 3.17 and Claim 3.18 that lim →+∞ ( ) -

=1 + (• -( )) 1 = 0. (3.56)
Step 3: Refinement of (3.56) Proposition 3.19 (Improvement of the 1 -convergence for asymptotic -soliton like solutions).

Let ∈ C (R, 1 (R)) be a solution of (gKdV) and let 0

< 1 < • • • < . Assume the existence of 0 ≥ 0, 0 > 0, and N functions 1 , . . . , : R → R of class C 1 satisfying for all ≥ 0 , ∀ = 1, . . . , -1, +1 ( ) -( ) ≥ 0 and ∀ = 1, . . . , , ( ) ≥ 0 , (3.57)
and such that

lim →+∞ ( ) - =1 (• -( )) 1 = 0. (3.58)
Then there exist > 0 and 1 , . . . , ∈ R such that

∀ ≥ 0, ( ) - =1 (• - -) 1 ≤ -1 8 3 2 0 . (3.59)
It was first observed by Martel [63,Proposition 4] that multi-solitons in the sense of Definition 4.1 do actually converge exponentially fast to their profile: this was a key to proving uniqueness of multi-solitons, in the 2 -subcritical case. In the above Proposition 3.19 we further refine this observation, by noticing that the conclusion still holds even if one gives some freedom to the center of mass of the soliton ( ) (instead of (3.57), the assumption in [63, Proposition 4] was ( ) = + ). The proof of Proposition 3.19 follows the lines of [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] and is postponed to the Appendix; we go on assuming it holds. Given (3.48), (3.54), and (3.56), we just have to apply the previous proposition (with replaced by and 0 by defined in Step 1) to conclude that is a multi-soliton. In other words, there exist

+ 1 , . . . , + ∈ R such that ∀ ≥ 0, ( ) - =1 + (• -+ -+ ) 1 ≤ - 1 8 3 2 . (3.60)
We recall then from Martel [63, proof of Proposition 5] that for all ∈ N * , ∈ C ( [0, +∞), (R)) and there exists ˜ ≥ 0 such that

∀ ≥ 0, ( ) - =1 + (• -+ -+ ) ≤ ˜ -1 32 3 2 . (3.61)
This concludes the proof of Theorem 3.4.

Step 4: Proof of smoothness and exponential decay of Apply Proposition 3.12 with ( ) := ( -) and ( ) := ( + ) to obtain ∈ C ∞ ( [0, +∞) × R) and for each ≥ 0,

∀ ≤ ( -) , ( , ) ≤ -| -( -) | ,
where , > 0 are independent of and .

Under the global non-dispersion assumption of Corollary 3.1, which we take as granted from now on, we have also

∀ ≥ ( + ) , ( , ) ≤ -( -( + ) ) .
(See Remark 3.13.) At this stage and as we explain just below, the desired exponential decay estimate (3.10) follows from the strong property (3.5) or (3.61). We distinguish three cases, depending on the position of with respect to ±2( + ) ; the moral being that (3.5) implies the expected pointwise estimate in each region | | ≤ (with an exponential decay rate depending on ) and even if it means taking large enough and reducing the decay rate , one can propagate the control by -| -+ | (respectively -| -+ 1 | ) to the region > ( + ) (respectively < ( -) ).

Let ≥ 0 and ˜ := 1 32 3 2 , for each ∈ N, there exists ˜ > 0 such that

( ) - =1 + (• -+ -+ ) ∞ ≤ ˜ -˜ .
Case 1: | | ≤ 2( + ) . We have

| -+ | ≤ (2( + ) + + ) that is, ˜ 2( + ) + + | -+ | ≤ ˜ , and thus
( ) - =1 + (• -+ -+ ) ∞ ≤ ˜ - ˜ 2( + )+ + | -+ | . (3.62)
Consequently for ≥ 0 and | | ≤ 2( + ) , using the triangular inequality and the exponential decay of + , we obtain

| ( , )| ≤ =1 + ( -+ -+ ) + ( ) - =1 + (• -+ -+ ) ∞ ≤ ˜ =1 -˜ | -+ | , (3.63)
where ˜ := min

+ 1 , ˜ 2( + )+ + .
Case 2: ≥ 2( + ) . Let us rewrite this as -( + ) ≥ 1 2 . In particular

-( + ) ≥ 1 2 -+
so that for ≥ 2( + ) , ( , ) ≤ -( -( + ) ) ≤ -2 ( -+ ) .

(3.64)

Case 3: ≤ -2( + ) . Arguing similarly as before, we have then

( , ) ≤ -( ( -) -) ≤ -( + 1 -) , (3.65)
for > 2 chosen such that

( -)-+ 1 -1
> -2( + ). Set finally := min , ˜ to obtain (3.10) in Corollary 3.1.

3.4

The integrable cases: proofs of Theorems 3.7 and 3.9 Theorem 3.20 (Eckhaus and Schuur [28]; Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF]). Let = 2 and 0 ∈ C 4 (R) be such that for some 0 > 0, for all = 0, . . . , 4, and for all ∈ R,

Let be the corresponding global solution of (KdV).

Then there exists a solution which is a multi-soliton or zero such that for all > 0, there exists ≥ 0 such that for all > 0

( ) -( ) ∞ ( > ) + ( ) -( ) 2 ( > ) ≤ -1 3 . (3.67)
Proof of Theorem 3.7. Due to the assumption on 0 in Theorem 3.7, we can apply Theorem 3.20 and we obtain a solution of (KdV) as above which fulfills (3.67). We claim first that is not the trivial solution. Otherwise, with := 2 > 0, we would have ( )

2 ( > ) = O( - 1
3 ) as tends to +∞. On the other hand, by the non dispersion assumption and namely by Proposition 3.11,

∀ ≤ , | ( , )| ≤ -| -| , (3.68)
so that

( ) 2 ( ≤ ) ≤ - . (3.69)
Then, we would obtain that

( ) 2 = ( ) 2 ( ≤ ) + ( ) 2 ( > ) → 0 as → +∞,
hence conclude that 0 2 = 0 by the mass conservation law. This contradicts our assumption in Theorem 3.7.

Thus there exist ≥ 1, 0

< 1 < • • • < , + 1
, . . . , + ∈ R, and a possibly smaller > 0 such that

( ) - =1 , + ( ) 1 = O -, as → +∞. (3.70)
Claim 3.21. We have

( ) - =1 , + ( ) 2 = O -1 3 , as → +∞.
Proof of Claim 3.21. Consider ∈ (0, min{ 1 , }) so that (by the non dispersion assumption and the sech-shaped profiles of the solitons , + )

( ) 2 ( ≤ ) + =1 , + ( ) 2 ( ≤ )
= O - even if it means reducing > 0. We perform then

( ) -( ) 2 = ( ) -( ) 2 ( ≤ ) + ( ) -( ) 2 ( > ) ≤ ( ) 2 ( ≤ ) + ( ) 2 ( ≤ ) + O -1 3 ≤ ( ) 2 ( ≤ ) + O -1 3 + - ≤ ( ) - =1 , + ( ) 1 + =1 , + ( ) 2 ( ≤ ) + O -1 3 = O -1 3 ,
by the embeddings 1 (R) ↩→ 1 ( ≤ ) ↩→ 2 ( ≤ ) and by (3.70). By means of the triangular inequality and once again (3.70), we deduce the expected estimate in Claim 3.21.

We are now able to finish the proof of Theorem 3.7. Indeed, let us make the following key observation.

Claim 3.22. The solution belongs to ∞ ( [0, +∞), 2 (R)).

Proof of Claim 3.22. This is an immediate consequence of the following conservation law for the KdV equation

∫ R 2 2 - 10 3
( ) 2 + 5 9 4 ( , ) = 0, (3.71)
of the Sobolev embedding 1 (R) ↩→ ∞ (R), and from the fact that belongs to ∞ (R, 1 (R)).

As a consequence of Claim 3.22, := -=1 , + belongs also to ∞ ( [ 1 , +∞), 2 (R)). Then, integrating by parts and using the Cauchy-Schwarz inequality and Claim 3.21, we infer that

∫ R ( ) 2 ( ) = - ∫ R ( ) 2 ( ) ≤ ( ) 2 ( ) 2 ≤ -1 3 ,
from which it results that

( ) - =1 , + ( ) 1 → 0, as → +∞. (3.72)
Hence is a multi-soliton in +∞. By means of the well-known theory concerning multi-solitons of the KdV equation (see for instance Miura [START_REF] Miura | The Korteweg-de Vries equation: A survey of results[END_REF]), we deduce that is also a multi-soliton in -∞. This ends the proof of Theorem 3.7.

Non dispersive solutions of the modified Korteweg-de Vries equation

Theorem 3.9 is obtained by using the same strategy as that developed in the previous subsection. Thus we will only sketch its proof.

As for the KdV case, we apply first the following decomposition result, obtained from [101, Chapter 5, Theorem 5.1] and from [START_REF] Chen | Soliton resolution for the modified KdV equation[END_REF]Theorem 1.10] where a more precise version can be found. Theorem 3.23 (Schuur [START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF], Chen and Liu [START_REF] Chen | Soliton resolution for the modified KdV equation[END_REF]). Let = 3 and 0 ∈ C 4 (R) be such that for some 0 > 0, for all = 0, . . . , 4, and for all ∈ R,

0 ( ) ≤ 0 | | -11 , (3.73)
and be generic as in Theorem 3.9, with scattering data (3.12). Let be the corresponding global solution of (mKdV). Then there exist signs = ±1, = 1, . . . , 1 , and parameters 0, , = 1, . . . , 1 , and 1, , 2, , = 1, . . . , 2 , such that for all + > 0 and -< 0, there exists ≥ 0 such that for all > 0, denoting

( ) := 1 =1 2 , 0, ( ) + 2 =1 √ 2 , √ 2 , 1, , 2, ( ), we have ( ) -( ) ∞ ( > + ) + ( ) -( ) 2 ( > + ) ≤ -1 3 , (3.74) and ( ) -( ) ∞ ( < -) ≤ -1 2 , (3.75)
Then, the non dispersion assumption (3.6) in Theorem 3.9 shows that 1 + 2 ≥ 1. Since the profiles of the breathers are sech-shaped, due to (3.6) and (3.75), we deduce that the breathers have positive (envelope) velocities. Now, proceeding as in the proof of Claim 3.21, we obtain in fact that

( ) -( ) 2 = O -1 3 .
Moreover, (mKdV) admits conservation laws of orders 2, 3, and 4 in the spirit of (3.71), which shows that belongs to ∞ ( [0, +∞), 4 (R)). Proceeding similarly to subsection 3.4.1, we obtain that ( ) -( ) 2 → 0, as → +∞.

Finally, by the uniqueness and smoothness results and the estimates in higher Sobolev spaces proven by Semenov [START_REF] Semenov | On the existence and uniqueness of multi-breathers of (mKdV)[END_REF] as far as multi-breathers are concerned, we deduce that belongs to C ([0, +∞), (R)) and that there exist > 0 and positive constants such that for all ∈ N,

( ) -( ) ≤ -, as → +∞.
This finishes proving Theorem 3.9.

Appendix: Proof of Proposition 3.19

The proof follows the same lines as that of Proposition 3 and paragraph 3.2 in Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] for the 2 -subcritical and critical cases, and that of Lemma 4.1 in Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF] for the supercritical case. For the sake of simplicity and for the reader's convenience, we present here the essential ideas and also the changes in the 2 -subcritical case only. Remark 3.24. We mention that in the 2 -critical and supercritical cases, the proof is basically changed in terms of the coercivity property we use to control the modulation function defined below in Step 1. The monotonicity properties of local mass and energy obtained in Step 2 are still valid in these cases. Concerning the critical case, the idea is to modulate the scaling parameter in addition to the translation parameter so as to ensure a second orthogonality condition satisfied by , namely ∫ R ( ) ˜ ( ) 3 = 0, and then to apply a localized version of the coercivity property available in this case, which leads to:

∃ 0 > 0, ∀ , ( ) 2 1 ≤ 0 H ( ), (3.76)
with H defined in Step 3.

In the supercritical case, it is known from Pego and Weinstein [START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF] that, considering the standard linearized operator on 1 (R) defined by := -2 + --1 , the composed operator has two eigenfunctions + and -related by -( ) = + (-), which decay exponentially, and such that ± = ± 0 ± for some 0 > 0. In this case, we only have to make modifications in Step 3 by using this time

∃ 0 > 0, ∀ , ( ) 2 1 ≤ 0 H ( ) + 1 0 ,± ∫ R ( ) ˜ ± ( ) 2 , (3.77)
where ˜ ± ( ) := ± (• -( ) -( )) and ± ( ) :=

-1 2 ± 1 2
.

(The functions are defined in Lemma 3.5 below.) The control of ∫ R ( ) ˜ ± ( ) by a function of which decreases with exponential speed follows the strategy of Combet (for full details, see [11, paragraph 4.1 Step 4]).

Step 1: Set up of a modulation argument Set := min{ 1 , 0 }. We claim the following Lemma 3.5. There exist ≥ 0 and 1 ∈ (0, 1] such that for all ˜ ≤ 1 , the following holds. There exist unique C 1 functions : [ , +∞) → R such that defining In addition, there exists > 0 such that for all ≥ , for all ∈ {1, . . . , },

:= - =1 ˜ , (3.78
( ) 1 + =1 | ( )| ≤ ˜ , (3.80)
| ( ) + ( ) -| ≤ ∫ R 2 ( ) - √ | -( ) | 1 2 + -1 4 3 2 . (3.81)
Proof. Recall that the proof of existence and uniqueness of the functions ( ) is based on the implicit function theorem. We refer to [80, proof of Lemma 8] and also to [67, paragraph 2.3] for a complete proof in the case of one soliton. Moreover, estimate (3.81) which involves ≤ 1 is obtained formally by writing the equation of , that is

+ 3 = =1 ( + -) ( ˜ ) - + =1 ˜ - =1 ˜ , (3.82)
by multiplying it by ( ˜ ) , and by using the following properties:

0 = ∫ R ( ˜ ) = ∫ R ˜ -( + ) ∫ R 2 ˜ ; (3.83) ∀ ≠ , ∀ ≥ 2 , | ˜ ( , )| + | ˜ ( , )| ≤ - √ | -( ) | ; (3.84) ∫ R ˜ ( , ) ˜ ( , ) + | ˜ ( , ) ˜ ( , )| ≤ - 3 2
2 .

(3.85)

Note that (3.85) is a consequence of the decoupling assumption (3.57). We refer to [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] and the references therein for more details.

Step 2: Monotonicity properties for localized mass and some modified energy of

Let : ↦ → 2 Arctan - √ 2
be defined on R so that for all ∈ R,

( ) ≤ 0, | ( )| ≤ √ - √ 2 | | , | (3) ( )| ≤ 4 | ( )|,
(we recall = min{ 1 , 0 }). Then define on R + × R:

∀ ∈ {1, . . . , -1}, : ( , ) ↦ → - ( ) + +1 ( ) 2
, : (
, ) ↦ → 1 (3.86)
and also

1 := 1 , ∀ ∈ {2, . . . , -1}, := --1 , := 1 - -1 . (3.87)
Remark 3.25. Note that by definition and by (3.57), for > 0, ( ) takes values close to 1 in a neighborhood of ( ) and takes values close to 0 around ( ) for ≠ .

Take ∈ 0, 1 4 and consider now for all ∈ {1, . . . , -1} the following quantities:

• (localized mass of at the left of ( )+ +1 ( )

2 ) M ( ) := ∫ R 2 ( , ) ( , ) (3.88)
• (modified localized energy of at the left of ( )+ +1 ( )

2 ) Ẽ ( ) := ∫ R 1 2 2 - 1 + 1 +1 + 2 ( , ) . (3.89)
Let also M ( )

:= ∫ R 2 ( , ) ( , ) (3.90)
(which is nothing but the mass of ) and

Ẽ ( ) := ∫ R 1 2 2 - 1 + 1 +1 + 2 ( , ) (3.91)
(which is a global quantity linked to the energy of ).

Remark 3.26. The reason why we have to choose small enough appears clearly in Step 3 (see Remark 3.27).

We claim now a monotonicity result on the preceding quantities.

Lemma 3.6.

There exist 1 ≥ 0 and 1 ≥ 0 such that for all ≥ 1 and for all ∈ {1, . . . , }, M ( ) ≥ -1 Proof. First we observe that

M = - ∫ R 3 2 + + +1 2 2 - 2 + 1 +1 + ∫ R 2 (3)
.

For all 0 > 0, there exists 0 ≥ 0 such that for all ≥ 0 , ( ) -

=1 (• -( )) 1 ≤ 0
and for all 0 > 0, for each ( , ) ∈ R + × R such that ( ) + 0 ≤ ≤ +1 ( ) -0 , we have

| ( , )| ≤ =1 ( -( )) + ( ) - =1 (• -( )) 1 ≤ =1 - √ 0 + 0 .
Thus, for 0 sufficiently large and for 0 > 0 small enough being fixed, we have for some 1 > 0 : for all ≥ 1 , for all ∈ [ ( ) + 0 , +1 ( ) -0 ],

2 + 1 | ( , )| -1 ≤ 4 . If > +1 ( ) -0 or < ( ) + 0 , then - ( ) + +1 ( ) 2
> +1 ( ) -( ) 2 -0 > 2 -0 .
Consequently, for ≥ 1 and ∉ [ ( ) + 0 , +1 ( ) -0 ], we obtain

( , ) ≤ √ - √ 2 +1 ( )-( ) 2 -0 ≤ - 3 2 4 .
We deduce that

M ( ) ≥ - ∫ R 3 2 + 0 - 4 2 - ∫ R 4 2 | | -- 3 2 4 ≥ - ∫ R 3 2 + 2 2 -- 3 2 4
≥ -- (3.93)

Similarly, we compute

Ẽ = - ∫ R 2 + 2 + 2 2 + + +1 2 2 - + +1 + 1 +1 ( ) + ∫ R 2 (3) + 2 ∫ R 2 -1 ( ) + M ≥ - ∫ R 2 ( ) + 4 ∫ R 2 ( ) + 2 ∫ R 2 -1 ( ) - ∫ R 3 2 ( ) -- 3 2 4 + + +1 2 ∫ R 2 + 1 +1 -2 ( ) . (3.94)
As before, we can increase 1 and reduce 0 to have

2 ∫ R 2 -1 ( ) ≤ 4 ∫ R 2 |( ) | + - 3 2 4 and 2 + 1 ∫ R +1 ( ) ≤ 2 ∫ R 2 |( ) | .
Eventually, this leads to

Ẽ ≥ - 3 4 ∫ R 2 ( ) -- 3 2 4 + ( + +1 ) 4 ∫ R 2 |( ) | ≥ 3 4 ∫ R 2 |( ) | + 0 2 ∫ R 2 |( ) | -- 3 2 4 ≥ -- 3 2 4
.

Step 3: A Weinstein type functional

Let the functional H be given by

H := =1 1 2 ∫ R 2 + 2 -˜ -1 2 , (3.95)
define

F := =1 1 2 ∫ R 1 2 2 - 1 + 1 +1 + 2 ∫ R 2 , (3.96)
and set ( ) := ( ) -=1 ( ), where for all = 1, . . . , ,

( ) := (• -( )).
We gather next some properties satisfied by H and F which are essential to obtain ( ) 1 = O ( -) as → +∞, for some > 0.

Lemma 3.7. We have 1. (coercivity property satisfied by H )

∃ 0 > 0, ∀ ≥ , ( ) 2 1 ≤ 0 H ( ) + 1 0 =1 ∫ R ( ) ˜ ( ) 2 ;
(3.97)

(expansion of H )

H = 2 F - =1 1 2 ∫ R 1 2 ( ) 2 - 1 + 1 +1 + 2 ∫ R 2 + , (3.98
+ | ( )| + 2 -1 =1 1 2 - 1 2 +1 ( ) ∞ ∫ R 2 ( ) - ∫ R ( ) ( ) ( ) + 1 2 ∫ R ( ) 2 ( ) - -1 2 ( ) ( ) + 2 -1 =1 1 - 1 +1 1 2 - 1 + 1 +1 2 ∫ R ( ) + ∫ R 2 ( ) + 2 2 ( ) ∞ ∫ R 2 ( ) + 1 2 ∫ R ( ) 2 - -1 2 ( ) ( ) + 2 1 2 - 2 ∫ R ( ) + ∫ R 2 ( ) - 2 ∫ R ( ), (3.105)
that is to: Martel [63, Step 3]) for a proof), there exists 0 > 0 such that for all ≥ ,

∀ ≥ ≥ , H ( ) ≤ =1 1 2 ( ) 2 + 2 ( ) - -1 ( ) 2 ( ) ( ) + ( ) ∞ ( ) 2 2 + ˜ ( )
∫ R ( ) ˜ ( ) 2 ≤ - 3 2 4 ( ) 2 + ˜ ( ) 2 2 (3.108) (see
( ) 2 1 ≤ - 3 2 4 + ˜ ( ) 2 1 + =1 ∫ R ( ) ˜ ( ) 2 ≤ 0 - 3 2 4 + 0 ˜ ( ) 2 1 .
At the stage of (3.111), it suffices to see a posteriori that 0 ≤ 1 in order to obtain exactly (3.59). Let us justify it briefly. Combining this result with (3.112) and due to the fact that the speeds are distinct two by two, we obtain successively for describing the integers from -1 to 1:

=1 √ - √ +1 ( ) 2 → 0, as → +∞.
Hence 1 ( ) 2 -→ →+∞ 0 and even 1 ( ) 1 -→ →+∞ 0 judging by the expression of 1 . This implies 1 ( ) -1 -1 -→ →+∞ 0. Now it is clear that condition 1 ( ) ≥ 0 forces to have 1 ≥ 0 .

If > 5, there exists a one-to-one map Φ from R to the set of all 1 -solutions of (gKdV) defined in a neighborhood of +∞ such that is a multi-soliton in +∞ associated with the , if and only if there exist ∈ R and 0 ≥ 0 such that | [ 0 ,+∞) = Φ( ) | [ 0 ,+∞) . Moreover, in each case, belongs to C ( [ 0 , +∞), (R)) for all ≥ 0, and there exist > 0 (depending on the ) and positive constants such that for all ≥ 0, for all ≥ 0 ,

( ) - =1 , ( ) ≤ -. (4.3)
In [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF], we linked the multi-solitons of (gKdV) with the concept of non dispersive solutions which had already appeared in some form in early papers by Martel and Merle [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF] in the context of the study of asymptotic stability of the solitons. For the Korteweg-de Vries equation in particular, we showed that non dispersion is a necessary and sufficient condition for an 1 -solution with sufficiently regular initial data to be a multi-soliton. More generally, we saw that an 1 -solution of (gKdV) which is non dispersive and remains close to a sum of decoupled solitary waves for positive times is a multi-soliton in +∞.

Non dispersion is actually significant insofar as it self improves to smoothness and exponential decay in space. That being said, it appears that non dispersion only allows for exponential decay of the solution on a half line < for some > 0; to conclude to exponential decay everywhere, as for a sum of solitons, we had to assume in addition that the solution is global and that non dispersion holds also for negative times, which is a priori not verified by multi-solitons.

In this article, we go on studying qualitative properties of the multi-solitons by particularly concentrating on the behavior in space (at fixed time) of the multi-solitons of (gKdV). Let us fix the parameters 0 < 1 < • • • < and 1 , . . . , ∈ R. We consider ∈ R and ∈ C ([ , +∞), 1 (R)) a multi-soliton of (gKdV) associated with the solitons , , = 1, . . . , ; we recall that is unique if ≤ 5.

Let us state our main result.

Theorem 4.2. Let 0 < < 1 and let > . Then there exist > 0, > 0 and , > 0 such that for all ∈ N, there exists > 0 such that for all ≥ , if ≤ , (exponential decay at the left of the first soliton)

( , ) ≤ -| -| ; (4.4) if ≤ ≤ , (exponential decay in the soliton region) ( , ) ≤ =1 -, | -| . (4.5)
For all ∈ N and for all ∈ N, there exists , > 0 such that for all ≥ , for all > , (algebraic decay at the right of the last soliton)

( , ) ≤ , ( -) . (4.6)
Moreover, if is global and 1 -uniformly bounded (for instance if 1 < < 5), then there exists > 0 such that for all ∈ N, there exists > 0 such that for all ≥ , for all ≥ , (exponential decay at the right of the last soliton)

( , ) ≤ -| -| . (4.7)
Remark 4.2. The constant can be chosen in 0, √ 2 . The constant , depends on , , and the velocity parameters , = 1, . . . , ; with as in (4.3), one can take

, = min √ 1 , 1 - , - , min =1,..., -1 +1 - .
Theorem 4.2 shows in particular that the multi-solitons of (gKdV) belong to the Schwartz space S (R). To our knowledge, no similar result has yet emerged with regard to the multi-solitons of a nonintegrable partial differential equation. Since a sum of solitons and its derivatives decrease exponentially in space, obtaining exponential decay, at least at the left of the soliton region, is actually not surprising. As a corollary, we point out that multi-solitons are in particular non dispersive in the sense of [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF]. If the multi-soliton is globally defined (this is guaranteed if 1 < < 5), then exponential decay of all derivatives holds everywhere. Estimates (4.6) suggest that exponential decay holds also on the right in general, which would be in full conformity with the integrable setting, but this conclusion seems currently out of reach.

Comments

Exponential decay of the multi-solitons of (gKdV) at the left (on the region corresponding to ≤

) follows from revisiting a strong monotonicity argument set up in [30, section 2] and originally developed by Laurent and Martel [START_REF] Laurent | Smoothness and exponential decay of 2 -compact solutions of the generalized KdV equations[END_REF]. We underline that, rather than arising from non dispersion (which turns out to be a consequence of (4.4)), the produced monotonicity property holds for large values of due to (4.2) and the decay of the solitons in the present context.

On an interval like [ , ], estimate (4.5) is established by means of the exponential decrease in time of the quantities ( ) -=1 , ( ) , ∈ N, as stated in Theorem 4.1. On the half line > , it appears to be more difficult to show that multi-solitons decrease with exponential speed in space independently of . The monotonicity argument, linked to the dynamic of the flow of (gKdV), does not apply anymore if the multi-soliton is not assumed to be global. It is interesting to emphasize that multi-solitons (along with their derivatives) are proved to decrease faster than each polynomial function by exploiting the strong convergence result (4.3). This is achieved by a triangular process by considering the family of integrals

, ( ) := ∫ > 2 ( , ) ( -)
for , ∈ N. By induction on , we succeed in establishing that, if for all ∈ N, , < +∞, then for all ∈ N, , +1 < +∞. More precisely, assuming that 0, , . . . , , , +1, are finite quantities, we prove that , +1 is also finite. From a technical point of view, this consists in one main new observation provided by this paper.

Actually this triangular way of obtaining algebraic decay for and its derivatives comes from one term which carries a spatial derivative of of order + 1 and which appears in the derivative with respect to of a functional of the form

∫ R 2 ( , ) ,
where ( ) denotes a certain weight function. This phenomenon is obviously related to the structure of (gKdV).

DECAY OF THE MULTI-SOLITONS ON THE LEFT 123

The algebraic decay then implies non dispersion of the multi-soliton on the right, which allows us to conclude to (4.7) by monotonicity if is defined on R.

Yet, we can expect the multi-solitons to decrease exponentially with respect to the space variable on the full line even though multi-solitons are not global. This is still a very natural conjecture all the more so as multi-solitons are in some ways to be considered as special solutions in view of Theorem 4.1. Starting from (4.6), one possible first direction of research in order to prove this conjecture would be to "track" the constants that appear in (4.3). From another perspective, one could also rely on Kato smoothing effect [START_REF] Kato | On the Cauchy problem for the (generalized) Korteweg-de Vries equation[END_REF] by using the gain of derivatives in space which arises from this concept.

Furthermore, by a monotonicity argument in the spirit of [30, section 2], we would show that it suffices to obtain exponential decay on the region ≥ 0 for some time 0 in order to obtain exponential decay on the region ≥ for all time ≥ 0 . This observation is in line with general statements linked with persistence of regularity and decay of solutions to (gKdV) previously developed by Kato [START_REF] Kato | On the Cauchy problem for the (generalized) Korteweg-de Vries equation[END_REF] and Isaza, Linares and Ponce [START_REF] Isaza | On decay properties of solutions of the k-generalized KdV equation[END_REF][START_REF] Isaza | On the propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation[END_REF]. But similarly the question of obtaining exponential decrease everywhere in space for one time 0 remains unresolved, even for the particular multi-soliton solutions; actually, it is unclear whether one can improve (4.6) to an analog of (4.4). Additionally, the decay on the left-hand side of the real line does not in general propagate forward in time; conversely, one can thus wonder if decay on the right-hand side does really propagate backward in time.

For the time being, estimate (4.6) is thus meaningful in the 2 -critical and supercritical cases. Besides, our argument could be adapted to other models such as the nonlinear Schrödinger equations (see the main theorems in [START_REF] Côte | On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations[END_REF]) in order to prove algebraic decay (in the sense of (4.6)) for the corresponding multi-solitons, especially when smoothness and asymptotic estimates hold and monotonicity properties are missing.

The next sections are devoted to the proof of Theorem 4.2.

Decay of the multi-solitons on the left 4.2.1 Decay of the multi-solitons on the left of the first soliton

Let 0 < < 1 and ∈ 0, Proof. The proof follows the ideas of [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF] and [START_REF] Laurent | Smoothness and exponential decay of 2 -compact solutions of the generalized KdV equations[END_REF]. To reach the conclusion, we show the existence of ∈ R such that for each ∈ N, there exists > 0 such that, with := 2 ,

∀ ≥ , ∫ ≤ ( , ) 2 ( -)
≤ .

(4.9)

The first step is to obtain (4.9) for = 0. For this, we claim a strong monotonicity property which is the purpose of Lemma 4.4 and Lemma 4.5 below.

Let us introduce, for some > 0 to be determined later, the function defined by

( ) = 1 2 - 1 arctan( ).
It satisfies the following properties

∃ 0 > 0, ∀ ∈ R, 0 -| | < -( ) < 1 0 -| | , (4.10)
∀ ∈ R, | (3) ( )| ≤ -2 ( ). (4.11) ∃ 1 > 0, ∀ ≥ 0, 1 - ≤ ( ). (4.12)
Moreover, let us observe that

∫ < 2 ( , ) ( -) = ∫ <0 2 ( , + ) - , (4.13)
and that, for all 0 < 0,

∫ 0 ≤ <0 2 ( , + ) - ≤ -0 ∫ ≥ 0 2 ( , + ) -( -0 ) ≤ 1 1 -0 ∫ ≥ 0 2 ( , + ) ( -0 ) ≤ 1 1 -0 ∫ R 2 ( , + ) ( -0 ) . (4.14)
Since 2 < , one can choose ∈ (0, -2 ). We consider ∈ R to be determined later. Then, for fixed 0 ≥ and 0 ∈ R, we define

( 0 , 0 ) : [ , +∞) → R + ↦ → ∫ R 2 ( , + ) -0 + ( -0 ) . We have ∀ ≥ , ( 0 , 0 ) ( ) = ∫ R 2 ( , ) -0 + ( -0 ) - , (4.15)
so that by derivation with respect to , we obtain

( 0 , 0 ) ( ) = -3 ∫ R 2 ( , ) ( ˜ ) -( -) ∫ R 2 ( , ) ( ˜ ) + ∫ R 2 ( , ) (3) ( ˜ ) + 2 + 1 ∫ R +1 ( , ) ( ˜ ) , (4.16)
where ˜ := -0 + ( -0 ) -. We then claim Lemma 4.4. There exists 0 > 0 such that for all 0 ∈ R, and for all 0 , ≥ , Proof. Due to property (4.11) of , we have

( 0 , 0 ) ( ) ≥ -0 -(-0 + ( -0 )) . (4.17
∫ R 2 ( , ) (3) ( ˜ ) ≤ -2 ∫ R 2 ( , ) ( ˜ ) . (4.18)
Furthermore we control the nonlinear part by considering

1 ( ) := ∫ | ˜ |>-0 + ( -0 ) +1 ( , ) ( ˜ ) and 2 ( ) := ∫ | ˜ | ≤-0 + ( -0 ) +1 ( , ) ( ˜ ) .
On the one hand, we have due to (4.10)

1 ( ) ≤ 1 0 --0 + ( -0 ) ∫ R | | +1 ( , ) ≤ --0 + ( -0 ) , (4.19)
where we have used the Sobolev embedding 1 (R) ↩→ +1 (R) and the fact that belongs to ∞ ([ , +∞), 1 (R)). Note that > 0 is independent of 0 , 0 , and . On the other, we observe that

2 ( ) ≤ ( ) -1 ∞ ≤ ∫ ≤ 2 ( , )| ( ˜ )| ≤ √ 2 -1 ( ) -1 2 2 ≤ ( ) -1 2 2 ≤ ∫ R 2 ( , )| ( ˜ )| ≤ √ 2 -1 ( ) -1 2 2 ≤ sup ≥ ( ) -1 2 1 ∫ R 2 ( , )| ( ˜ )| . (4.20)
Since is a multi-soliton, we can choose ≥ 0 such that for all ≥ ,

√ 2 -1 ( ) -1 2 2 ≤ sup ≥ ( ) -1 2 1 ≤ + 1 2 ( --2 ). (4.21)
Let us justify it briefly (it consists in the main change with respect to previous proofs based on non dispersion [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF] or 2 -compactness [START_REF] Laurent | Smoothness and exponential decay of 2 -compact solutions of the generalized KdV equations[END_REF]): we have

∫ ≤ 2 ( , ) ≤ 2 ∫ ≤ - =1 , 2 ( , ) + 2 ∫ ≤ =1 , 2 ( , ) ≤ 2 2 0 -2 + 2 =1 ∫ ≤ 2 , ( ,
)
and for all = 1, . . . , , since < , we have for ≥ 0:

∫ ≤ 2 , ( , ) ≤ ∫ ≤ - √ | --| - √ | --| ≤ ∫ ≤ - √ ( -) - √ | --| ≤ - √ ( -) ∫ R - √ | --| ≤ - √ ( -) .
where denotes a positive constant which can change from one line to the other and which only depends on (see expression (4.1)). Thus, we can pick up ≥ 0 such that for all ≥ 0,

∫ ≤ 2 ( , ) ≤ -2 + =1 - √ ( -) .
Hence this leads to the existence of satisfying (4.21).

Taking into account (4.19), this eventually leads to the following estimate

2 + 1 ∫ R +1 ( , ) ( ˜ ) ≤ -( --2 ) ∫ R 2 ( , ) ( ˜ ) + 0 --0 -+ ( -0 ) , (4.22)
where 0 := 2 +1 is independent of 0 , 0 , and . Gathering (4.18) and (4.22) in (4.53), we finally deduce

( 0 , 0 ) ( ) ≥ -3 ∫ R 2 ( , ) ( ˜ ) -0 --0 -+ ( -0 ) .
Thus Lemma 4.4 is established.

As a consequence of the preceding lemma,

∃ 1 > 0, ∀ 0 ∈ R, ∀ ≥ 0 , ( 0 , 0 ) ( 0 ) ≤ ( 0 , 0 ) ( ) + 1 0 , (4.23)
with 1 independent of the parameters 0 and 0 . Next, we claim the following: Lemma 4.5. For fixed 0 ∈ R and 0 ≥ , ( 0 , 0 ) ( ) → 0 as → +∞.

Proof. This lemma is shown by adapting the proof in [57, paragraph 2.1, Step 2] and in [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF]. Let be a positive real number. As in the previous proof, because is a multi-soliton, we can find

1 ≥ large such that for all ≥ 1 , ∫ < 2 ( , ) ≤ 2
.

Since 0 ≤ ≤ 1, this enables us to see that

∫ <0 2 ( , + ) -0 + ( -0 ) ≤ ∫ < 2 ( , ) ≤ 2 . (4.24)
Now, recall that is decreasing so that

∫ ≥0 2 ( , + ) -0 + ( -0 ) ≤ -0 + ( -0 ) ( ) 2 2 ≤ -˜ -0 + ( -0 ) , (4.25)
with = ( ) 2 2 for all ∈ . Moreover, since ( ) → 0 as → +∞, there exists 2 ∈ R such that for all ≥ 2 , -0 + ( -0 ) ≤ 2

.

Then, for all ≥ max{ 1 , 2 },

( 0 , 0 ) ( ) ≤ 2 + 2 = .
Hence, we have finished proving Lemma 4.5.

At this stage, we deduce from (4.23) and Lemma 4.5 that

∀ 0 ≥ , ∀ 0 ∈ R, ( 0 , 0 ) ( 0 ) ≤ 1 0 . (4.26)
Thus, (4.14) leads to

∀ ≥ , ∫ 0 ≤ <0 2 ( , + ) - ≤ 1 1 . Letting 0 → -∞, we infer that ∀ ≥ , ∫ <0 2 ( , + ) - ≤ 1 1
.

which proves (4.9) with = 0. Now, to conclude to (4.9) for all ∈ N, one actually proves by induction on ∈ N the existence of ˜ ≥ 0 such that for all ≥ ,

∫ R 2 ( , + ) - + ∫ +1 ∫ R 2 ( , + ) - ≤ ˜ . (4.27)
For = 0, this is in fact a consequence of (4.9) and of the following estimate: for all ≥ 0 ≥ ,

( 0 , 0 ) ( 0 ) -( 0 , 0 ) ( ) ≤ 1 1 0 + 3 ∫ 0 ∫ R 2 ( , + ) ( -0 + ( -0 ))
(which follows from the proof of Lemma 4.4). Indeed, we notice that by (4.10) and since is decreasing, for ∈ [ 0 , ],

0 -| -0 | < -( -0 ) ≤ -( -0 + ( -0 ))
so that for = 0 + 1 in particular, we have

∫ 0 +1 0 ∫ 0 < 2 ( , + ) - ≤ -0 ( 0 , 0 ) ( ) -( 0 , 0 ) ( 0 ) ≤ -0 ( 0 , 0 ) ( 0 + 1) ≤ -0 ( 0 +1, 0 ) ( 0 + 1) ≤ .
where the last inequality results from (4.26). Passing to the limit when 0 → -∞, we obtain the desired inequality (4.27).

The rest of the induction argument closely follows [57, paragraph 2.3 and paragraph 2. 2 Step 2]. Since it does not depend on the properties of the multi-soliton and for the sake of brevity, we will not detail the proof (4.27) for higher values of .

Decay of the multi-solitons between the solitary waves

We consider 0 < < 1 and > . Let us prove Proposition 4.6 (Exponential decay in the soliton region). There exists , > 0 such that for all ∈ N, there exists > 0 such that for all ≥ , for all ≤ ≤ ,

( , ) ≤ =1 -, | -| . ( 4

.28)

Proof. We denote ( ) := ( ) -=1 , ( ). For all ∈ N, for all ≥ , we have by (4.3) and the Sobolev embedding 1 ≤ ≤ , we have -≤ -, ( -) because ≤ + , . Thus, we obtain that for all ≥ and for all ≤ ≤ ,

(R) ↩→ ∞ (R) ( ) ∞ ≤ ( ) 1 ≤ ( ) +1 ≤ +1 -. Now, let , := min √ 1 , 1 -, -,
( , ) ≤ ( ) ∞ ≤ +1 - ≤ +1 =1 -, | -| .
Moreover, for all = 1, . . . , , , . We can check that ∀ ∈ R,

( ) ≤ ( ); (4.31) ∀ ∈ R, 0 ≤ ( ) ≤ √ ; (4.32) ∀ ≤ 0, 0 ≤ ( ) ≤ 2 √ . ( (3)
For 0 > 0 and ≥ , we consider ˜ = ˜ ( ) := -0 -. Notice that the choice of is made in order to obtain the following interaction estimate, which roughly expresses that the growth of ↦ → √ is weaker than the decay of the solitons. ≤ .

On the one hand, we have

∫ ≤ - √ | -| √ ( -) ≤ ∫ ≤ - √ | -| ≤ ∫ R - √ | | ≤ 2 √ .
On the other, since > and

√ < √ , ∫ > - √ | -| √ ( -) ≤ ( √ -√ ) ∫ > ( √ - √ ) ≤ ∫ > ( √ - √ ) ≤ ( √ - √ ) √ - √ .
Hence, noticing that we also have ( , ) ≤ ( , ) -√ | -| , Claim 4.3 holds.

For all ∈ N, let us define for some 0 > 0 and for all ≥ :

, 0 ( ) := ∫ R 2 ( , ) ( -0 -) .
We first show the following recurrence formula which makes the link between the functions , 0 , ∈ N.

Lemma 4.8. For all ∈ N, there exists ≥ 0 (independent of 0 ) such that

, 0 ( ) ≤ ∫ R +1 =1 2 ( , ) ( ˜ ) + - -1 =0 , 0 ( ) + - √ 0 -. (4.35)
Proof. Let us observe that

, 0 ( ) = -3 ∫ R +1 2 ( ˜ ) + ∫ R 2 (3) ( ˜ ) - ∫ R 2 ( ˜ ) + 2 ∫ R + =1 - =1 . (4.36)
By (4.31), we have

∫ R 2 (3) ( ˜ ) - ∫ R 2 ( ˜ ) ≤ 2 ∫ R 2 ( ˜ ) . ( 4

.37)

Let us control the nonlinear term ∫ R ( ) which does not contain any soliton. For = 0, we observe that ∫

R ( ) = + 1 ∫ R +1 ( ˜ ) . Thus ∫ R ( ) ≤ + 1 ( ) -1 ∞ ∫ R 2 ( ˜ ) ≤ ∫ R 2 ( ˜ ) . (4.38)
For ≥ 1, we can write

∫ R ( ) = ∫ R ( ) ( ˜ ) - ∫ R -1 ( ) +2 + +1 .
We have

( ) = 1 +•••+ = 1 , . . . , ( 1 )
. . .

( ) so that ∫ R -1 ( ) +2 ≤ +2 ( ) ∞ 1 +•••+ = -1 ∫ R ( 1 )
. . .

( ) ( ˜ ) ≤ ( ) +3 ∫ R -1 =0 2 ( ˜ ) ≤ - -1 =0 , 0 , 0 ( ), (4.39)
where we have used (4.30).

Similarly we obtain

∫ R -1 ( ) +1 ≤ - ∫ R -1 =0 2 ( ˜ ) (4.40) and ∫ R ( ) ≤ - ∫ R =0 2 ( ˜ ) . (4.41) Moreover,
∫ R + =1 - =1 - ≤ ( ) +1 =1 =0 ∫ R ( , ) ( ( ˜ ) + ( ˜ )) ≤ ( ) +1 - √ 0 , (4.42)
where the last line follows from Claim 4.3. We obtain Lemma 4.8 by gathering the above estimates.

Then, we obtain the following control of , 0 ( ): Lemma 4.9. For all ∈ N, there exists ≥ 0 such that for all ≥ :

, 0 ( ) ≤ ∫ +∞ ∫ R +1 =1 2 ( , ) ( ˜ ( )) + - √ 0 -. (4.43)
Proof. The preceding lemma follows from (4.35) and an induction argument. Notice that for all ∈ N, , 0 ( ) tends to 0 as tends to +∞. Thus, for = 0, (4.43) follows by integration of (4.35) between and +∞. Now assume that (4.43) is proved for 0, . . . , -1 for some particular ∈ N * . Then, by integration of (4.35) between and +∞ (for ≥ ), it results:

, 0 ( ) ≤ ∫ +∞ ∫ R +1 =1 2 ( , ) ( ˜ ) + - √ 0 ∫ +∞ - + -1 =0 ∫ +∞ - ∫ +∞ ∫ R +1 =1 2 ( , ) ( ˜ ) + -1 =0 - √ 0 ∫ +∞ - ≤ ∫ +∞ ∫ R +1 =1 2 ( , ) ( ˜ ) + -1 =0 ∫ +∞ ∫ R +1 =0 2 ( , ) ( ˜ ) ∫ +∞ - + max { , max{ , = 0, . . . , -1}} -√ 0 -,
hence the existence of ≥ 0 such that:

, 0 ( ) ≤ ∫ +∞ ∫ R +1 =1 2 ( , ) ( ˜ ) + - √ 0 -.
Now, we show how to deduce from Lemma 4.9 the polynomial decay of and its derivatives. This is the object of Claim 4.4 and Lemma 4.10 below.

Let us define Φ [-1] ( ) := ( ) and for all ∈ N,

Φ [ ] ( ) := ∫ -∞ Φ [ -1] ( ) .
The following claim justifies that Φ [ ] is well-defined for all ∈ N and in fact motivates the introduction of Φ [ ] .

Claim 4.4. We have for all

∈ N ∀ ≤ 0, 0 ≤ Φ [ ] ( ) ≤ 1 √ √ ∀ ≥ 0, 1 2 ! ≤ Φ [ ] ( ) ≤ =0 1 √ -! (4.44)
Proof. We argue by induction on . Note that Φ [0] = is an increasing function and that ∀ ≥ 0, arctan ≤ .

Thus

∀ ≤ 0, 0 ≤ Φ [0] ( ) ≤ 2 √ ≤ √ ∀ ≥ 0, 1 2 = (0) ≤ Φ [0] ( ) ≤ 1.
Now assume that (4.44) holds for some ∈ N being fixed. In particular, Φ [ ] is positive on R and is integrable in -∞; thus Φ [ +1] is well-defined. Moreover, by definition of Φ [ +1] and by the induction assumption, we have for all ≤ 0

0 ≤ Φ [ +1] ( ) ≤ ∫ -∞ 1 √ √ ≤ 1 √ +1 √ .
In particular,

0 ≤ Φ [ +1] (0) ≤ 1 √ +1 .
By the induction assumption, we then infer that for all ≥ 0,

Φ [ +1] ( ) = Φ [ +1] (0) + ∫ 0 Φ [ ] ( ) satisfies 0 + ∫ 0 1 2 ! ≤ Φ [ +1] ( ) ≤ 1 √ +1 + ∫ 0 =0 1 √ -! .
Thus for all ≥ 0, 1 2

+1 ( + 1)! ≤ Φ [ +1] ( ) ≤ 1 √ +1 + =0 1 √ +1- +1 ( + 1)! = +1 =0 1 √ +1-! .
This finishes the induction argument, hence the proof of Claim 4.4.

Lemma 4.10. For all ∈ N, for all ∈ N, there exists , ≥ 0 such that for all ≥ ,

∫ R 2 Φ [ ] ( -0 -) ≤ , - .
Proof. The following inequality holds true: for all ∈ N,

∫ R 2 Φ [0] ( -0 -) ≤ ,0 -. (4.45)
Assume that for some ∈ N, we have:

∀ ∈ N, ∃ , ≥ 0, ∀ ≥ , ∀ > 0, ∫ R ( , ) 2 Φ [ ] ( --) ≤ , -. (4.46)
Then take ∈ N and +1 > 0. Integrate estimate (4.43) provided by Lemma 4.9 on [ 1 , +∞) with respect to 0 , for some 1 > 0. We obtain by Fubini theorem: for ≥ ,

∫ R 2 ( , ) ∫ +∞ 1 Φ [0] ( -0 -) 0 ≤ ∫ +∞ ∫ R +1 =0 2 ( , ) ∫ +∞ 1 Φ [-1] ( -0 -) 0 + √ -√ 1 -
and then by an affine change of variable

∫ R 2 ( , )Φ [1] ( -1 -) ≤ ∫ +∞ ∫ R +1 =0 2 ( , )Φ [0] ( -1 -) + √ -√ 1 -.
Then integrating the preceding estimate on [ 2 , +∞) with respect to 1 , for some 2 > 0 leads to

∫ R 2 ( , )Φ [2] ( -2 -) ≤ ∫ +∞ ∫ R +1 =0 2 ( , )Φ [1] ( -2 -) + √ 2 - √ 2 -.
Iterating several integrations ( + 1 in total), we finally obtain:

∫ R 2 ( , )Φ [ +1] ( -+1 -) ≤ ∫ +∞ ∫ R +1 =0 2 ( , )Φ [ ] ( -+1 -) + √ +1 - √ +1 - ≤ +1 =0 , ∫ +∞ - + √ +1 - √ +1 - ≤ +1 =0 , -+ √ +1 - √ +1 -,
where the second line follows from the induction assumption (4.46). Thus, there exists , +1 ≥ 0 such that for all ≥ , for all +1 > 0,

∫ R ( , ) 2 Φ [ +1] ( -+1 -) ≤ , +1 -.
This finishes proving Lemma 4.10.

Proposition 4.7 follows now from Lemma 4.10, from the Sobolev embedding 1 (R) ↩→ ∞ (R), and the decay of all derivatives of the .

Exponential decay on the right for global multi-solitons

In this subsection, we assume moreover that the multi-soliton (in +∞) is global; in other words, belongs to C (R, ∞ (R)). This is in particular satisfied in the 2 -subcritical case, when 1 < < 5.

We will show that also decays exponentially on the right of the last solitary wave. Let > and ∈ 0, • for all ≥ 1, ( ) =

• for all ∈ R, ( ) ≥ max{ 1 , . . . , }

• for all ∈ R, 1 ≤ ( ) ≤ .

The choice of is essentially motivated by the next claim, the main observation being the uniform (in time on the whole real line) polynomial decay of . As a direct consequence of this claim we obtain non dispersion on the right in space for all time ∈ R. Proof of Claim 4.5. The proof is done by adapting the previous subsection. On the one hand, we note that there exists ˜ ≥ 0 such that for all ∈ R,

( ) - =1 ( ) 1 ≤ ˜ -; (4.48)
this is by definition of a multi-soliton and due to the fact that -=1 ∈ ∞ (R, 1 (R)).

On the other, taking again the notation introduced in subsection 4.3.1 with ˜ ( ) := -0 -( ) for some fixed parameter 0 > 0, the analog of Claim 4.3 writes as follows: for all ∈ N, there exists > 0 such that for all = 1, . . . , , for all ∈ R, ∫

R ( , ) ( ( ˜ ) + ( ˜ )) ≤ - √ 0 . (4.49)
Indeed, it suffices to check that for all ∈ R,

∫ > ( ) - √ | -| √ ( -( )) = 3 2 - √ ( ) ∫ > ( ) -( √ - √ )
Proof of Lemma 4.12. We argue as in [57, paragraph 2.1, proof of Claim 1] and as in the proof of Lemma 4.4 in the present article. Property (4.51) allows us to control the integral ∫ R 2 ( , ) (3) ( ˜ ) ; see (4.18). The nonlinear term is treated by cutting it into two pieces, depending on the position of | ˜ | with respect to 0 --( -0 ), being fixed so that

∀ ∈ R, | ( , • + ( ))| 2 ( ≥ ) sup ∈R ( ) -1 2 1 ≤ + 1 2 ( 1 --2 ),
which is indeed possible by Claim 4.6 and the choice of .

By means of the non dispersion property provided by Claim 4.6, we deduce (with the same proof as that of Lemma 4.5) that for all 0 ∈ R, ( 0 , 0 ) ( ) → 0 as → -∞.

Hence, Lemma 4.12 leads to

∀ 0 ∈ R, ( 0 , 0 ) ( 0 ) ≤ -0 ,
with independent of 0 and 0 . By (4.52), we obtain

∫ < 0 2 ( , + ( )) ( -0 ) ≤ ∫ < 0 2 ( , + ( )) ( -0 ) ≤ -0 .
In other words, for all 0 ∈ R,

∫ < 0 2 ( , + ( )) ≤ .
Letting 0 tend to +∞, we finally obtain by monotone convergence

∫ R 2 ( , + ( )) ≤ .
With the same strategy as that developed in [57, paragraph 2.1 Step 3 and paragraph 2.2] and that used in order to prove (4.9), we then deduce that for all ∈ N, for all ∈ R,

∫ ≥ ( ) ( , ) 2
( -( ))

< .

(4.55)

In particular, for large values of ,

∫ ≥ ( , ) 2
( -) < , (4.56)
which allows us to conclude to (4.47).

We notice that this theorem has been extended to solutions describing multi-bound states by Côte and Martel [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF], that is to multi-traveling waves made of any number of decoupled general (excited) bound states. In the present paper, we will however only focus on multi-solitary waves in the above sense.

Since solitons are unstable, a solution of (NLKG) which behaves as a soliton in large time is not expected to be necessarily a soliton. One of our goals is thus to precise the dynamic of the flow of (NLKG) near a soliton. Similarly, the dynamic near a sum of solitons is also supposed to be more complex as time goes to infinity.

Main results

Given distinct velocity parameters, we aim at proving the existence of a whole family of multisolitons which turns out to be the unique family of multi-solitons in a certain class of solutions. Our first result reads as follows.

Theorem 5.2. Assume that is of class

C 2 and 0 < | | < • • • < | 1 | < 1. There exist > 0, 0 < 1 < • • • < , +, ∈ C (R, 1 (R ) × 2 (R )) ∩ ∞ (R, 1 (R ) × 2 (R )) for = 1, .
. . , and an -parameter family ( 1 ,..., ) ( 1 ,..., ) ∈R of solutions of (NLKG) such that, for all ( 1 , . . . , ) ∈ R , there exist 0 ∈ R and > 0 such that

∀ ≥ 0 , Φ 1 ,..., ( ) - =1 , ( ) - =1 - +, ( ) 1 × 2 ≤ -( + ) , (5.2)
where Φ 1 ,..., :=

.

Remark 5.3. The parameters and the functions +, ( = 1, . . . , ) are defined in Proposition 5.9 and in subsection 5.1.5. One can moreover precise the value of in Theorem 5.2; for this, we refer to (5.6).

Below is about classification in a class with polynomial decay. We emphasize that the corresponding polynomial decay rate is explicit and independent of the soliton parameters, which is undeniably a breakthrough with regard to the thorny question of the classification of multi-soliton solutions. As a comparison, we proved uniqueness of a multi-soliton in a class of solutions with decrease faster than any power of 1 in the context of the nonlinear Schrödinger equations [START_REF] Côte | On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations[END_REF].

Theorem 5.4. Under the assumptions of Theorem 5.2 and keeping the same notations, if is a solution of (NLKG) such that

( ) - =1 , ( ) 1 × 2 = O 1 as → +∞, (5.3)
where = and where > 3, then there exist 1 , . . . , ∈ R and 0 ∈ R such that for all ≥ 0 , ( ) = Φ 1 ,..., ( ).

Remark 5.5. Notice that Theorem 5.2 and Theorem 5.4 apply in dimension ≤ 5 only. Indeed, assuming (H'1) and of class C 2 forces to have > 2, hence 2 < +2 -2 if ≥ 3.

In the case where only one soliton is considered, one can moreover improve the preceding Theorem by completely characterizing solutions which converge to a soliton in large time.

Theorem 5.6. Let ∈ R , | | < 1 and assume that is of class C 2 . There exist > 0, +, ∈ C (R, 1 (R ) × 2 (R )), and a one-parameter family ( ) ∈R of solutions of (NLKG) such that for all ∈ R, there exists 0 = 0 ( ) ∈ R such that for all ≥ 0 ( ) -( ) -- +, ( )

1 × 2 ≤ -2 , (5.4)
where := . In addition, if ≠ , then ≠ .

Moreover, if is a solution of (NLKG) such that

( ) -( ) 1 × 2 → 0 as → +∞, (5.5)
where = , then there exist ∈ R and 0 ∈ R such that for all ≥ 0 , ( ) = ( ).

Remark 5.7. The parameter (which depends on ) and the function +, are defined in Proposition 5.9; they are intimately related to the spectral theory dealing with the flow around .

It is interesting to remark that there are only three special solutions among the elements of the preceding family ( ) ∈R , up to translations in time and in space. This is the object of the following Corollary 5.1. Consider the family of solutions ∈R defined in Theorem 5.6.

1. If > 0, there exists ∈ R such that for all possible , ( ) = 1 ( + , • + ).

2. If < 0, there exists ∈ R such that for all possible , ( ) = -1 ( + , • + ).

3. For all ∈ R, 0 ( ) = ( ).

Remark 5.8. Let us observe that Remark 5.5 is valid for Theorem 5.6 and Corollary 5.1 too.

Theorem 5.6 provides the behavior of the solutions converging to solitons at the order O -2 in 1 (R ) × 2 (R ). The instability direction due to the existence of one negative eigenvalue (related to ) for the linearized operator around yields infinitely many possibilities (described by the real line) to perturb the soliton and to retrieve another solution of (NLKG) which however remains exponentially close in time (with decay rate ) to . Let us emphasize that this phenomenon appears quite naturally in the study of unstable solitons. In a stable mode such as for the 2 -subcritical generalized Korteweg-de Vries (gKdV) equation, a solution which converges to a soliton as time goes to infinity is pretty well-known to be exactly the corresponding soliton.

What is more, Theorem 5.6 is built exactly as in [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF]. It reminds us of similar classification results, previously obtained for the 2 -supercritical gKdV equations by Combet [10, Theorem 1.1] and for the three-dimensional cubic Schrödinger equation by Duyckaerts and Roudenko [26, Proposition 3.1], both inspired from pioneering works of Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF][START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical NLS[END_REF].

As far as the multi-soliton case is concerned, Theorem 5.2 provides an -parameter family of solutions to (NLKG) which behave as a sum of (differently boosted) solitons in large time. Here again, the existence result looks like that of Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF][START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF] and reinforces the idea that, in general, non-uniqueness holds for multi-solitons in an unstable context. Furthermore, Theorem 5.4 opens the way to treat the question of the classification of multi-solitons for other models, at least in the restraint class of solutions with algebraic convergence in the sense of (5.3). Regarding the global approach developed in this article, we shed new light on the construction of the one-parameter family ( ) ∈R by a compactness procedure. Note that the special solution -1 in [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF] has also been obtained in a first rigorous way by a compactness method (the starting point was the proof of instability of the soliton), but the question of obtaining 1 (as well as the other solutions for > 0) by such method was raised (see Remark 4.15 in [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF]) and, to our knowledge, remained unanswered. Obviously, our process can be thought and used as an alternative to prove similar results in the context of other partial differential equations which involve unstable solitons, and for which the spectral theory around the ground states is well understood (and actually analogous to the present case). This is a nice feature of our paper.

A second interesting point to be discussed is about proving uniqueness of multi-solitary waves. Exploring the possibility of obtaining a "weak" monotonicity property like (5.42) in Corollary 5.11 would be a promising direction of research in order to classify multi-solitons of other models.

Besides, a significant issue would be to know to what extent the classification obtained in Theorem 5.6 could be transcribed to the multi-soliton topic. Indeed, contrary to the gKdV setting [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF] and especially as for the NLS case [START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF][START_REF] Côte | On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations[END_REF], proving general uniqueness in the sense of (5.5), where is replaced by a sum of several solitons , , still remains unclear. Another question to be addressed could be related to the potential generalizations of the previous theorems to multi-bound states.

Outline

In the spirit of [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] (dealing with the construction of multi-soliton solutions), our approach to constructing the families in Theorem 5.2 and Theorem 5.6 is based on backward uniform 1 × 2estimates satisfied by well-chosen sequences of solutions of (NLKG) which aim to approximate the desired solutions. We entirely exploit a coercivity property available in the present matter (see Proposition 5.10 stated below) in order to establish those estimates and, in fact, to obtain the expected exponential convergences to zero in time. We finally use continuity of the flow of (NLKG) for the weak 1 × 2 -topology to obtain special solutions which fulfill (5.2) or (5.4).

The proof of Theorem 5.2 is thus based on compactness and energy methods; it follows the strategy of [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF][START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF]. The construction of Φ 1 ,..., is done by iteration, by means of Proposition 5.12 which roughly asserts that each multi-soliton can be perturbed slightly at the order - around the soliton , . In order to establish this key proposition, we particularly rely on the topological ingredient set up originally by Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF] for the construction of one given multi-soliton in unstable situations.

For the construction of in the one-soliton case (Theorem 5.6), we also substantially rely on the spectral theory available for (NLKG), built on the linearized operator of the flow around the boosted ground state. In particular, we point out that our approach differs from previous articles [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF][START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF][START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical NLS[END_REF][START_REF] Duyckaerts | Threshold solutions for the focusing 3D cubic Schrödinger equation[END_REF] where the construction centers around the contraction principle.

Regarding the question of classification, from (5.4) and orthogonality properties exposed in the next subsection, we notice that corresponds to the limit of , -, as → +∞ in Theorem 5.6. This is precisely useful for the uniqueness part of this theorem, where the goal is to identify with an element of the one-parameter family ( ) ∈R . Actually, to prove the second part of Theorem 5.6, we follow [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF] in the first instance (up to the obtainment of an exponential control of -). Then, a refined version of the coercivity argument considered in [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF], and indeed an elementary but careful analysis of the available estimates, allows us to reach the conclusion, that is to show that equals some (already constructed in the first part of the theorem), and that without making use of any supplementary tool. We underline once again that we do not need any fixed point argument to conclude. Yet, we do not exclude the possibility to prove Theorem 5.6 via the contraction principle; if the nonlinearity is sufficiently regular (C +1 for instance), one could precise by this means the behavior of in large time, and indeed expand the solution at the order O -( +1) in the Sobolev space , in line with [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF]. For clarity purposes, we will not explore this path further and anyway, our description of the family ( ) ∈R is sufficient to characterize solutions verifying (5.5).

Concerning Theorem 5.4, the identification of the solution satisfying (5.3) is done step by step as in [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]. The core of the proof is the obtainment of an almost monotonicity property, inspired by Martel and Merle [START_REF] Martel | Construction of multi-Solitons for the energy-critical wave equation in dimension 5[END_REF]. By means of a technical lemma of analysis (we refer to Lemma 5.26 in Appendix), this monotonicity property allows us to see that any multi-soliton in the class with polynomial convergence to zero converges in fact exponentially (see subsection 5.3.1), provided one assumes suitable integrability conditions in the neighborhood of +∞ (and indeed > 3). Such a "weak" monotonicity property has a priori been used so far only for the construction of multisolitons or multi-bound states of the energy-critical wave equation [START_REF] Martel | Construction of multi-Solitons for the energy-critical wave equation in dimension 5[END_REF][START_REF] Yuan | On multi-solitons for the energy-critical wave equation in dimension 5[END_REF][START_REF] Yuan | Construction of excited multi-solitons for the 5D energy-critical wave equation[END_REF]. We also underline that, by Lemma 5.26, we directly obtain the adequate exponential convergence rate which allows us to identify 1 ; this is in contrast with [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]. The monotonicity property is also used as a key ingredient to identify the other parameters 2 , . . . , .

This paper is organized as follows. In Section 2, we introduce essential notations and tools which are used throughout our article. In the following sections, we establish the proofs of our main results; for ease of writing, each proof will be made in dimension 1. Section 3 is devoted to the construction of the family of multi-solitons described in Theorem 5.2. Section 4 deals with the classification of multi-solitons, which is the object of Theorem 5.4. In Section 5, we study the existence part of Theorem 5.6 focusing on one soliton. Section 6 aims at proving the second part of this latter theorem, that is general uniqueness of the one-parameter family previously constructed. In the appendix, we explain how to adapt the proof to all dimensions and we justify Corollary 5.1, and we state and prove the lemma of analytic theory of differential equations used in Section 4.

As usual, denotes a positive constant which may depend on the soliton parameters and change from one line to the other, but which is always independent of and .

Acknowledgments

The author would like to thank his supervisor Raphaël Côte for his constant encouragements and for fruitful discussions.

Construction of a family of multi-solitons for ≥ 2

In this section, we give a detailed proof of Theorem 5.2 in the one-dimensional case. Let ≥ 2 and 1 , . . . , , 1 , . . . , be 2 parameters as in Theorem 5.2. Denote by a multi-soliton solution associated with these parameters, satisfying (5.7) and consider Φ := .

As it was firstly observed in [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF], the existence of ( 1 ,..., ) ( 1 ,..., ) ∈R verifying (5.2) in Theorem 5.2 is a consequence of the following crucial Proposition 5.12. Let ∈ {1, . . . , } and ∈ R. Then there exist 0 > 0, > 0, and a solution of (NLKG), defined on [ 0 , +∞) such that ∀ ≥ 0 , ( ) -Φ( ) --+, ( ) 1 × 2 ≤ -( + ) , where := .

By (5.7), assuming that the preceding proposition holds, and considering 1 , . . . , ∈ R, we indeed obtain a solution 1 of (NLKG) and 1 > 0 such that

∀ ≥ 1 , Φ 1 ( ) -Φ 0 ( ) -1 -1 +,1 ( ) 1 × 2 ≤ -( 1 + )
with obvious notations. We notice that 1 is a multi-soliton. Now, assume that we have constructed, for some ∈ {1, . . . , -1}, a family of multi-solitons 1 , . . . , 1 ,..., such that there exists > 0 such that for all = 1, . . . , , ∀ ≥ , Φ 1 ,..., ( ) -Φ 1 ,..., -1 ( ) --+, ( )

1 × 2 ≤ -( + ) ,
where Φ 1 ,..., -1 = Φ 0 if = 1. Hence we can apply Proposition 5.12 with Φ 1 ,..., instead of Φ and there exist Φ 1 ,..., +1 and +1 > 0 such that ∀ ≥ +1 , Φ 1 ,..., +1 ( ) -Φ 1 ,..., ( ) -+1 -+1 +, +1 ( ) 1 × 2 ≤ -( +1 + ) .

Thus, by induction on , we obtain a family of multi-solitons 1 , . . . , 1 ,..., such that for all At this stage, we conclude to (5.2) in Theorem 5.2, using once more (5.7) and the triangular inequality.

Let us already justify also that for all ( 1 , . . . , ) ≠ ( 1 , . . . , ), we have ). Then, we denote 0 := min{ ∈ {1, . . . , }| ≠ }.

From the construction of 1 ,..., , there exists > 0 such that for large Φ 1 ,..., ( ) -Φ 1 ,..., 0 -1 ( ) - (5.9)

Using that Φ 1 ,..., ( ) = Φ 1 ,..., ( ) and Φ 1 ,..., 0 -1 ( ) = Φ 1 ,..., 0 -1 ( ), we deduce from (5.8) and (5.9) that for all sufficiently large

-0 | 0 - 0 | ≤ -( 0 + ) .
Hence, letting → +∞, we obtain 0 -0 = 0, which leads to a contradiction. This ends the proof of Theorem 5. (5.15)

Now, we observe that ± +, , ±, + +, , ±, = O( -4 ).

This is clear if ≠ and for = , we have ± +, , ±, + +, , ±, =

0 + 0 = 0 if ± = + -+ = 0 if ± = - ;
indeed, we recall from Proposition 5.9

+, , +, = 0 and +, , -, = 1.

In addition, we have by the well-known inequality ( + )

2 ≤ 2( 2 + 2 ), -Φ 2 1 × 2 ≤ 2 1 × 2 + -2 .
Considering that 2 ≥ + 4 , we have thus finished the proof of the claim.

Step Proof. The corollary immediately follows from (5.31) and bounds for the derivatives of and +, which are analogous to (5.37).

In the spirit of [76, Proposition 4.2], we will state an almost monotonicity property satisfied by F . Let us define

F ,Ω( ) ( ) := ∫ Ω( ) 2 + 2 2 + 2 2 .
Let ∈ (1, -1). The choice of is linked with the integrability of particular quantities and will appear naturally later. Proposition 5.23. There exists > 0 and 0 > 0 such that for all ≥ 0 , -F ( ) ≤ F ,Ω( ) ( ) + O -

2 1 × 2 + 3 1 × 2 .
(5.40)

Proof. Since (0) = 0 and is continuous, there exists 0 > 0 such that for all ∈ [0, 0 ], | ( )| ≤ 1. There exists > 0 (independent of ) such that for all ≥ 0 and for all ∈ Ω( ), | ( , )| ≤ -. Even if it means increasing 0 , we can assume that -≤ 0 for all ≥ 0 . In addition ≥ 0. Thus, for ≥ 0 , -

∫ Ω( ) 2 + ∫ Ω( ) 2 + 2 2 -2 + ( ) 2 ≤ -2 ∫ Ω( ) 2 + ∫ Ω( ) 2 + 2 2 . Moreover, - 2
∫ Ω( ) 2 + ∫ Ω( ) 2 + 2 2 = 1 (1 -2 ) ∫ Ω( ) 2 2 + 2 + 2 2 = 1 (1 -2 ) F ,Ω( ) + 2 ∫ Ω( ) - 2 . 2
Now, for ∈ Ω( ), we have

-( , ) ≤ 1 - 1 1 -2 + 1 -2 × 2 max | | ≤ 2 + ≤ .
Thus,

2 (1 -2 ) ∫ Ω( ) - 2 = O 2 1 × 2 .
Noticing moreover that (5.55)

F ,Ω( ) ≥ ∫ Ω( ) 2 + 2 2 -2 ( ) ∞ ∫ Ω( ) 2 ≥ ∫ Ω( ) 2 + 2 2 -( ) ∞ ∫ Ω( ) 2 + 2 2 ≥ (1 -( ) ∞ )
Moreover, denoting ±, , := , ±, for all = 1, . . . , , we have ∀ ∈ {1, . . . , }, -, , ( ) → 0, as → +∞.

(5.56)

Proof. We proceed by induction on . First, we focus on the case where = 1.

We have ( ) 1 × 2 ≤ -1 by Proposition 5.26. Thus, by Lemma 5.6 and given that < 1 ,

( 1 -,1 ) ≤ -.

Since ↦ → -is integrable in +∞, there exists 1 ∈ R such that 1 -,1 ( ) → 1 , as → +∞.

We then define 1 := -Φ 1 . We notice that 1 = + (Φ -Φ 1 ) so that

1 ( ) 1 × 2 ≤ ( ) 1 × 2 + Φ( ) -Φ 1 ( ) 1 × 2 ≤ -1 + Φ 1 ( ) -Φ( ) -1 -1 +,1 ( ) 1 × 2 + 1 -1 +,1 ( ) 1 × 2 ≤ -1 .

Moreover

-,1,1 = , -,1 + Φ -Φ 1 -1 -1 +,1 , -,1 + 1

-1 +,1 , -,1 = --,1 + 1 -1 + O -( 1 + ) = o -1 ,
the last line resulting from the definition of 1 . Thus Proposition 5.27 is true for = 1.

We now assume that there exist 1 , . . . , -1 ∈ R such that -1 ( ) 1 × 2 ≤ --1 and for all = 1, . . . , -1, -, -1, ( ) → 0 as → +∞. Let us show Claim 5.28. We have

-1 ( ) 1 × 2 ≤ -.
• To prove this claim, we show that, if -1 ( ) ≤ -0 with -1 < 0 < -, then For all = , . . . , , we have + 0 -≤ + 0 -< 0, thus by integration on [ 0 , ], we obtain -, -1, ( ) -0 -, -1, ( 0 ) ≤ ( -0 -) .

Eventually, we obtain (by a "cut-and-paste" of the argument exposed in subsection 5.3.1)

-1 ( ) 1 × 2 ≤ sup ≥ =1 | -, -1, ( )| ≤ -( 0 + ) ,
which is what was expected.

• Now, from the preceding induction, there exists ˜ 0 ∈ -, such that In order to avoid the supremum in front of the expression 2 +, + 2 -, and hence to obtain the second way of estimating ±, ∓ ±, which is described by Lemma 5.17, we can rewrite (5.79) more simply as follows |F ( )| ≤ .

Then the analog of (5.80) takes the following form:

2 1 × 2 ≤ - + 2
+, ( ) + 2 -, ( ) ; the last arguments remain unchanged. However note that, in this case, the obtained estimates are at the cost of a worse exponential term.

Proposition 5.32. There exists > 0 such that for all 0 ≥ 0 sufficiently small, for all 0 sufficiently large, for all such that ≥ 0 , for all ∈ [ * , ], the following inequalities hold: (5.82)

Proof. Let us start with the system (5.77) obtained in Lemma 5.17. We have in particular At this stage, we can deduce that for all in [ * , ], ℎ( ) ≤ 0. Assume for the sake of contradiction that there exists ˜ * ∈ [ * , ] such that ℎ(˜ * ) > 0. Then, we can define := sup{ ∈ [˜ * , ], ℎ( ) > 0}.

           +, - +, ≤ 2
We necessarily have ℎ( ) = 0. Indeed, ℎ( ) < 0 is excluded by continuity of ℎ in (on the left side) and by definition of the supremum; if ℎ( ) > 0, then < (since ℎ( ) = -˜ - ) which leads once more to a contradiction, using the continuity of ℎ in (on the right side) and the definition of as a supremum. It follows that ℎ ( ) ≥ 0; thus ℎ is non-decreasing in the neighborhood of and in particular, ℎ( ) ≤ 0 on [ -, ] for some > 0. This again contradicts the definition of . Hence, for all in [ * , ], ℎ( ) ≤ 0. Given that -+, satisfies the same differential system as +, , we finally obtain (5.84). We claim that this implies the estimates in Proposition 5.32. The preceding inequality rewrites as follows: Step 4: Improvement of the exponential decay rate

The goal of this paragraph is to optimize the exponential decay rate in the estimate of 1 × 2 . We actually prove Proposition 5.33. The following estimate holds for all ∈ [ * , ]:

( ) 1 × 2 ≤ -2
.

  , multi-solitons et théorie non-linéaire des équations aux dérivées partielles dispersives 1.1.1 Les solitons dans le cadre des équations de Korteweg-de Vries généralisées et de Schrödinger non-linéaires D'un point de vue historique, l'équation de Korteweg-de Vries (non-linéaire)

  Des formules analogues peuvent être obtenues pour (mKdV) [101, chapitre 6] et (NLS) en dimension 1 avec la non-linéarité | | 2 .

4 - 2 dans

 42 )) tel que pour toute suite ( 0 ) qui converge vers 0 dans (R ), la solution de (NLS) telle que (0) = 0 est définie sur [-T,T] pour large et ( ) converge vers dans C ( [-, ], (R )).L'énoncé démontré par Cazenave et Weissler est en réalité plus précis en ce qui concerne le résultat d'existence et d'unicité locales, dans la mesure où les auteurs considèrent également des espaces auxiliaires (de Besov). Notons qu'en passant cette fois par des espaces de Lebesgue, Kato[START_REF] Kato | On nonlinear Schrödinger equations. II, solutions and unconditional wellposedness[END_REF] a généralisé le théorème précédent aux non-linéarités de classe C (au sens des fonctions différentiables R 2 → R 2 ) dont les dérivées partielles d'ordre ∈ {0, . . . , } croissent en O pour un certain ∈ , 1 + le cas où ≤ 2 .

Théorème 1 . 8 .

 18 Pour tout segment [-, ] inclus dans (-( 0 ), ( 0 )), pour toute suite ( 0 ) qui converge vers 0 dans (R ), la solution de (NLS) telle que (0) = 0 est définie sur [-, ] et converge vers dans C ( [-, ], (R )).

  où a pour expression ( , ) := ( , + ( )) -( ) ( ) pour de certaines fonctions de modulation ( ) et ( ) de classe C 1 .

  ) resulting from the local boundedness of the distributional derivative of ↦ → 1 2

Lemma 2 . 13 .

 213 For large enough, and for all = 1, . . . , , ( ) and ( ) are uniquely determined. Moreover, ↦ → ( ) and ↦ → ( ) are differentiable in the sense of distributions and |

  ,1 , . . . , ,1 , . . . , 1, , . . . , ,and( ) = -[ 0 , 1 , . . . , ] ,

  (2.81) follow from the differentiability in the sense of distributions (and the expressions of the differentials) of ↦ → Im ∫ for ∈ {1, . . . , } and ∈ {1, . . . , }. Let us explain how to show the differentiability of ↦ → ∫ R ( , ) ( , ) . This is essentially due to a density argument and the local well-posedness of (2.76) with continuous dependence on compact sets of time (as for (NLS)). Let us consider a C 1 function defined for large values of and with compact support, say included in [ 0 , 1 ]. Since ( 0 ) ∈ 1 (R ), there exists ( ( 0 )) ∈ C ∞ (R ) converging to ( 0 ) in the sense of the 1 -norm. The solution of (2.76) with initial data ( 0 ) in time 0 is defined on [ 0 , 1 ]) for large, belongs to C [ 0 , 1 ], S (R ) , and satisfies sup ∈ [ 0 , 1 ]

  and | ( )|, | ( )|, | ( )| ≤ ( ) 1 .

4 and 4 .

 44 -, ( ) = -Δ + -Let us prove Proposition 2.25, following the results and ideas of Weinstein[START_REF] Michael | Modulational stability of ground states of nonlinear dispersive Schrödinger equations[END_REF].Due to Weinstein [108, Proposition 2.7], inf ∈ 1 (R ), ,=0+,

Corollary 3 . 1 .

 31 Let ∈ C (R, 1 (R)) satisfy the assumptions (3.7), (3.8) of Theorem 3.4, and assume (to replace (3.6)) the existence of two constants 0 < < such that

  and ∀ ( , ) ∈ {1, . . . , 1 } × {1, . . . , 2 }, ≠ 2 -3 2 . (3.14) We refer to Schuur [101, Chapter 4], Chen and Liu [8, Paragraph 1.2], and the references therein for more details concerning genericity.

3. 4 . 1

 41 Non dispersive solutions of the Korteweg-de Vries equationThe strategy to prove Theorem 3.7 takes inspiration in [57, Proof of Theorem 2]. We use the following result of Eckhaus and Schuur [28, Section 5], which is also a consequence of a generalized version by Schuur [101, Chapter 2, Theorem 7.1 and (7.23)].

  ) where ˜ ( , ) := -( ) -( ) , we have for all ≥ , ∀ ∈ {1, . . . , },

-

For 2 .

 2 all ∈ {1, . . . , }, set ( , ) A direct computation yields∀ ∈ R, ( ) =ℎ ( ) ( ),from which we have for all ∈ {1, . . . , } ( ,

√ 2 .Proposition 4 . 3 (

 243 The goal of this section is to prove Exponential decay in large time on the left of the first soliton). There exists ≥ such that for all ∈ N, there exists > 0 such that for all ≥ , for all ≤ ,

  )

1 . 4 . 3 4 . 3 . 1 4 . 7 ( 1

 143431471 | for some , > 0 depending on and . Hence, we conclude to (4.28) by the triangular inequality and by the fact that , ≤ √ Decay of the multi-solitons on the right of the last soliton Rapid decrease on the right Let > . In this subsection, we prove the general Proposition Polynomial decay in large time on the right of the last soliton). For all ∈ N and for all ∈ N, there exists , > 0 such that for all ≥ , for all > , Let us denote ( ) := ( ) -=1 , ( ) and = , . By Theorem 4.1, there exists > 0 such that for all ∈ N, there exists > 0 such that for all ≥ , and introduce the function : R → R defined by ( ) := 2 arctan √

Claim 4 . 3 . 34 )

 4334 For all ∈ N, there exists = ( ) > 0 such that for all = 1, . . . , and for all ≥ , Proof of Claim 4.3. By (4.32) and (4.33), it suffices to show that ∫ R

√ 1 2 . 4 . 11 (

 2411 Proposition Exponential decay on the right). Let us assume that belongs to C (R, ∞ (R)).There exists ∈ R such that for all ∈ N, there exists > 0 such that for all ≥ , for all ≥ ,( , ) ≤ -| -| .(4.47)Proof. Let us consider a function : R → R of class C 1 such that:

Claim 4 . 5 .

 45 There exists > 0 such that for all ∈ R,∀ > ( ), | ( , )| ≤ -( ) .

- 4 . 0 ∈ 7 )

 407 What is more, due to Theorem 5.1, there exist 0 ∈ R and > 0, only depending on the sets ( ) , ( ) , and a solutionΦ 0 = 0 C ( [ 0 , +∞), 1 (R ) × 2 (R )) of (NLKG) such that for all ≥ 0 ,When dealing with the multi-soliton case, we will need to consider in the present article the euclidean space R , | • | and euclidean balls and spheres of radius > 0 in R , = 1, . . . , ; in particular we define:R ( ) := { ∈ R | | | ≤ } R ( ) := { ∈ R | | | = }.

Φ 1

 1 ≥ 0 := max{ | = 1, . . . , }, Φ 1 ,..., ( ) -Φ 0 ( ) -,..., ( ) -Φ 1 ,..., -1

  sake of contradiction that 1 ,..., = 1 ,..., for some -uples ( 1 , . . . ,) ≠ ( 1 , . . . ,

2 : Control of the stable directions Claim 5 . 17 . 38 ) 2 2 + 2 - 2 + ( ) 2 + O - 2 1 × 2 + 3 1 × 2 .

 25173822222232 We have for all ∈ {1, . . . , }, for all ∈ [ ( ), ],| +, ( )| ≤ -( +4 ) .(5.16)Proof. Due to Claim 5.16 and (5.12), we obtain+, ( ) -+, ( ) ≤ -( +4 ) ,that is, for all ∈ [ ( ), ],Finally we gather (5.35),(5.36), and (5.38) in order to obtain(5.31).Let us introduce the components and 2 of the vector = (5.39)

2 =

 2 O F ,Ω( ) .

2 1 × 2 2 1 × 2 .Proposition 5 . 27 .

 2222527 and G ( ) = O - Proof. Let be the matrix -2 + -( ) 0 0 . For all = 1, . . . , , we have the decom-For all = 1, . . . , , there exist ≥ 0, 0 ≥ 0, and 1 , . . . , ∈ R such that, defining := -Φ 1 ,..., , we have:( ) 1 × 2 ≤ -.

- 1 (

 1 ) 1 × 2 ≤ -( + 0 ) . As ±, -1, ( ) ∓ ±, -1, ( )| ≤ --1 ( ) 1 × 2 + -1 ( ) 2 ≤ -( + 0 )(by the same calculations and arguments as those developed in the proof of Lemma 5.6), we have for all = 1, . . . , -1,( -, -1, ) ≤ -( + 0 -) .Since ↦ → -( + 0 -) is integrable in the neighborhood of +∞ (since ≤ -1 ), and by assumption, -, -1, ( ) → 0 as → +∞, we have by integration | -, -1, ( )| ≤ -( + 0 ) .

- 1 (- 1 (

 11 ) 1 × 2 ≤ -˜ 0 , from which we deduce ( -, -1, ) ≤ -( + ˜ 0 -) .Now, for ∈ {1, . . . , -1}, --˜ 0 ≤ -1 --< 0, we thus have| -, -1, ( )| ≤ -( 0 + ˜ ) ≤ -.For = , we have |( -, -1, ) | ≤ ( -˜ 0 -) . Thus, there exists ∈ R such that -, -1, ( ) → , as → +∞.For ∈ { + 1, . . . , }, we have+ ˜ 0 -) 1 × 2 ≤ sup ≥ =1| -, -1, ( )| ≤ -.

3 1 ×

 1 2 + -+ -3

83 ) 2 - 2 - 2 - 2 -, + ˜ - 2 ≤ 2ℎ 2 + 4 2 4 -, + 4 ˜ 2 - 2 lead 2 +.

 8322222244222 Taking some inspiration in[10, paragraph 4.4.2], we will first show the existence of ≥ 0 such that for all 0 large enough, for all ∈ [ * , ], prove (5.84), let us consider, for some positive constants and ˜ , the functionℎ : ↦ → +, ( ) -, ( ) -˜ -and show that it is always negative on [ * , ], provided and ˜ are well chosen. We computeℎ ( ) = +, ( ) -, + | -, | 2 +, + 2 -, + - + ˜ -.Replacing +, by its expression in terms of ℎ, 2 -, , and -and using that ℎ + to the following estimate of ℎ :ℎ ≥ ℎ -2 1 + 2 | -, | ℎ ˜ --2 | -, | -4 ˜ 2 --8 ˜ 2 | -, | -Now we choose := and ˜ := 2 so that the expressions3 --2 | -, | -4 2 2 -, -8 3 | -, | 2 -,and˜ --2 | -, | -4 ˜ 2 --8 ˜ 2 | -, | -are positive for 0 sufficiently small and 0 large enough. Thus for such values of 0 and 0 , there exists > 0 such that for all ∈ [ * , ],ℎ ( ) ≥ ℎ( )ℎ( ) 2 .

2 -

 2 Now, we have for all ∈ [ * , ],By using Lemma 5.17 and even if it means reducing 0 and increasing 0 , we obtain that for all∈ [ * , ], | -, ( ) + -, ( )| ≤ sup ∈ [ , ] , ( ) + -2 ≤ 10 | -, ( )| + -2 .

  For belonging to [ * , ], we obtain by integration on [ , Passing to the supremum and defining ( ) := sup ∈ [ , ] | -, ( )| on [ * , ], this leads to standard Grönwall argument allows us to see that ( ) ≤ -, which precisely provides the expected estimate of the parameter -, in Proposition 5.32. Then, the similar estimate of the parameter +, follows from the integration of the inequality | +, ( ) -+, ( )| ≤ -2 and finally (5.82) follows from (5.80) and (5.81). For the reader's convenience, let us write explicitly the Grönwall argument. The function ( ) := 10 ∫ ( ) is C 1 on [ * , ] and for all ∈ [ * , 85). Observing that ( ) = 0, it follows that ( ) ≤ -

  1 -uniformes obtenues par Martel et Merle et par Côte, Martel et Merle, et en s'inspirant de Martel [63, section 3], la preuve des estimées se fait par récurrence sur l'indice de régularité de la fonction . Le point clé de la récurrence est d'observer que la fonctionnelle suivante

1.3 Solutions non-dispersives des équations de Korteweg-de Vries gé- néralisées 1.3.1 Solutions non-dispersives et théorèmes de rigidité pour (gKdV)

	Pour l'heure, il est malgré tout remarquable d'observer que le théorème 1.10 permet de dépasser le
	cadre de la classe à décroissance exponentielle, laquelle demeure communément la classe naturelle
	de construction des multi-solitons.
	Les équations de Korteweg-de Vries généralisées présentent la caractéristique d'admettre des pro-
	priétés de type Liouville au voisinage des solitons. Un des premiers résultats en la matière remonte
	à [65, Theorem 1] dans le contexte de l'équation 2 -critique : l'approche repose sur des consi-
	dérations et des techniques propres à (gKdV).
	Puis, ce résultat a été étendu par Martel et Merle au cas 2 -sous-critique [66, 70] et, en faisant
	l'hypothèse d'une proximité suffisante d'un soliton en tout temps, au cas instable avec des non-
	linéarités générales [72].
	Le théorème suivant regroupe les différentes situations.
	Théorème 1.11 (Propriété de Liouville au voisinage d'un soliton; Martel et Merle [65,66,70,72]).
	Soit 0 > 0. Il existe > 0 tel que si ∈ C (R, 1 (R)) est une solution de (gKdV) qui satisfait,
	pour une certaine fonction : R → R de classe C 1 ,
	Néanmoins, nous conjecturons que le résultat d'unicité précédent s'étend à toute la classe des
	solutions telles que
	( ) -( ) 1 → 0 lorsque	→ +∞.

  rappelons que le breather , , 1 , 2 , caractérisé par la vitesse d'enveloppe := 2 -3 2 , la vitesse de phase := 3 2 -2 et les paramètres de translation 1 , 2 ∈ R, a pour expression :

	, , 1 , 2 ( , ) := 2 √	2	arctan	sin( ( --1 )) cosh( ( --2 ))	.	(1.19)
	On pourra consulter notamment Alejo and Muñoz [1] pour l'introduction et l'étude de ces solutions
	dans l'espace de Sobolev 2 (R).
	Un résultat de décomposition en termes de solitions et breathers pour (mKdV) a été développé
	dans [101, Chapter 5, Theorem 5.1] et récemment dans [8, Theorem 1.10] pour des données initiales
	0 génériques (en un sens que nous préciserons au chapitre 3).
	Exploitant ce résultat de décomposition, nous obtenons l'énoncé suivant en ce qui concerne les
	solutions non-dispersives de (mKdV).
	Théorème 1.15 (F.). Soit = 3, soit 0 ∈ S (R) \ {0} générique, telle que la solution globale
	correspondante de (mKdV) soit non-dispersive pour les temps positifs (c'est-à-dire satisfasse
	(1.16)).

  Le résultat précédent est issu de la considération d'intégrales de la forme Par récurrence sur , nous montrons que si pour tout ∈ N, , < +∞, alors pour tout ∈ N, , +1 < +∞. Précisément, sous l'hypothèse que chacune des quantités 0, , . . . , , , +1, est finie, nous établissons que , +1 est finie. Autrement dit, la convergence des intégrales , est démontrée de proche en proche, par procédé « triangulaire » ; par l'intermédiaire des cases grisées, le tableau ci-dessous permet de visualiser toutes les valeurs des indices et à considérer pour montrer que +1, < +∞.

	, ( ) :=	∫	2 ( , ) ( -)
				>
	pour et entiers naturels. H
	H	H	H	1 2 . . .	+ 1
		0		H H
		1
		. . .
		+ 1
		+ 2
		. . .
		+
	D'un point de vue technique, cette obtention triangulaire de la décroissance polynomiale de a
	pour origine l'apparition d'un terme contenant une dérivée d'ordre + 1 en espace sur lorsqu'on
	dérive en temps une fonctionnelle de la forme fonction « poids ». Ce phénomène est évidemment lié à la structure de l'équation (gKdV). ∫ 2 ( , ) , où ( ) désigne une certaine R

  H4) are intimately related to the stability of . Regarding the stable case, we have the following result by Grillakis, Shatah and Strauss [38, p. 341-345] (see also the work by Weinstein [108, 109] and by Maris [62, Lemma 2.4]). Assume that O is open and that the map ↦ → is of class C 1 . Let 0 ∈ O. Under the non-degeneracy assumption that

	2
	. (2.16)
	Assumptions (H3) and (Proposition 2.4.

  .38)

	Remark 2.6. Note that, considering (2.34), (2.36), and (2.37), the property that allows us to obtain
	(2.38) is in fact Im	= Re	. Indeed, this assumption suffices to have: for all ∈ N with

  or in other words locally Lipschitz on C), it results from (2.40) and (2.41) that

  1,∞ (C),

	, +3 -	, , , ˜ 1 ,..., ˜ ( ) ≤	-2 .	(2.51)
	=1

Hence, we conclude from (2.46), (2.47), (2.48), (2.50), and (2.51) that (2.39) holds true.

  Remark 2.9. Note that in (2.140), ˜ seems to depend on (or equivalently on ) but in fact it does not (even if it means changing which does actually depend on ). Indeed, ˜ depends only on the

	constants appearing in (2.74), (2.86), (2.87), (2.94) (linked with the parameters used to define the
	solitons ), on the constants appearing in (2.96), (2.100), (2.127), (2.128) (linked with , with
	the parameters used to define the solitons, and with ( ) 1 which can be chosen less or equal to
	1 provided is sufficiently large, depending on ), and also on universal constants which enable us
	to pass from (2.135) to (2.136), from (2.137) to (2.138), and from (2.139) to (2.140) (on condition
	that is once more sufficiently large, which depends on ). Thus one should read the following
	assertion: there exists ˜ > 0 such that for all satisfying (2.76) and ( ) 1 = O 1 , there
	exists ( ) > 0 such that for all ≥ ( ), (2.140) holds.
	Now take in (2.77) such that > 4 ˜ + 1 (in this way, does not depend on , as emphasized
	in Remark 2.9). Even if it means taking a larger , we can assume

.140)

+ | | 2

˜ = O | | 4 -2 , lorsque | | → +∞ ;

-1 1 in the neighborhood of +∞ (provided is chosen sufficiently large); once more, we observe here that the condition ≤ 3 is important.

˜ 1 +

˜ 1 .

( ) ≤ 0 | | -11 . (3.66)

×

×

+ -2

Remerciements

and also functions on R × R by ∀ ∈ {1, . . . , }, := --1 .

We can check that, for large values of , ( , •) has a smooth profile localized at the "neighborhood" of the -th solitary wave; more precisely we have

for all = 2, . . . , -1:

Besides, for large values of , the following inequalities hold owing to the decay properties of and the support properties of and its derivatives. Proof. The proof, postponed in Appendix, is similar to that of Combet [12, Proof of Lemma 3.9, Appendix A].

Let us introduce the following Weinstein energy functional which is inspired from Martel, Merle and Tsai [START_REF] Martel | Stability in 1 of the sum of solitary waves for some nonlinear Schrödinger equations[END_REF] for dimensions 1 to 3:

One of the main features concerning is the following coercivity property, which turns out to be a key ingredient in our matter.

Proof of Lemma 2.14

Assume that 1 ≤ < ≤ . If 1 < -1 + ( -1 -0 ) , we have ( , ) = 0 . If 1 ≥ -1 + ( -1 -0 ) , then 1 > ,1 for large values of and thus, by (2.5), we have

Assume now that 1 ≤ < ≤ . If 1 > + ( + 0 ) , we have ( , ) = 0 . If 1 ≤ + ( + 0 ) , then 1 < ,1 for large values of and thus we have as before

Thus for all ≠ ,

and of course the same estimate is valid for | 1 ( , ) ( , )|. This proves (2.89). In a similar way, one proves (2.90). Now, let us show how to obtain (2.91). First, notice that it is sufficient to prove (2.91) with instead of . Then,

Hence,

∞ , and

which leads to (2.91).

To finish with, let us observe that 1 ( , ) = 0 if -1 + ( -1 + 0 ) ≤ 1 ≤ + ( -0 ) (and in fact also if 1 ≤ -1 + ( -1 -0 ) or 1 ≥ + ( + 0 ) ). Thus, for ≠ , the proof of (2.92) is just a copy of that of (2.89). Moreover, if 1 ≤ -1 + ( -1 + 0 ) , then 1 < ,1 for large and thus, as before, we obtain

(3.99)

4. (consequence of the monotonicity properties) For all ≥ ≥ 1 , for all = 1, . . . , , (3.100)

(3.101)

Proof of Lemma 3.7. We only give some indications, and particularly the key ingredients. Property (3.99) is obtained by Abel transformation. Now, we focus on the other lines. Estimate (3.97) is a consequence of a localized version around of the coercivity property of the linearized operator around (for all = 1, . . . , ), which holds under the orthogonality conditions (3.79) satisfied by ; we refer to [80, proof of Lemma 4]. To prove (3.98), one has obviously to replace by its definition (3.78). To finish with, integrate the almost monotonicity properties as expressed in Lemma 3.6 between and and use the expression of ( ) in terms of ( ) in order to obtain (3.100) and (3.101). Let us mention furthermore that (3.97), (3.98), (3.100), and (3.101) rely all on classical inequalities used in studying quantities which are localized near the solitons, and which write in the present context as follows: Hence, using (3.109), the triangular inequality, the following estimate

(which is a consequence of Lemma 3.8 below), and (3.110), we have

Lemma 3.8. For all = 1, . . . , , for all ≥ 0, and for all ∈ N * , (• -) -

where := max +1 -1 ({0}) ∩ R * + .

Proof of Lemmma 3.8. By the mean value theorem, we have:

, where -≤ ≤ for all ∈ R. Now, split the preceding integral into three regions: ≤ -, -≤ ≤ + , and ≥ + . In the first and third regions, use the monotonicity of +1 2 . We have:

∞ .

Thus, we obtain:

which puts an end to the proof.

Chapter 4

Pointwise decay of the multi-solitons of the generalized Korteweg-de Vries equation Abstract Focusing on the decay properties of the multi-solitons of the generalized Korteweg-de Vries equations, we obtain that these solutions and their derivatives decrease exponentially in space on the left and inside the soliton region, by exploiting the general dynamics of the flow and asymptotic estimates available for the multi-solitons, respectively. On the right of the last solitary wave, we prove rapid decrease for all derivatives by a novel approach, based on an induction process and the good knowledge of the behavior of the multi-solitons in large time. For global 1 -bounded multi-solitons in particular, exponential decay is also proved to hold on the right.

Introduction

Multi-solitons and main result

We are interested in multi-solitons of the generalized Korteweg-de Vries equations

where ( , ) are elements of R × R and > 1 is an integer.

Recall that (gKdV) admits a family of explicit traveling wave solutions indexed by R * + × R. Let be the unique (up to translation) positive solution in 1 (R) (known also as ground state) to the following stationary elliptic problem associated with (gKdV)

given by the explicit formula

.

CHAPTER 4. POINTWISE DECAY OF THE MULTI-SOLITONS OF (GKDV)

Then for all 0 > 0 (velocity parameter) and 0 ∈ R (translation parameter),

is a global traveling wave solution of (gKdV) classically named soliton solution, where 0 ( )

In this article, we explore some properties of the multi-solitons, built upon solitons and which are defined as follows: Definition 4.1. Let ≥ 1 and consider solitons , as in (4.1) with speeds 0

, is an 1 -solution of (gKdV) defined in a neighborhood of +∞ (resp. -∞) and such that

Recall that several results have been established so far concerning the multi-solitons of (gKdV).

For the original Korteweg-de Vries equation (with = 2) and the modified Korteweg-de Vries equation (corresponding to = 3), given solitons 1 , . . . , with velocity parameters 0 < 1 < • • • < and translation parameters 1 , . . . , ∈ R, the inverse scattering transform led to the existence of a solution which is a multi-soliton both in ±∞ [START_REF] Miura | The Korteweg-de Vries equation: A survey of results[END_REF], with the same velocity parameters in ±∞ but with distinct translation parameters; namely, there is a shift which one can quantify in terms of the . Besides this solution is explicit; for instance, for = 2, it writes (see [START_REF] Gardner | Kortewegde Vries equation and generalizations. VI. Methods for exact solutions[END_REF], [89, section 6], or [START_REF] Hirota | Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[END_REF]): = 6 2 2 ln det , where ( , ) is the × -matrix with generic entry

and , is the Kronecker delta. We refer to Schuur [101, chapter 5, (5.5)] and to Lamb [55, chapter 5] for a formula when = 3.

The study of multi-solitons is motivated by decomposition results obtained for general solutions of (gKdV) with = 2 and = 3; we refer to the soliton resolution properties stated in [START_REF] Chen | Soliton resolution for the modified KdV equation[END_REF][START_REF] Eckhaus | The emergence of solutions for the generalized Korteweg-de Vries equation from arbitrary initial conditions[END_REF][START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF] for instance. Though multi-solitons have already been considered for various models, it is undoubtedly for the gKdV equations that the knowledge about multi-solitons has been most developed. One central result concerns the exhaustive classification of the multi-solitons according to the value of .

Theorem 4.1 (Martel [63]; Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]; Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]). Let > 1 be an integer and let ≥ 1, 0 < 1 < • • • < , and 1 , . . . , ∈ R. If ≤ 5, there exists 0 ≥ 0 and a unique multi-soliton ∈ C ( [ 0 , +∞), 1 (R)) associated with the , , ∈ {1, . . . , }.

(by the choice of and since < ) and that for all ∈ R,

With (5.32) and (5. Set := 2 . In a similar way as in subsection 4.2.1, we take

Then properties (4.10), (4.11), and (4.12) become

We can choose ∈ (0, 1 -2 ). We then introduce for all 0 ∈ R and ≤ 0 0 , 0 ( )

By derivation with respect to , we have

where this time ˜ := -0 + ( -0 ) -( ).

We then obtain Lemma 4.12. There exists 0 > 0 such that for all 0 ∈ R, for all 0 ∈ R and ≤ 0 ,

Chapter 5

Asymptotic -soliton-like solutions of the nonlinear Klein-Gordon equation

Abstract

We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in R 1+ , ≥ 1, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an -parameter family of solutions of (NLKG) which converges exponentially fast to a sum of given (unstable) solitons. For = 1, this family completely describes the set of solutions converging to the soliton considered; for ≥ 2, we prove uniqueness in a class with explicit algebraic rate of convergence.

Introduction

Setting of the problem

We consider the following nonlinear Klein-Gordon equation

where is a real-valued function of ( , ) ∈ R × R and is a C 1 real-valued function on R.

Let us denote by the unique primitive of on R which vanishes in 0. We make the following assumptions:

Ce chapitre fait l'objet d'un article soumis pour publication [START_REF] Friederich | On existence and uniqueness of asymptotic -soliton-like solutions of the nonlinear Klein-Gordon equation[END_REF].

CHAPTER 5. ASYMPTOTIC -SOLITON-LIKE SOLUTIONS OF (NLKG)

The (NLKG) equation classically rewrites as the following first order system in time:

where is the two-vector .

Assumption (H1) for = 1 or assumption (H'1) for ≥ 2 on the nonlinearity ensures that the Cauchy problem is locally well-posed in the energy space 1 (R ) × 2 (R ) [START_REF] Ginibre | The global Cauchy problem for the nonlinear Klein-Gordon equation[END_REF][START_REF] Nakamura | The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces[END_REF]. It is even globally well-posed if one assumes further sufficient smallness on the initial condition.

Recall also that the following quantities are conserved for 1 × 2 -solutions ( , ) of (NLKG'):

• the energy

• the momentum ∫ R { ∇ } ( , ) . Moreover, the structure of the equation is left invariant under the action of R × R by (time and space) translation, and under the action of the Lorentz group (1, ) which consists of the linear automorphisms of R 1+ that preserve the quadratic form ( , 1 , . . . , ) ↦ → 2 -=1 2 . In other words, precising this latter action, for all ∈ R with R -euclidean norm | | < 1 and :

is still a solution to (NLKG), where Λ is the linear transformation with matrix

in the canonical basis of R 1+ . We observe in particular that

We refer to [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF] for further details concerning the Lorentz transformations in all dimensions.

It is well-known that (NLKG) admits a family of solitons indexed by two parameters: the velocity parameter ∈ R with | | < 1 and the translation parameter 0 ∈ R . Let denote the unique (up to translation) positive 1 solution of the following stationary elliptic problem, associated with (NLKG):

which we take as radial; for the record, existence of follows from a standard result of Berestycki and Lions [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF] due to (H2) or (H'1) and uniqueness has been proved in Kwong [START_REF] Kwong | Uniqueness of positive solutions of Δ -+ = 0 in R[END_REF] (in the case where ( ) = | | -1 is the particular power nonlinearity) and in Serrin and Tang [START_REF] Serrin | Uniqueness of ground states for quasilinear elliptic equations[END_REF]. We recall that and its partial derivatives up to order 3 decay exponentially. Then for all ∈ R such that | | < 1, for all 0 ∈ R , the boosted ground state

is the canonical projection R 1+ → R on the last coordinates, is a solution of (NLKG) known as soliton. In the one-dimensional case, this soliton rewrites

Soliton theory concerning (NLKG) has extensively been studied in many articles. One major result is linked to the classification of the solutions with energy near that of the ground state. Dynamics of the solutions of (NLKG) on the threshold energy ( ) = ( ) has been investigated in Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF]. More generally, classification of the solutions with energy less than a quantity slightly larger than the energy of the ground state has been done by Nakanishi and Schlag [START_REF] Nakanishi | Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation[END_REF] and by Krieger, Nakanishi and Schlag [START_REF] Krieger | Global dynamics above the ground state energy for the one-dimensional NLKG equation[END_REF].

Let us also mention that solitons of (NLKG) are known to be orbitally unstable in 1 (R ) by a general property by Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF].

We further develop soliton analysis by exploring solutions which behave as a soliton or a sum of solitons as time goes to infinity.

For all ∈ R such that | | < 1 and 0 ∈ R , let us denote

.

When 0 = 0, we will write instead of ,0 for the sake of simplification.

Drawing on the work by Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], Côte and Muñoz [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF] have developed and proved spectral results adapted to the unstable dynamic around the (vector) soliton . Essential properties which are needed in this paper, as well as the introduction of useful notations, are presented in the next subsection. Note that a similar spectral theory was firstly considered by Pego and Weinstein [START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF] in the context of the generalized Korteweg-de Vries equations.

Starting from this point of view, we are interested in solutions which converge to a soliton or a sum of solitons for large values of ; these solutions are classically known as multi-solitons.

Let us consider an integer ≥ 1 and 2 parameters

We recall the following theorem by Côte and Muñoz which states the existence of at least one multi-soliton.

Theorem 5.1 ( [20]

). There exist 0 , 0 ∈ R and 0 > 0, only depending on the sets ( ) , ( ) , and a solution = ∈ C ( [ 0 , +∞), 1 (R ) × 2 (R )) of (NLKG) such that for all ≥ 0 , ( ) -

=1

, ( )

Notations, review of spectral theory, and multi-solitons

Elements of spectral theory concerning (NLKG)

, we define the scalar product:

and the energy norm

.

Under assumption (H'1) (or (H1) and (H2) in the particular one-dimensional case), the operator := -Δ + -( ) admits a unique simple negative eigenvalue, which we denote by -0 . The kernel of is spanned by ( ) =1,..., [START_REF] Mihai | Existence of nonstationary bubbles in higher dimension[END_REF][START_REF] Mc | Uniqueness of positive radial solutions of Δ + ( ) = 0 in R[END_REF]. Note that for a general nonlinearity and for ≥ 2, the operator possibly counts several and multiple negative eigenvalues. We refer to Côte and Martel [START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF] for the detail of the spectral properties in this case.

With a slight abuse of notation, we still denote by the function defined on R by

that for all , ( ) = ( , ). In the sequel, we sometimes omit the variables and when there is no ambiguity (we work with functions which either depend on time or not).

For all ∈ R with | | < 1, we consider the matrix operator

the matrix := 0 1 -1 0 , and the operator

We define for all = 1, . . . ,

Proposition 5.9 (Côte and Muñoz [20]). We have H , = 0 for all ∈ {1, . . . , } and there exist two functions ±, whose components decrease exponentially in space and such that

where := √ 0 . Moreover there exist unique functions ±, (whose components are exponentially decreasing in space) such that ±, ∈ Span{ ±, }, , , ±, = 0, and ±, , ∓, = 1.

In addition, the following orthogonality properties hold:

±, , ±, = 0 and 0, , ±, = 0.

The following coercivity property turns out to be a crucial tool in our paper.

Proposition 5.10 (Almost coercivity of ; Côte and Muñoz [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF]). There exists > 0 such that for all

.

Multi-soliton results

Let us consider a set of 2 parameters as given in Theorem 5.2 and the associated (vector) solitons = := , , = 1, . . . , . We introduce moreover the vectors:

In particular, let us observe that for all = 1, . . . , , ±, belongs to

There exists ℓ ∈ R such that

we postpone the argument in the appendix. (If = 1, one can take obviously ℓ = 1.) Let us consider the permutation of {1, . . . , } such that

We denote also

We can quantify the interactions between the solitons and the functions ±, and ±, , for = 1, . . . , in terms of the parameter . This is the object of the following Proposition 5.11. We have for all ≠ , for all , ∈ {0, 1, 2}, and for all ≥ 0, ( ), ( ) = O -4 .

±, ( ), ±, ( ) = O -4 .

Compactness argument assuming uniform estimate

The goal of this subsection is to explain how to prove Proposition 5.12; for this, we follow the strategy of Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF] and Côte and Muñoz [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF], both inspired from pioneering work by Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] and Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF]. One key ingredient in the construction is the obtainment of uniform estimates satisfied by a sequence of approximating solutions of (NLKG). We fix ∈ {1, . . . , } and ∈ R. Let ( ) be an increasing sequence of time such that → +∞. Let us consider = ( , ) < ≤ ∈ R -the generic term of a sequence of parameters to be determined, and let be the maximal solution of (NLKG) such that

where := .

Concerning , we claim: Proposition 5.13. There exist 0 ≥ 0 and 0 > 0 (independent of ) such that for each ≥ 0 , there exists ) and such that is defined on [ 0 , ] and satisfies

The take the role of modulation parameters and are to be determined (if indeed possible) so that fulfills (5.11), thus is a natural candidate in order to "approximate" the desired solution which is the object of Proposition 5.13.

We postpone the proof of the previous statement at the next subsection; for the time being, let us assume that Proposition 5.13 is satisfied and let us show how it implies Proposition 5.12. In fact, the existence of is due to the continuity of the flow of (NLKG) for the weak 1 × 2 topology. We explicit the construction of below, following the same strategy as [19, paragraph 2.2, step 2] or [20, section 4].

Proof of Proposition 5.12. We observe that the sequence ( ( 0 ) 1 × 2 ) ∈N is bounded; thus there exist a subsequence of ( ( 0 )) ∈N , say ( ( 0 )) ∈N , and 0 ∈ 1 (R) × 2 (R) such that ( ( 0 )) ∈N converges to 0 in the sense of the weak topology in 1 (R) × 2 (R). Let us consider , defined as the maximal solution of (NLKG) such that ( 0 ) = 0 . Let ≥ 0 . For sufficiently large, ≥ and thus is defined on [ 0 , ]. By a standard result (we refer to [20, Lemma 10] and [104, Theorem 1.2]), is defined on [ 0 , ] and ( ( ))

Moreover, by property of the weak limit,

Now, the remainder of Section 5.2 is devoted to the proof of Proposition 5.13.

Proof of Proposition 5.13

For ease of reading, we will drop the index for the rest of this subsection (except for ), that is, we will write for , for , etc. Let us introduce the following variable (which depends on )

and for all ∈ {1, . . . , }, ±, ( ) := ( ), ±, ( ) (which depends on in particular by definition of = (5.10)). We denote also -( ) := ( -, ( )) < ≤ .

Modulated final data and strategy of the proof of Proposition 5.13

We make the first step in order to determine the appropriate modulation parameter . We obtain as the solution of a well-chosen equation; this is the object of the following Lemma 5.2. There exists 0 ≥ 0 such that for all ≥ 0 and for all ∈ R -, there exists a unique ∈ R -such that ≤ 2 and -( ) = .

Proof. Let us consider the linear application

Its matrix in the canonical basis of R -has generic entry , := +, + ( ), -, + ( ) where ( , ) ∈ {1, . . . , } 2 . Since , = 1 if = and | , | ≤ 0 for ≠ , with 0 > 0 independent of , we have Ψ = + with ≤ 1 2 for large values of . Thus Ψ is invertible (for large) and Ψ -1 ≤ 2. We deduce the content of Lemma 5.2 by taking 0 large enough and by considering, for a given ∈ R -, the element := Ψ -1 ( ).

Roughly speaking, Lemma 5.2 reflects that estimate (5.11) is to be proven by choosing a relevant vector = -( ). The reason why we determine according to the value of -( ) essentially comes from the directions -, , which yield "instability" in some sense (given Claim 5.16 below), and also from definition (5.12) below.

At this stage, we notice that we already have: Claim 5.14. We have:

Let 0 > 0 independent of to be chosen later and ∈ R -( -( +2 ) ) to be determined. We consider the associated data given by Lemma 5.2 and defined in (5.10). Let us define

(5.12) We observe that Proposition 5.13 holds if for all , we can find such that ( ) = 0 . In the rest of the proof, our goal is thus to prove the existence of such an element .

To this end, we will first of all improve the estimate on ( ) 1 × 2 which falls within the definition of ( ). This is the object of the following paragraph. Then, we will only need to care about the second condition, which implies a control of -( ).

Improvement of the estimate on

For notation purposes and ease of reading, we sometimes omit the index and also write O ( ( )) in order to refer to a function which a priori depends on and such that there exists ≥ 0 (independent of ) such that for all large and for all ∈ [ * , ], | ( )| ≤ | ( )|.

Lemma 5.3. There exists

The whole subsection consists of the proof of this lemma.

Step 1: Estimates on ±,

Let us begin with the computation of the time derivative of .

Claim 5.15. We have for all ∈ {1, . . . , },

Proof. Claim 5.15 follows from the fact that both and Φ satisfy (NLKG') and is also a consequence of the following Taylor inequality ( is

Now, we are in a position to prove the following estimate on ±, .

Claim 5.16. For all ∈ {1, . . . , } and for all ∈ [ ( ), ], we have

(5.14)

Proof. Let ∈ {1, . . . , }. By means of (5.13) and since ±, = -±, , we compute

Let us notice first that

We have

and

Similarly, we have

and

Indeed, we notice that

Step 3: Control of the unstable directions for ≤ Claim 5.18. We have for all ∈ {1, . . . , }, for all ∈ [ ( ), ],

| -, ( )| ≤ -( +4 ) .

(5.17)

Proof. As in the preceding step, we have for all ∈ {1, . . . , } and ∈ [ ( ), ],

-, ( )

which writes also

For ≤ , we have ≤ , and so by integration, we obtain

But again from Claim 5.14 and Lemma 5.2, we infer

Step 4: Control of a Lyapunov functional satisfying a coercivity property Let us consider

We define for all = 1, . . . , -1,

, and then

Recall that the permutation has been chosen so that -1

Now, let us introduce for all ∈ {1, . . . , }

, and

By means of Proposition 5.10 and a usual localization argument [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of solitons for subcritical gKdV equations[END_REF], we obtain that F is coercive on a subspace of 1 × 2 of finite codimension. More precisely, there exists > 0 such that

(5.19)

We state the following control about the derivative of F : Claim 5.19. For 0 large and for all ∈ [ ( ), ],

(5.20)

Proof. Let us rewrite F , differently, using the notations developed in the introduction. Relying on integrations by parts, our computations lead to:

.

Thus

(5.21)

We immediately have

, .

Besides

Since ( ) is a self-adjoint operator, we have

By a straightforward calculation, we have moreover

At this stage, we thus obtain

Now, by (5.13), we write

where

Let us deal with 1 : we observe that

We have

In addition, we have

Note that the last line of the previous equality is a consequence of the following observation: if ( ) = , we have H +, = +, and

Gathering the preceding computations yields

hence the expected claim, by summing on .

Step 5: Control of the directions

To obtain a control of the scalar products , which is more precise than the a priori control by 1 × 2 , let us introduce the following modulated variable ˜ :

where ( ) ∈ R, = 1, . . . , are chosen so that for all = 1, . . . , , ˜ ( ), ( ) = 0. Existence and uniqueness of the family ( ( )) ∈ {1,..., } are justified by the fact that the (interaction) × -matrix with generic entry ( ), ( ) is invertible for large enough. Notice that

(5.23)

The functional F ˜ ( ), defined as F ( ) by changing in ˜ , satisfies the following coercivity property:

.

(5.24)

We have

and we have moreover by Proposition 5.11 and (5.22).

Claim 5.20 (Estimate on ˜ ). We have

Proof. Let belong to [ ( ), ]. We obtain by (5.24) and by integration of (5.20) on [ , +∞) (which is indeed possible by definition of ( )) that

Using the estimate on ±, provided by the definition of ( ) and Claim 5.17, we then infer:

This concludes the proof of the claim.

Claim 5.21 (Control of the modulation parameters).

We have for all = 1, . . . , ,

Proof. By definition of the modulation parameters , we have ˜ , = 0. Thus, we have by differentiation with respect to :

By Proposition 5.11, we have for ≠ ,

and for all ,

We deduce that

We have in addition

What is more,

again by Proposition 5.11. Hence,

Now, gathering (5.22), Claim 5.20, and Claim 5.21, we immediately deduce the expected estimate of 1 × 2 , which ends the proof of Lemma 5.3.

Control of the unstable directions for > and end of the proof

To control -= -, < ≤ and eventually obtain the following statement, we resort to a classical topological argument, already set up in [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF] and initially developed by Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the 2 -supercritical gKdV and NLS equations[END_REF].

Lemma 5.4. For 0 large enough, there exists

The proof follows that of Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]. We write it below for the sake of completeness.

Proof. We first choose 0 sufficiently large such that 0

Then, we have by Lemma 5.3

Assume, for the sake of contradiction, that for all ∈ R -( -( +2 ) ), ( ) > 0 . As ( ( )

, by definition of ( ) and continuity of the flow, we have:

(We recall that -( ) = ( -, ( )) < ≤ .) In other words, the map

is well-defined. Now, we aim at showing that M is continuous and that its restriction to R -( -( +2 ) ) is the identity.

Let ∈ [ 0 , ( )] be such that is defined on [ , ] and, by continuity,

We consider, for all ∈ [ , ]:

Claim 5.22. For 0 large enough, and for all ∈ [ , ] such that N ( ) = 1, we have:

Proof of Claim 5.22. Let us start from estimate (5.18): for all ∈ { +1, . . . , }, for all ∈ [ , ],

-, + -, ≤ -( +4 ) .

Thus we obtain for all ∈ { + 1, . . . , }, -,

Then, summing on ∈ { + 1, . . . , } leads to

Therefore we can estimate:

Hence we have for all

where = 2( +1 --2 ) > 0 by definition of . In particular, for all ∈ [ , ] satisfying N ( ) = 1, we have:

Now, we fix 0 large enough such that -2 0 ≤ 2 . Thus for all ∈ [ , ] such that N ( ) = 1, we have N ( ) ≤ -2

.

Finally, we claim that ↦ → ( ) is continuous. Indeed, let > 0. By definition of ( ) and by Claim 5.22, there exists > 0 such that for all ∈ [ ( ) + , ], N ( ) < 1 -, and such that N ( ( ) -) > 1 + . But from continuity of the flow, there exists > 0 such that for all ˜ satisfying ˜ -≤ , we have

.

We finally deduce that ( ) -≤ ( ˜ ) ≤ ( ) + .

Hence, ↦ → ( ) is continuous.

We then obtain that the map M is continuous. What is more, for ∈ R -( -( +2 ) ), as N ( ) ≤ -( +1 --2 ) < 0, we then deduce by definition of ( ) that ( ) = , and thus, M ( ) = .

The existence of such a map M contradicts Brouwer's fixed point theorem. Thus, we have finished proving Lemma 5.4.

Classification under condition of the multi-solitons of (NLKG)

Let ≥ 2 and 1 , . . . , , 1 , . . . , be 2 parameters as in Theorem 5.2. Let be a solution of (NLKG) such that

for some > 3.

The goal of this section is to prove the existence of 1 , . . . , ∈ R such that = Φ 1 ,..., .

Here again, we make the proof for = 1.

We denote by a multi-soliton solution associated with these parameters, satisfying (5.7) and Φ := . Let us consider := -Φ = . Obviously,

Our first objective is to improve this comparison, and namely to pass from the polynomial decay to an exponential one.

Exponential convergence to 0 at speed

Introduction of a new variable by modulation

In a standard way, we modulate the variable in order to obtain suitable orthogonality properties, making it possible to obtain crucial estimates when we apply the spectral theory available for (NLKG). Lemma 5.5. There exists 0 > 0 and C 1 functions : [ 0 , +∞) → R and : [ 0 , +∞) → R for all = 1, . . . , such that, defining

we have for all = 1, . . . , and for all ≥ 0 : ( ), ( ) = 0 (5.26) ( ), -, ( ) = 0.

(5.27)

Moreover, we have for all = 1, . . . , :

(5.28)

(5.29)

Proof. This lemma follows from the consideration of the system with unknown variables and which is obtained by replacing by its definition in (5.26) and (5.27). See also [START_REF] Côte | On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations[END_REF] for similar considerations in the case of modulation for the nonlinear Schrödinger equations.

Control of the +, and -, directions

Define ±, := , ±, for all = 1, . . . , . We claim: Lemma 5.6. The following bounds hold: for all = 1, . . . , , for all ≥ 0 ,

Proof. The proof is in a similar fashion as that of Claim 5.16. We note that

Control of the remaining modulation parameters

Lemma 5.7. For all = 1, . . . , , we have

(5.30)

Proof. We do not detail the proof of this lemma which is similar to Claim 5.21. It suffices to start by differentiating the orthogonality relation , ( ) = 0 with respect to and then to control terms by means of

Study of a Lyapunov functional

Taking some inspiration in [START_REF] Martel | Construction of multi-Solitons for the energy-critical wave equation in dimension 5[END_REF][START_REF] Yuan | On multi-solitons for the energy-critical wave equation in dimension 5[END_REF][START_REF] Yuan | Construction of excited multi-solitons for the 5D energy-critical wave equation[END_REF], we consider for all ≥ 0 :

where is defined as follows.

To begin with, recall that the parameters are ordered in such a way: -1 < (1) < • • • < ( ) < 1; let us denote, for some small > 0 which will be determined later:

We then define for all ≥ 0 and for all ∈ R:

For all ≥ 0 , ( ) is a piecewise C 1 function.

Set Ω( ) := =1 ( ¯ , +1 ). It follows from the definition of that

Lemma 5.8. There exists > 0 such that

Proof. We essentially have to use the identity 2 = 2 -+ ( ) -( ) in the expression of F ( ). We compute

(5.34)

Hence, collecting (5.32), (5.33), and (5.34),

(5.35)

Lastly, observe that

( ) (( ) + ( ) ) .

On the one hand,

(5.36)

Indeed, for all ∈ [ ( ), ¯ ], we have ( ( ) ) ( , ) + ( , ) ( ( ) ) ( , ) = 0. On the other, for ∈ Ω( ), there exists ∈ {1, . . . , } such that ¯ ≤ ≤ +1 . Then ( ¯ -) ≤ -≤ ( +1 -)

As a consequence, we have for all ∈ Ω( )

where 0 < < min =1,..., -1 { ( +1) -( ) }.

Then we infer

with

Integrating by parts, we obtain

Consequently,

We deduce from Proposition 5.23 and Lemma 5.10 the following "weak" monotonicity property.

Corollary 5.11. We have for all ≥ 0 , -F ( ) ≤ F ( )

(5.42)

Proof. From Lemma 5.6, we obtain: for all = 1, . . . , , for all ≥ 0 ,

(5.43)

Thus, we have

We now make use of the following property satisfied by F which is a consequence of a localized version of Proposition 5.9 (we refer to [76, proof of (4.12) and (4.21)] for similar considerations in the case of the energy-critical wave equation):

to deduce that

We are now in a position to prove Proposition 5.24. Even if it means taking a larger 0 , we have for all ≥ 0

| -, ( )|.

(5.45)

Proof. Multiplying the estimate obtained in Corollary 5.11 by , we have for all ≥ 0 ,

1 × 2 are integrable functions of on [ 0 , +∞) and F ( ) → 0 as → +∞, we infer that

By the coercivity property satisfied by F , we thus obtain

In other words, there exists ≥ 0 such that for all ≥ 0 ,

+, ( )

+, ( )

(5.47)

From Lemma 5.6 we recall the estimate

which is equivalent to

Integrating the preceding inequality (which is indeed possible), we deduce:

(5.48) Now,

By Lemma 5.7, it follows that

(5.49) Observing that the quantity ∫ +∞ ( ) 1 × 2 makes sense and tends to 0 as → +∞ (because > 1) and that

we deduce that for sufficiently large

Now, in order to obtain an estimate of ( ) 1 × 2 and of ∫ +∞ ( ) 1 × 2 in terms of ( ) 1 × 2 and sup ≥ ( ) 1 × 2 , we replace (5.48) in (5.47). We notice that the following well-defined quantities tend to 0 as → +∞ (because > 3 and by the choice of < -1):

.

and use (5.50). We then obtain for sufficiently large

Then, we claim Proposition 5.25. We have for all ≥ 0 , for all = 1, . . . , ,

(5.51)

Proof. From Lemma 5.6 and Proposition 5.24, it results that for all = 1, . . . , , for all ≥ 0 ,

(5.52)

Then, for all = 1, . . . , ,

(5.53) Let us denote A := =1 2 -, . Summing on = 1, . . . , , we have in particular:

≤ A, we obtain the existence of > 0 such that for all ≥ 0 ,

Lastly we observe that : ↦ → -+ A ( )

.

By Lemma 5.26 in Appendix, we obtain A ( ) ≤ -2 1 for ≥ 0 . Consequently, for all = 1, . . . , , | -, ( )| ≤ -1 .

Gathering Propositions 5.24 and 5.25, we deduce Proposition 5.26. There exists ≥ 0 such that for sufficiently large,

Identification of the solution

Recall that we have constructed in Section 5.2 a family of multi-solitons ( 1 ,..., ) such that for all = 1, . . . , , for all ≥ 0 ,

(5.54)

(We can always assume that < min 1 , min =2,..., { --1 } ). Following the strategy of Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF], our goal is to establish Let us conclude the proof of Proposition 5.27. We define at this stage := -Φ 1 ,..., . We immediately have ( ) = -1 ( ) + Φ 1 ,..., -1 ( ) -Φ 1 ,..., ( ).

Then,

What is more, -, , ( ) = ( ), -, ( )

For = 1, . . . , -1, we have:

This finishes the induction argument.

Finally we obtain that = Φ 1 ,..., by means of Corollary 5.12. For sufficiently large, ( ) 1 × 2 = 0.

Proof. As in the preceding proofs, the following bounds hold:

Since for all = 1, . . . , , -, , ( ) → 0 as → +∞, we obtain by integration of (5.58) on [ , +∞):

Then, using (5.57), we obtain

This implies that ( ) 1 × 2 = 0 for sufficiently large.

Construction of a one-parameter family of solutions converging to a soliton

The goal of this section is to prove the existence part in Theorem 5.6. Once again we restrict our focus to = 1.

Outline of the construction

Let ∈ R.

Let ( ) ∈N be an increasing sequence of real numbers which tends to +∞ and, for all ∈ N, define as the maximal solution of (NLKG) such that

with obvious notations. We aim at proving the following key proposition:

Proposition 5.29. There exist 0 ≥ 0 and 0 ≥ 0 such that for large,

To this end, we will set up a bootstrap argument and show Proposition 5.30. There exist 0 > 0, 0 > 0, and 0 ≥ 0 such that for sufficiently large, if there exists * ∈ [ 0 , ] such that for all ∈ [ * , ],

( ) -( ) -- +, ( ) 1 × 2 ≤ 0 , (5.61)

(5.62)

Let us show how to deduce Proposition 5.29 from Proposition 5.30.

Proof of Proposition 5.29. Assume momentarily that Proposition 5.30 holds true. Let us consider 0 and 0 as in Proposition 5.30 and suppose (even if it means enlarging 0 ) that 0 -2 0 ≤ 0 2 . We define for all such that > 0 :

By (5.59) and by continuity in time of , , and +, , * is indeed well-defined and we necessarily have 0 ≤ * < . Since (5.61) implies (5.62), for all ∈ [ * , ],

Let us assume for the sake of contradiction that * > 0 for some . Then, observing the preceding inequality, we obtain (again by continuity in time of , , and +, ) the existence of > 0 such that * -≥ 0 and for all ∈ [ * -, ],

( ) -( ) -- +, ( )

This contradicts the definition of * as an infimum. Hence * = 0 and (5.61) (and thus (5.62)) holds on [ 0 , ] for all . This achieves the proof of Proposition 5.29.

The existence of (and ), as stated in Theorem 5.6 is a consequence of Proposition 5.29 and the continuity of the flow of (NLKG) for the weak 1 × 2 topology. We will not detail the construction of considering that it is a sort of cut and paste of what was done in order to prove Proposition 5.12 in the context of multiple solitons.

Similarly, we do not repeat the arguments exposed at the beginning of section 5.2 which justify that the map ↦ → is one-to-one. We devote the next subsection to the proof of Proposition 5.30.

Proof of Proposition 5.30

We assume that ( ) is defined on some interval [ * , ] and satisfies (5.61). We want to show that (5.62) holds, provided that the parameters 0 and 0 are well chosen.

In this subsection again, for notation purposes and ease of reading, we sometimes omit the index and also write O ( ( )) in order to refer to a function which a priori depends on and such that there exists ≥ 0 (independent of ) such that for all large and for all ∈ [ * , ], | ( )| ≤ | ( )|.

Step 1: Set up of a modulation argument Lemma 5.13. For 0 ≥ 0 sufficiently large and 0 > 0 sufficiently small, there exists a unique C 1 function : [ * , ] → R such that if we set

( ), ˜ ( ) = 0.

(5.63)

Moreover there exists 1 > 0 such that for all ∈ [ * , ], Proof. The existence of such that (5.63) is granted and the existence of 2 > 0 such that

are standard consequences of the implicit function theorem. Now, let us prove (5.65). For this, we notice that = satisfies the following equation:

) where ˜ ( , ) = ( , -( )).

Since

, ˜ = 0, we have:

Observing moreover that

by Taylor formula ( is C 2 ), we have thus:

Notice that we have used +, , = 0 (see Proposition 5.9). We now observe that

where ˜ is the matrix operator defined like by replacing by ˜ , that is

We obtain:

Finally we can choose large enough such that

and we can take 0 > 0 sufficiently small such that

and such that 2 0 ≤ 1 (then

. Consequently for large and 0 small, we have:

for some constant 3 > 0. Finally, take 1 := max( 2 , 3 ) to obtain Lemma 5.13.

Step 2: Control of particular directions

Let us denote

where ˜ ±, ( ) := ±, ( , • -( )).

Lemma 5.14. We have for all ∈ [ * , ],

Proof. Observe that

By (5.65), we obtain

In addition, defining H like H by replacing by ˜ , we have by (5.66)

Since

we always obtain

We have in addition

and ˜ , ˜ +, by Proposition 5.9. Hence, gathering (5.69), (5.70), (5.71), and (5.72), we infer:

which proves Lemma 5.14.

Step 3: Exponential control of

Let us introduce the functional

(We recall that ˜ is defined in (5.67).)

Lemma 5.15 (Control of F ).

There exists > 0 such that for 0 sufficiently large, we have for all such that ≥ 0 , for all ∈ [ * , ]:

Proof. We have

Notice that we have used ˜ ˜ +, = ˜ +, and ˜ +, , ˜ +, = 0. Now let us focus on the quadratic term ˜ ( -˜ ), -˜ . This term rewrites

where 1 and 2 are defined as follows:

In a compact manner, we can write:

Considering that and are solutions of (NLKG), the energy and the momentum of these solutions as defined in introduction are conserved. Thus, there exists ∈ R such that

Thus

) .

(5.75)

The last integral is zero by the equation satisfied by . By means of Taylor inequality, we claim

(5.76) Hence, collecting (5.74), (5.75), and (5.76), we have:

On the other hand, we immediately have

Corollary 5.16. There exists > 0 such that for 0 sufficently large, we have for all such that ≥ 0 , for all ∈ [ * , ]:

Proof. From Lemma 5.14 we deduce that

Given that - +, ( ) = 0, we deduce by integration of the preceding inequality that:

Now the corollary follows from the combination of this last result with Lemma 5.15.

Lemma 5.17. There exists > 0 such that for 0 > 0 sufficiently small and 0 ≥ 0 sufficiently large, for all such that ≥ 0 , for all ∈ [ * , ]:

-, ( ) + -, ( ) ≤ min sup

, 2 +, ( ) + 2 -, ( ) + - .

(5.77)

Proof. Due to Proposition 5.10, we have on the one hand

On the other,

Now, inserting (5.79) into (5.78), we obtain

provided 0 is chosen small enough such that sup ∈ [ , ] ( ) 1 × 2 ≤ 1 4 (for all ) and 0 is chosen large enough such that -0 ≤ 1 4 .

Then, combining (5.80) and the estimates obtained in Lemma 5.14, we deduce:

Proof. Let us consider this time the derivative of F (and not the functional F itself). From the definition (5.73) and the symmetric property of ˜ for •, • , we immediately obtain:

, where 1 (resp. 2 ) is the first (resp. the second) component of . Replacing by its expression obtained in (5.66) and noticing that

we have:

Let us analyze each term appearing in the preceding decomposition. Since ˜ is self-adjoint, we infer:

Moreover,

We notice that (5.65) implies

(5.89)

Defining ˜ := -2 + -( ˜ ), it remains us to examine

On the one hand, we observe that

and on the other

Gathering the above lines, we deduce

Using the estimate of 1 × 2 obtained in Proposition 5.32, we infer:

(5.90)

By integration of (5.90), we obtain for all ∈ [ * , ]:

(5.91) At this stage, using once again the coercivity property provided by Proposition 5.10 and the estimates on ±, given in Proposition 5.32, and taking 0 large enough so that -< 1 2 for all ≥ 0 , we have:

( ) 1 × 2 -2 + -4 .

We now deduce the existence of > 0 such that for all , for all ∈ [ * , ],

. Now, Proposition 5.30 is obtained as a corollary of Proposition 5.33. The triangular inequality implies ( ) -( ) -- +, ( )

and since ( ) = 0, the result follows from the integration of

.

Classification of the asymptotic soliton-like solutions

Let us consider a solution of (NLKG), denote = , and assume that

We want to show that equals to , for some ∈ R. In this section again, we consider the one-dimensional case. .

Modulation of and coercivity property

Then, introduce the functional F defined as follows: for all ≥ ,

With analogous notations as that employed in the previous section, we denote ˜ ±, ( , ) := ±, ( , -( )).

Proposition 5.34. There exists > 0 such that for all ≥ :

(5.96)

Proof. We observe that F ( ) = ˜ ( ), ( ) so that Proposition 5.34 follows from Proposition 5.10 and from (5.92).

Exponential control of

We improve the control of ( ) 1 × 2 , proceeding as in section 5.4.

Control of the functional F

Proposition 5.35. We have

(5.97)

Proof. Let us take again the proof of Lemma 5.15 (replacing in that proof by 0 here). We obtain in the same way the existence of ∈ R such that

By Taylor inequality,

It follows that

On the other hand, F ( ) = O ( ) 2 1 × 2 ; thus = 0 and Proposition 5.35 is proved.

Control of the unstable directions

Let us denote ±, ( ) := ( ), ˜ ±, ( ) .

Lemma 5.19. We have

Proof. The proof follows the same lines as that of Lemma 5.14 (by taking = 0). We compute .

(5.98)

Proof. For large enough, we obtain as a consequence of Lemma 5.34 and Proposition 5.35:

Now, we deduce from Lemma 5.19 the following differential system, for some constant ≥ 0 and for all large enough:

Then, the argument exposed in [10, paragraph 4.4.2] shows that

and allows us to conclude to (5.98).

Proposition 5.37. There exists ≥ 0 such that for large enough,

Proof. This is an immediate consequence of (5.98) and (5.99). (5.102)

Identification of with

Proof. We have from (5.100):

-, ( ) ≤ -.

(5.103)

Thus the derivative of ↦ → -, ( ) is integrable in the neighborhood of +∞. Hence, there exists ∈ R such that -, ( ) → as → +∞. We finally obtain Lemma 5.20 by integration of (5.103). (

Now, we have

(5.106)

Hence, Lemma 5.22 is obtained as a consequence of Proposition 5.37, Lemma 5.21, and the estimate of ( ) 1 × 2 given by (5.4), that is

Let us decompose ( , ) as follows:

where +, ( ) = ( ), +, ( , • -∞ ) , -, ( ) = ( ), -, ( , • -∞ ) and

Then, we have

Thus, by Proposition 5.10, there exists ≥ 0 such that ⊥ ( ) 2 1 × 2 ≤ ⊥ ( ), ⊥ ( ) .

(5.108)

We claim Lemma 5.23. The following assertions hold:

1.

( ), ( ) = ⊥ ( ), ⊥ ( ) + 2 +, ( ) -, ( ).

( ), ( )

Proof. The first assertion in Lemma 5.23 is obtained by the decomposition of in terms of ⊥ (5.107) and by means of the following properties (see Proposition 5.9):

±, , ±, = 0, ±, , ∓, = 1, and = 0.

Let us now prove assertion [START_REF] Berestycki | Non linear scalar field equations, I. Existence of a ground state[END_REF]. Let := -; recall that = and = .

We observe that

Using the fact that and satisfy (NLKG), we obtain ( ), ( )

By Lemma 5.22, assertion (2) is thus proved.

In order to prove (3) and ( 4), let us write

Then,

Note that we have used that ±, ( , • -∞ ) = -±, ( , • -∞ ). We then deduce, in a similar way as for Proposition 5.36:

In addition we deduce -, ( ) ≤ ( ) 1 × 2 .

Arguing similarly as for +, ( ), we then obtain -, ( ) ≤ -∫ +∞ ( ) 1 × 2 . This is due to the fact that ↦ → ( ) 1 × 2 is integrable in +∞ and the fact that, by (5.105), (5.106), and (5.102),

which explains that -, ( ) → 0 as → +∞. It remains to prove [START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF]. By definition of ( ) and the decomposition (5.107) of ,

since ±, , = 0 (we refer to Proposition 5.9). Hence,

Now, using that = 0 and 2 -+ ( ) = , we have:

Finally, let us notice that

and assertion (4) indeed holds.

It follows from (5.108) and ( 1), (2), and (3) in Lemma 5.23 that for large,

we deduce that for large,

.

(5.111) Lemma 5.24 (Estimate of ( ) 1 × 2 in terms of ∫ +∞ ( ) 1 × 2 and ⊥ ( ) 1 × 2 ). We have the existence of ≥ 0 such that for sufficiently large:

Proof. Using the decomposition (5.107) of , we have:

By Lemma 5.23 which gives estimates of | +, ( )|, | -, ( )|, and ( ) in terms of ( ) 1 × 2 and ⊥ ( ) 1 × 2 , we obtain:

Hence, even if it means taking larger values of , this finishes proving Lemma 5.24.

Lemma 5.25. We have

Proof. From (5.111), it follows that

.

(5.115) Gathering Lemma 5.24, (5.111), and (5.115), we obtain:

. Now integrating the preceding inequality (which is obviously possible) leads to:

Or more simply

.

Even if it means considering larger values of , we obtain:

which immediately implies the expected result.

Now, let us show

Proposition 5.38. For sufficiently large, ( ) = 0.

Proof. Gathering Lemma 5.24 and Lemma 5.25, we infer that for large:

From (5.111) and (5.115), we deduce

Now, it results from Lemma 5.25 again that

Thus we deduce that ( ) 1 × 2 = 0 for large values of .

Finally let us observe below that ∞ = 0 so that = .

Proposition 5.39. There exists 0 ≥ 0 such that for all ≥ 0 , ( ) = ( ).

Proof. On the one hand, we have

On the other hand, we have ( ) -( ) 1 × 2 → 0 as → +∞. Since we have ( ) = ( , • -∞ ), it follows from the triangular inequality that

Hence ∞ = 0 by the following claim, which is a consequence of Taylor formula.

Claim 5.40.

There exist ℎ 0 > 0, 0 > 0, and > 0 such that 1. if |ℎ| ≤ ℎ 0 , then ℎ 2 ≤ (• + ℎ) -2 1 ≤ 4 ℎ 2 ;

2. if |ℎ| > ℎ 0 , then (• + ℎ) -2 1 > 0 .

Appendix

Extension of the proofs to higher dimensions

The main parts of the proofs remain obviously unchanged. Essentially three notable adaptations are to be made, passing from the one-dimensional case to higher dimensions.

In a first instance, one has to be careful about how establishing several estimates. Although all estimates we have proved in the previous sections (in dimension 1) are identical for general , the way we establish them when ≥ 2 can be altered. For example, we point out that it is no longer possible to use the Sobolev embedding 1 ↩→ ∞ when ≥ 2. Particularly, multi-solitons in dimension ≥ 2 do not necessarily take values in ∞ (R ) and in order to estimate a quantity like Secondly, in view of Proposition 5.9, one has to take into account, for all = 1, . . . , the directions which generate the kernel of the operator H when we practice modulation in dimension . For instance, in Lemma 5.5, we would define as follows: A third change to be done concerns the way we define the different Lyapunov functionals which are studied throughout the article. To deal with dimensions greater or equal than 2, we reduce the problem to the case of a one-dimensional variable. For instance, let us explain how to generalize Step 4 in subsection 5.2.2 to all dimensions. The subset

of R is of zero Lebesgue measure. Hence, there exists ℓ ∈ R such that for all ≠ , ℓ • ( -) ≠ 0.

In particular ℓ ≠ 0 and, even if it means considering ℓ |ℓ | , we can assume that |ℓ| = 1, so that ∀ = 1, . . . , , |ℓ • | < 1. Now, defining ˜ := ℓ • , and even if it means changing the permutation , we have

Then, the direction described by ℓ is to be favored: we consider the following cut-off functions:

.

CHAPTER 5. ASYMPTOTIC -SOLITON-LIKE SOLUTIONS OF (NLKG)

At this stage, the definition of the functions in terms of the is kept unchanged and the corresponding Lyapunov functional is to be written:

Proof of Corollary 5.1

The proof is an immediate adaptation of that of Proposition 4.12 in [START_REF] Combet | Construction and characterization of solutions converging to solitons for supercritical gKdV equations[END_REF].

Let > 0 and denote := -ln( ) . In the sense of the 1 × 2 -norm, we have: Then, 1 ( + , • + ) -( ) 1 × 2 → →+∞ 0 so that there exist ˜ ∈ R and 0 = 0 ( ˜ ) ∈ R such that for all ≥ 0 , ˜ ( ) = 1 ( + , • + ).

But on the other hand,

Hence, ( -˜ ) - +, ( ) = O -2 , which implies = ˜ . Consequently, ( ) = 1 ( + , • + ).

If < 0, we have just to repeat the above argument withinstead of .

Lastly, let us identify 0 . Given that is a solution of (NLKG) which satisfies (5.5), Theorem 5.2 provides the existence of ∈ R and of 0 ∈ R such that for all ≥ 0 , ( ) = ( ). Since satisfies (5.4), we deduce that -+, ( ) 1 × 2 ≤ -2

.

Thus = 0 and 0 = is defined for all ∈ R. (5.121) By a standard Grönwall argument, we conclude to the existence of > 0 such that for all ≥ 1 , ( ) ≤ , which implies the desired result. For the sake of completeness, let us explicit this argument. We define ( ) := exp -∫ We then infer This achieves the proof of Lemma 5.26.

A result of analytic theory of differential equations