Je Salue Également

Benjamin L Candice

Clement G Chifaa

Diego Clement C Denis

Fu-Hsuan Fanny

Mingmin, Mitja, Virgile Lucas C Louis

Viviana Je

À ma mère, dont l'amour et les sacrifices ont rendu cette réussite possible. À mon père, dont la mémoire m'inspirera toujours à viser les étoiles.

iii Remerciements J'exprime tout d'abord ma profonde gratitude à mon directeur de thèse Stefan Le Coz et mon co-directeur Tai-Peng Tsai, pour leur accompagnement tout au long de cette thèse.

Stefan, tu m'as encouragé et guidé avec patience, tout en faisant preuve de qualités humaines et scientifiques exceptionnelles. Ta présence bienveillante a été une source d'inspiration constante pour moi, et tes connaissances approfondies ont grandement contribué à l'enrichissement de mon travail. Ton soutien m'a permis de surmonter les difficultés et de repousser mes limites. Je te suis profondément reconnaissante d'avoir assumé la direction de mon doctorat ainsi que pour tout le savoir que tu m'as transmis et pour tout ce que j'ai appris grâce à toi.

Tai-Peng, your remarkable scientific expertise, valuable recommendations, and support have greatly contributed to the quality of my work. I want to offer my heartfelt thanks for the knowledge and insights that I gained during my time in Vancouver under your guidance. Moreover, I sincerely thank you for your commitment and dedication during this thesis.

J'exprime mes sincères remerciements à Mathieu Colin et Nabile Boussaïd pour avoir consacré leur temps à la lecture de mon manuscrit et à l'évaluation de mon travail. Je tiens également à remercier chaleureusement Hajer Bahouri et Mihai Maris d'avoir accepté de faire partie de mon jury.

Je remercie également le personnel administratif, les agents d'entretien ainsi que les chercheurs et enseignants-chercheurs de l'Institut de Mathématiques de Toulouse.

Pendant ces 4 années passées au laboratoire, la présence des doctorants a été une véritable source de joie et de plaisir. Je tiens à vous remercier du fond du coeur pour le temps que nous avons partagé ensemble. Votre compagnie a fait de ces années parmi les plus merveilleuses de ma vie. Je suis infiniment reconnaissante pour les rires partagés, les discussions enrichissantes, votre humour, votre intelligence, votre soutien et votre gentillesse. Merci sincèrement pour toutes les expériences, les voyages, les soirées et tous les autres moments précieux que nous avons partagés.

J'adresse mes plus affectueux remerciements à Paola d'avoir été une amie précieuse et véritable, je ne pourrai jamais te remercier assez, Corentin d'être celui sur qui je peux toujours compter, tu es le meilleur, Javi qui apporte une dose de joie et de folie à chaque instant, Alain le plus sympa et toujours souriant. Votre amitié m'a apporté tellement.

Merci à Alexandre (et sa passion pour les jeux), Anthony (et ses blagues), Maxime (qui surveille mon heure d'arrivée), Nicolas O, Etienne, Armand, Joachim, Nicolas E, Sophia (qui me met la pression pour m'habiller

Chapter 1

Introduction

Nonlinear Schrödinger equation

We consider the one dimensional nonlinear Schrödinger equation

i∂ t u + ∂ 2 x u + f (u) = 0, u(0, x) = u 0 (x), (1.1.1)
where u : R t × R x → C, the nonlinearity f : C → C is defined for any z ∈ C by f (z) = g(|z| 2 )z with g ∈ C 0 ([0, +∞), R) ∩ C 1 ((0, +∞), R), g(0) = 0 and lim s→0 sg (s) = 0. Moreover, we assume that there exist C > 0 and 1 < p < ∞, such that |s 2 g (s 2 )| Cs p-1 for s 1. A typical example for f is the sum of power type nonlinearities given by

f (u) = m j=1 a j |u| p j -1 u, a j ∈ R, 1 < p 1 < ... < p m < ∞.
There are two types of solutions that we will investigate: standing waves with a non-periodic spatial profile and standing waves with a spatially periodic profile. In the first case we can represent the spatial domain as K = R and in the second case as K = [0, T ], where T is the period of the profile function. Equation (1.1.1) arises in various physical and biological contexts, for example in nonlinear optics, for Bose-Einstein condensates, in the modeling of the DNA structure, etc. For more details on the physical background see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF].

The nonlinear Schrödinger equation (NLS) is invariant under the following transformations:

-Translation: u(t, x) → u(t, x + x 0 ), ∀x 0 ∈ R.

-Phase shift: u(t, x) → e iα u(t, x), ∀α ∈ R.

-Galilean invariance: u(t, x) → e -i β 2 x+ β 2 2 Chapter 1. Introduction

The local Cauchy problem

For standing waves and periodic waves, we have the following result concerning the local well-posedness of the Cauchy problem for (1.1.1) (see, e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references therein).

Proposition 1.1.1. For any initial condition u 0 ∈ H 1 (K), there exists T max > 0 such that the Cauchy problem of (1.1.1) admits a unique maximal solution u ∈ C([0, T max ), H 1 (K)∩C([0, T max ), H -1 (K)). Moreover we have the blow-up alternative: either T max = +∞ or lim t→Tmax u(t) H 1 = +∞.

Note that the local well-posedness of the Cauchy Problem is independent of the nature (focusing or defocusing) of the nonlinearity .

For the physical properties of the model as well as for the mathematical study of the equation, it is interesting to look for quantities conserved along the time. Solutions u to (1.1.1) conserve the mass M , the momentum P and the energy E :

M (u) = 1 2 K |u| 2 dx, P (u) = 1 2 Im K uu x dx, E(u) = 1 2 K |u x | 2 dx - K F (u)dx, where F (u) = |u| 0 f (s)ds.
Formally, the conservation of the mass is obtained from multiplying (1.1.1) with u, integrating over K and taking the imaginary part. The second conserved quantity is the momentum obtained by multiplying (1.1.1) by ∂ x u, integrating over K and taking the real part. Finally multiplying (1.1.1) by ∂ t u, integrating over K and taking the real part we obtain the conservation of the energy.

The model case of the pure power nonlinearity

This section is focused on the investigation of the nonlinear Schrödinger equation with a single power nonlinearity. Within this context, we will present important results such as the global existence of solutions, the occurrence of blow-up, and the possibility of scattering. To simplify the notation, we will use p = p 1 .

Global well-posedness

For the defocusing case, using the conservation of the mass and the energy we may prove that the H 1 -norm of u is uniformly bounded in time. This implies that the solution exists globally in time. In the focusing case, the situation is more complex. The proof relies on the Gagliardo-Nirenberg's 1.1. Nonlinear Schrödinger equation 3 inequality (see [START_REF] Adams | Sobolev spaces[END_REF]). If 1 < p < 5 or p = 5 and u 0 L 2 small enough then the solution is global. We only expect the existence of blow up solutions for p 5. Thus in the focusing case, the case p = 5 is a threshold between global existence and blow up. This result is applicable to both standing waves and periodic solutions.

Blow-up of solution for the focusing case

Concerning the blow-up, we have the following result in the focusing case.

Proposition 1.1.2. Assume that p 5 and let u 0 ∈ H 1 (R) be such that

|x|u 0 ∈ L 2 (R), E(u 0 ) < 0.
Then the solution u of (1.1.1) corresponding to u 0 blows up in finite time.

The proof of Proposition 1.1.2 relies on the Virial Theorem.

Proposition 1.1.3. (Virial Theorem) Let u 0 ∈ H 1 (R) be such that |x|u 0 ∈ L 2 (R) and let u be the solution of (1.1.1) corresponding to u 0 . Then |x|u(t) ∈ L 2 (R) for all t ∈ [0, T ) and the function t → xu(t) 2 L 2 (R) is of class C 2 and we have the following identities:

d dt R |x| 2 |u(t, x)| 2 dx = 4Im R u(t)x.∇u(t)dx, 1 2
d dt Im R u(t)x.∇u(t)dx = R |∇u| 2 dx -a 1 1 2 - 1 p + 1 R |u| p+1 dx.
The Virial Theorem comes from the work of Glassey [42] in which the identities above were formally derived.

Scattering of solution for the defocusing case

In this section we present the cases where the scattering occurs. Let Σ be the weighted space defined by

Σ = {u ∈ H 1 (R) : |.|u(.) ∈ L 2 (R)}.
(1.1.1) has the following equivalent integral equation u(t) = S(t)u 0 + ia p t 0 S(ts)|u| p-1 u(s)ds, where S(t) is the Schrödinger group. We have the following result. Proposition 1.1.4. Let u 0 ∈ Σ and u ∈ C([0, +∞), Σ) the global solution of (1.1.1) for the defocusing case. Therefore there exists u +∞ ∈ Σ such that lim t →+∞ u(t) -S(t)u +∞ H 1 = 0.

Standing waves

According to the theory of nonlinear Schrödinger equations, solutions can display either dispersion, caused by the linear aspect of the equation, or concentration at specific points due to nonlinear effects. However, in exceptional cases, these behaviors balance each other out, resulting in special solutions known as standing waves that do not disperse or focus. Standing waves belong to a broader category of solutions that arise in various nonlinear equations, such as the Korteweg-de Vries or Klein-Gordon equations. Solitary waves or solitons are another type of special solution that can be found in these equations, where their profile remains unchanged over time. (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Dauxois | Physics of Solitons[END_REF][START_REF] Drazin | Solitons: an introduction[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF]).

A standing wave of (1.1.1) is a solution of the form e iωt φ(x), in which the frequency ω ∈ R and the profile φ ∈ H 1 (R) satisfies

-φ + ωφ -f (φ) = 0.
(1.2.1)

Notice that φ = 0 is always a trivial solution of (1.2.1), we are interested in nontrivial solutions, that is φ = 0.

Existence of standing waves

We consider the following problem

u = g(u) = ωu -f (u), ω > 0, lim u→0 f (u) u = 0.
The following is a general result.

Proposition 1.2.1. Let g ∈ C(R; R) be a locally Lipschitz continuous function with g(0) = 0 and let G(t) = T 0 g(s)ds. A necessary and sufficient condition for the existence of a solution φ of the problem

φ ∈ C 2 (R), lim t→±∞ φ(t) = 0, φ(0) > 0, φ = g(φ),
is that φ 0 = inf{t > 0 : G(t) = 0} exists, φ(0) > 0, g(φ 0 ) < 0.

More precisely we have the following results for the single and double power case.

Standing waves 1.2.1.1 Existence for the single power case

For the single power nonlinearity case, no standing waves exist for (1.1.1) in the defocusing case. In the focusing case, standing waves only exist for ω > 0. The statement that there are no standing wave solutions of (1.1.1) for ω 0 can be derived from the identity of Pohozhaev in [START_REF] Pohožaev | On the eigenfunctions of the equation ∆u + λf (u) = 0[END_REF], which is given by:

∇φ 2 L 2 (R) -a 1 (p -1) 2(p + 1) φ p+1 L p+1 (R) = 0.
Therefore we only consider the focusing case where we assume that ω > 0. In this case we know the explicit formula for the standing wave solution of

-φ + ωφ -φ p = 0,
it is given by

φ(x) = (p + 1)ω 2 sech 2 (p -1) √ ω) 2 x 1 p-1 .

Existence for the double power case

We define ω * by ω * = sup{ω 0 : ∃s > 0, such that ω 2 s 2 -F (s) < 0}.

It is well known (see [START_REF] Berestycki | Nonlinear scalar field equations. I. Existence of a ground state[END_REF]) that existence and uniqueness of non-trivial solutions of (1.2.1) with lim |x|→∞ φ(x) = 0 hold if and only if 0 ω < ω * when a 1 < 0, a 2 > 0, 0 < ω < ω * otherwise .

The case ω = 0 corresponds to zero mass case in elliptic equations, and associated problems are more delicate in many cases compared with the case ω > 0.

For the case p 2 = 2p 1 -1, we know the explicit formula of the standing wave solution of -φ + ωφa 1 φ p 1a 2 φ p 2 = 0, it is given by

φ(x) = ω A + √ A 2 + Bω cosh(β -1 √ ωx) β , where A = a 1 2 + 1 β , B = a 2 2 + 1 β .

Orbital stability

To analyze the dynamics of (1.1.1), it is essential to have a comprehensive understanding of the dynamical properties of standing waves, particularly their stability. Several stability concepts are available for standing waves. The one most commonly used is orbital stability, which is defined as follows. The standing wave e iωt φ(x) solution of (1.1.1) is said to be orbitally stable if the following holds. For any ε > 0 there exists δ > 0 such that if

u 0 ∈ H 1 (R) verifies u 0 -φ H 1 < δ,
then the associated solution u of (1.1.1) exists globally and verifies

sup t∈R inf y∈R,θ∈R u(t) -e iθ φ(• -y) H 1 < ε.
The groundwork for studying the orbital stability of standing wave solutions were established by Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF], Cazenave and Lions [25], and Weinstein [START_REF] Michael | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Michael | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. There are two approaches to obtaining stability or instability results: the variational approach of [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]25] and the spectral approach of [START_REF] Michael | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Michael | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. Grillakis, Shatah and Strauss [43,44] later developed an abstract theory that, under certain assumptions, reduces the stability analysis of a branch of standing waves ω → φ ω to the study of the sign of the quantity (usually called slope) ∂ ∂ω M (φ ω ). Using the above techniques, the orbital stability of positive standing waves has been fully characterized for the single power case in any dimension N 1 in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]25,[START_REF] Michael | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Michael | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. In dimension N = 1, we have the following results.

Theorem 1.2.2. Let φ be a ground state of (1.2.1). If 1 < p < 5 then the standing wave e iωt φ(x) is orbitally stable.

We give a precise definition of instability by blow-up. Suppose that φ is a solution of (1.2.1). We say that the standing wave e iωt φ(x) is unstable by blow-up in finite time if for any given > 0, there exists u ,0 ∈ H 1 (R) such that u ,0φ H 1 (R) < , but the corresponding maximal solution u of (1.1.1) in the interval [0, T ) satisfies T < +∞, which means that

lim t→T u (t) H 1 (R) = +∞.
We then have the following theorem.

Theorem 1.2.3. If 5 p < ∞ then the standing wave e iωt φ(x) is unstable by blow-up in finite time.

Note that the scaling properties in the single power case play an important role in the proof of stability and instability results. These properties ensure in particular that stability and instability are independent of the value of the frequency parameter ω.

The absence of scaling invariance for double power nonlinearities makes the study of stability more challenging. Consequently, in higher dimensions, only limited results are currently available. However, in dimension 1, the ODE structure of the profile equation (1.2.1) can be exploited, which leads to some favorable outcomes in the analysis. In this regard, Iliev and Kirchev [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF] conducted preliminary investigations for the stability of standing waves in dimension 1 for a generic nonlinearity, deriving a formula for the slope condition. Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] was the first to devote work to the stability of standing waves for the double power nonlinearity in dimension 1, using the integral expression for the slope condition from [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF] to establish the stability/instability of standing waves in various cases. Maeda [START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF] later refined Ohta's approach, extending the stability/instability results to most of the situations not covered in [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. Fukaya and Hayashi [36] improved Ohta's results on instability for small ω with a condition on the powers, but the stability picture is still not complete. We also mention the work of Colin and Ohta [START_REF] Mathieu | Stability of solitary waves for derivative nonlinear Schrödinger equation[END_REF], on the stability of solitary waves for derivative nonlinear Schrödinger equation which is related to the double power nonlinearity.

Although stability and instability studies have been widely conducted, to the best of our knowledge, the contributions of Liu, Tsai, and Zwiers [START_REF] Fei | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF] and Tin [START_REF] Tin | Instability of algebraic standing waves for nonlinear Schrödinger equations with triple power nonlinearities[END_REF] are the only known research works that specifically focuses on the stability and instability of the triple power nonlinearity.

Periodic waves

In this section we are interested in the standing waves e -iat φ(x) where the profile function φ(x) is spatially periodic and satisfies the ordinary differential equation (1.2.1) with a = -ω.

We define the spatially periodic setting

P T = {f ∈ L 2 loc (R) : f (x + T ) = f (x), ∀x ∈ R},
and the spatially anti-periodic setting

A T = {f ∈ L 2 loc (R) : f (x + T ) = -f (x), ∀x ∈ R}.
As previously mentioned, the Cauchy problem for (1.1.1) is locally well-posed in H 1 loc ∩ P T . Moreover in the cubic case the Cauchy problem is globally well posed in H 1 loc ∩ P T .

In this section, we will focus on presenting the results obtained from investigating the properties of periodic waves for the one dimensional nonlinear Schrödinger equation with cubic nonlinearity f (u) = b|u| 2 u.

For this type of equation, non-constant, real-valued, periodic solutions of (1.2.1) are well known to be given by the Jacobi elliptic functions: dnoidal (dn), cnoidal (cn) (for b > 0), and snoidal (sn)(for b < 0). The Jacobi elliptic functions are standard forms of elliptic functions. The three basic functions are denoted cn(x, k), dn(x, k), and sn(x, k), where k ∈ (0, 1) is known as the elliptic modulus. The incomplete elliptic integral of the first kind is defined by

x = F(τ, k) := τ 0 dθ 1 -k 2 sin 2 (θ)
,

where τ is called the Jacobi amplitude, and the Jacobi elliptic functions are defined through the inverse of F (., k):

sn(x, k) := sin(τ ), cn(x, k) := cos(τ ), dn(x, k) := 1k 2 sin 2 (τ ).

The period of the elliptic functions can be expressed in terms of the complete elliptic integral of the first kind

K(k) := F π 2 , k , K(k) → π 2 , k → 0 ∞, k → 1.
This will be further developed in the subsequent sections.

Variational characterization

A global variational characterization for each of the periodic waves described by Jacobi elliptic functions, constant solutions, and plane wave solutions has been provided by Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF]. This approach offers alternative proofs of their stability. Specifically, the following results have been obtained. -For all 0 < m π 2

bT , the unique (up to translation and phase multiplication) minimizer of the energy, under the constraint of fixed mass alone (without momentum) or fixed mass with fixed momentum equal to 0, among periodic functions is the constant function u min = 2m

T . -For all π 2 bT < m < ∞, the unique (up to translation and phase multiplication) minimizer of the energy, at fixed mass alone (without momentum) or fixed mass with fixed momentum equal to 0, among periodic functions is a (appropriately rescaled) dnoidal function.

-The unique (up to translation and phase multiplication) global minimizer of the energy, with fixed mass (without momentum) or fixed mass with fixed momentum equal to 0, among half-anti-periodic functions is a (appropriately rescaled) cnoidal function.

Theorem 1.3.2. Let b < 0.

-The unique (up to translation and phase multiplication) minimizer of the energy, with fixed mass, (and fixed momentum equal to 0) among periodic functions is the constant function u min = 2m T . -The unique (up to translation and phase multiplication) minimizer of the energy, with fixed mass, among half-anti-periodic functions is the plane wave u min = 2m T e iπx T . -The unique (up to phase shift) minimizer of the energy, with fixed mass, among odd half-anti-periodic functions is a (appropriately rescaled) snoidal function.

Moreover we have the following conjecture.

Conjecture 1.3.3. Assume b < 0. The unique (up to translation and phase multiplication) minimizer of the energy at fixed mass and fixed momentum equal to 0, among half-anti-periodic functions is a (appropriately rescaled) snoidal function.

Orbital stability

By Grillakis, Shatah and Strauss [43,44] type methods, orbital stability against energy H 1 loc -norm perturbations for the same period is known for dnoidal waves [START_REF] Angulo | Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations[END_REF], and for snoidal waves [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF] under the additional constraint that perturbations are anti-symmetric with respect to the half-period.

In the work of Gallay and Haragus [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF], they proved that periodic and quasi-periodic solutions are orbitally stable with respect to disturbances having the same period. They also showed that the cnoidal wave solution are stable with respect to perturbations of twice their period. We also mention the work of Antonelli and Shakarov [START_REF] Antonelli | Stability of Cnoidal Waves for the Damped Nonlinear Schr\" odinger Equation[END_REF], who proved the stability of cnoidal waves for the damped nonlinear Schrödinger equation.

The following results were obtained in the work of Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF]. Theorem 1.3.4. The standing wave e -iat φ(x) is a solution of (1.1.1), and is orbitally stable in X in the following cases. For Jacobi elliptic functions, for any k ∈ (0, 1):

a = 1 + k 2 , b = -2k 2 , φ = sn(., k), X = H 1 loc ∩ A - 2K ; a = 1 -2k 2 , b = 2k 2 , φ = cn(., k), X = H 1 loc ∩ A 2K ; a = -(2 -k 2 ), b = 2, φ = dn(., k), X = H 1 loc ∩ P 2K .
For constants and plane wave:

(b = 0) a = - 2bm T , -∞ < b π 2 T m , u = 2m T , X = H 1 loc ∩ P T ; a = 4π 2 T 2 - 2bm T , b < 0, u = e ± iπx T 2m T , X = H 1 loc ∩ A T /2 .
The proof follows the standard approach introduced by Cazenave and Lions [25].

Remark 1.3.5. In [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF], the orbital stability of cn with respect to half-antiperiodic perturbations was obtained only for small amplitude cn and the orbital stability for sn in H 1 loc ∩ A T /2 was proved using the Grillakis-Shatah-Strauss approach.

Remark 1.3.6. Using the complete integrability of (1.1.1), Bottman and Deconinck and Nivala [START_REF] Bottman | Elliptic solutions of the defocusing NLS equation are stable[END_REF] and Gallay and Pelinovsky [40] thus showed that sn is in fact a minimizer of a higher-order functional in H 2 loc ∩ P nT for any n ∈ N and thus showed it is orbitally stable in these spaces.

Spectral stability

Spectral stability with respect to long-wave disturbances has been examined first by Rowlands [START_REF] Rowlands | On the stability of solutions of the non-linear Schrödinger equation[END_REF], who found that periodic waves with real-valued profile are unstable in the focusing case. More recently we mention the work of Gallay and Haragus [START_REF] Gallay | Stability of small periodic waves for the nonlinear Schrödinger equation[END_REF], Gallay and Pelinovsky [40] and Ivey and Lafortune [50]. We also mention the work of Boussaïd and Comech [START_REF] Boussaïd | Nonlinear Dirac equation: Spectral stability of solitary waves[END_REF] on the spectral stability of solitary waves. To examine the spectral stability of a standing wave u(t, x) = e -iat φ(x) solution of (1.1.1), we consider the linearization of (1.1.1) around this solution: if u(t, x) = e -iat (φ(x) + h), then h verifies

i∂ t h -Lh + N (h) = 0,
where L denotes the linear part and N the nonlinear part. Assuming u is real-valued, we separate h into real and imaginary parts to get the equation

h R h I t = JL h R h I + N (h),
where

L = L + 0 0 L - , J = 0 1 -1 0 , L + = -∂ xx -a -3bφ 2 , L -= -∂ xx -a -bφ 2 .
We call

JL = 0 L - L + 0
the linearized operator of (1.1.1) around the standing wave e -iat φ(x). The spectral stability is concluded by examining whether the entire spectrum of JL is located on the imaginary axis. If this is the case, we consider the wave to be spectrally stable. We have the following results concerning the spectral stability (for more details see [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF]).

Theorem 1.3.7. Spectral stability in P T , T = 4K(k), holds for:

-φ = sn, k ∈ (0, 1), -φ = cn and k ∈ (0, k c ),
where k c is the unique k ∈ (0, 1) so that

K(k) = 2E(k), k c ≈ 0.908.
Moreover we have the following theorems.

Theorem 1.3.8. For k ∈ (0, 1), K = K(k), dn is spectrally stable in P 2K , while cn and sn are spectrally stable in A 2K .

Theorem 1.3.9. Cnoidal waves are unstable against perturbations whose period is a sufficiently large multiple of its own, i.e the spectrum of JL cn as an operator on P 4nK contains an eigenvalue with positive real part.

Remark 1.3.10. The orbital stability implies the spectral stability.

Main results

Double power nonlinearity

The results presented in this section have been published in "Comptes rendus mathématique" in collaboration with Stefan Le Coz and Tai-Peng Tsai [52].

As already mentioned the stability picture in the double power case was still not complete. Let p 1 = p, p 2 = q, a 1 = a p and a 2 = a q . The following case was left partially open: a p < 0, a q > 0, 1 < p < q < 5, p + q 6 or p 7 3 .

The above is the defocusing-focusing case where the stability of standing waves for large ω and their instability for small ω were established by Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. The condition p + q > 6 was shown to be sufficient for instability by Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] and later improved by Fukaya and Hayashi [36] to (p+3)(q+3) > 32. However, the stability behavior in the intermediate range of ω where (p + 3)(q + 3) > 32 and Chapter 1. Introduction p+q 6 or p 7 3 , as well as the stability for small ω when (p+3)(q +3) 32, were not fully understood in [36,[START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. The only exception is the case of p = 2 and q = 3 where explicit calculations were possible, revealing the stability of the wave for any ω > 0.

For the sake of simplicity, we will use the following convention. A standing wave is called type S if it is stable for any ω ∈ (0, ω * ). If there exists ω 1 ∈ (0, ω * ) such that the standing wave is unstable for ω ∈ (0, ω 1 ) and stable for ω ∈ (ω 1 , ω * ), we say it is type US. We define other types in a similar way. Note that the endpoint ω 1 may also fall within the range of instability, subject to regularity assumptions on the nonlinearity. This is possible due to the criterion established by Comech and Pelinovsky [START_REF] Comech | Purely nonlinear instability of standing waves with minimal energy[END_REF].

We aim to address the incomplete aspects of the earlier studies [36,[START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] and provide a complete stability picture for the standing waves of the Schrödinger equation with double power nonlinearity using the slope criterion of Grillakis, Shatah and Strauss.

Our main result is the following.

Theorem 1.4.1. Let (φ ω ) ω∈(0,ω * ) be the family of standing waves of (1.1.1).

The following gives the stability type of the family of standing waves.

1. Assume that a p > 0 and a q > 0.

(a) If q 5, then it is of type S.

(b) If p 5, then it is of type U .

(c) If p < 5 < q, then it is of type SU .

2. Assume that a p > 0 and a q < 0.

(a) If p 5, then it is of type S.

(b) If p > 5, then it is of type U S.

3. Assume that a p < 0 and a q > 0.

(a) If q 7 -2p, then it is of type S.

(b) If 7 -2p < q < 5, then it is of type U S. (c) If q 5, then it is of type U .
This theorem implies in particular that stability change occurs at most once, which is conjectured in [59, p. 265], and is in contrast to NLS with triple power nonlinearity considered in [START_REF] Fei | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF].

The cases (1), (2), and (3)(c) of Theorem 1.4.1 were previously addressed in [START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. However, for the sake of completeness, we also include their proofs in our work. Cases (3)(a) and (b), they were only partially solved in previous works. We provide a definitive result for these cases.

Note that our results do not extend to the case of zero frequency, ω = 0. However, it is conjectured that the stability or instability of the corresponding algebraic standing waves, if they exist, should be the same as for small ω > 0, which is consistent with the results of Fukaya and Hayashi [36].

Moreover, it should be noted that our statement does not include the critical frequency at which the stability change occurs. This frequency was shown by Comech and Pelinovsky to lead to instability under higher regularity assumptions on the nonlinearity. Alternative proofs with improved regularity assumptions are given in [START_REF] Maeda | Stability of bound states of Hamiltonian PDEs in the degenerate cases[END_REF][START_REF] Ohta | Instability of bound states for abstract nonlinear Schrödinger equations[END_REF], but they are still not sufficient to cover the entire range of double-power nonlinearities. Strategies have been developed to handle the critical frequency without relying on regularity assumptions for other types of equations, as seen in [START_REF] Guo | Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case[END_REF][START_REF] Wu | Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension[END_REF]. However, in our work, we only verify the validity of the stability criterion and do not attempt to cover the critical frequency.

Finally, for the defocusing-focusing case where a p < 0, a q > 0, and 1 < p < q < 5, we supplement our theoretical findings with numerical experiments. Initially, we first represent the critical surface, denoted by ω c (p, q), at which the stability transition occurs and examine the various shapes of the surface based on the ap aq ratio. We then simulate the dynamics of (1.1.1) around a standing wave with the Crank-Nicolson scheme with relaxation of Besse [START_REF] Besse | A relaxation scheme for the nonlinear Schrödinger equation[END_REF]. We observe three distinct types of behaviors based on the initial data: stability, growth followed by oscillations, and scattering. This problem will be clearly detailed in Chapter 2.

Quasi-periodic case

In chapter 3 we focus on the one-dimensional cubic nonlinear Schrödinger equation, given by

iψ t + ψ xx + b|ψ| 2 ψ = 0,
where ψ is a complex-valued function defined on R t ×R x , and b is a nonzero real constant. We investigate the simplest non-trivial solutions of the equation, which are the standing waves of the form

ψ(t, x) = e -iat u(x), a ∈ R,
where the profile function u satisfies the ordinary differential equation

u xx + au + b|u| 2 u = 0.
(1.4.1)

We are particularly interested here in those standing waves whose profile u is quasi-periodic and defined on the space

H θ T = {f ∈ H 1 loc (R) : f (x + T ) = e iθ f (x), ∀x ∈ R},
where θ is real and so-called Floquet multiplier.

In addition to the conserved quantities of the NLS, namely mass M , momentum P and energy E, we also consider the conserved quantities of the ordinary differential equation, namely momentum J and energy E, given by

J = Im(u x u), E = 1 2 |u x | 2 + a 2 |u| 2 + b 4 |u| 4 .
The object of our interest is the minimization problem given by

min{E(u) : u ∈ H θ T , M (u) = m, P (u) = p}, (1.4.2)
where m > 0 and p ∈ R.

The aim of this chapter is to establish a connection between the solutions of the minimization problem (1.4.2) and the solutions of the ordinary differential equation (1.4.1).

Starting from the ordinary differential equation, our objective is to establish a connection with the minimization problem. Specifically, we seek to find a correspondence between the solutions of the ordinary differential equation and the minimizers of the energy functional subject to fixed values of mass and momentum. Thus, we establish a diffeomorphic correspondence between them. We also study these conserved quantities and provide monotonicity results for some of them.

Next, we proceed in the opposite direction by starting from the minimization problem and establishing a link with the solutions of the ordinary differential equation. Our main result is the following.

Observation 1.4.2. For each defocusing and focusing case, for fixed (J, E), let u ode be the associated solution of the ordinary differential equation. Then (up to phase shift and translation) the minimizer of the problem

min{E(u) : u ∈ H θ T , M (u) = M (u ode ), P (u) = P (u ode ), u ∈ H θ T },
is given by u ode .

Our results were obtained through numerical computations. We first computed the ordinary differential equation (1.4.1) numerically and compared the resulting solution to the exact solution when available. We then utilized the gradient flow with discrete normalization method to find the minimizer of the energy E subject to fixed mass m > 0 and fixed momentum p. This method, also known as the "imaginary time method" in physics literature, has been applied to various equations, including nonlinear Schrödinger equations. We compared the minimizer obtained from the normalized gradient flow with the numerical solution of the ODE and made our conjecture based on the results.

General nonlinearity

In this section we present our work on the nonlinear Schrödinger equation with general nonlinearity:

iψ t + ψ xx + bf (ψ) = 0, (1.4.3)
where

ψ : R t × R x → C, the nonlinearity f : C → C is defined for any z ∈ C by f (z) = g(|z| 2 )z with g ∈ C 0 ([0, +∞), R) ∩ C 2 ((0, +∞), R) and b ∈ R \ {0}.
Similarly to the other case, the simplest non-trivial solutions of (1.4.3) are the standing waves given by

ψ(t, x) = e -iat u(x), a ∈ R.
The profile function u satisfies the ordinary differential equation

u xx + au + bf (u) = 0. (1.4.4)
We are particularly interested in the spatially periodic solutions ψ(t, .) ∈ H 1 loc ∩ P T , and anti-periodic solutions ψ(t, .) ∈ H 1 loc ∩ A T . Our goal is to extend previous work on the cubic case to the case of a general nonlinearity. Specifically, we will consider the real and complex-valued solutions of the profile equation and represent their phase portraits for various parameter regimes. We will study the minimization problems for the energy functional with fixed mass and momentum constraints for the periodic and anti-periodic cases, extending the previous results to this more general setting. Consequently, we investigate the properties of minimizers in each case, including their existence and uniqueness. We have the following results for the periodic case.

Proposition 1.4.3. In the focusing case, for all m > 0, there exists a unique (up to phase shift) minimizer of the energy, under the constraint of fixed mass alone (without momentum) or fixed mass with fixed momentum equal to 0, among periodic functions. Moreover, the minimal energy is finite and negative. Furthermore, there exists m > 0 such that if m > m, then the minimizer is not a constant, the associated Lagrange multiplier a < 0, the minimizer is positive, and it is a solution of the ordinary differential equation (1.4.4). Proposition 1.4.4. In the defocusing case, for all m ∈ (0, ∞) the minimizer of the energy, under the constraint of fixed mass alone (without momentum) or fixed mass with fixed momentum equal to 0, among periodic functions is the same and unique (up to phase shift). It is the constant function u ∞ ≡ 2m T .

Chapter 1. Introduction

For the anti-periodic case, we limit ourselves to the nonlinearity of the type f (u) = |u| p-1 u + |u| q-1 u, with p, q > 1. We prove that in this case there exists a unique (up to phase shift and complex conjugate) minimizer of the energy, under the constraint of fixed mass among anti-periodic functions. It is the plane wave u ∞ ≡ 2m T e iπx T . Finally, we investigate the minimization problem on the Nehari manifold. We restrict ourselves to the nonlinearity of the form f (u) = |u| p-1 u, with p > 1. We define a functional S :

H 1 (R) → R by setting for v ∈ H 1 (R) S(v) := 1 2 ∇v 2 L 2 (R ) + ω 2 v 2 L 2 (R) - a 1 p + 1 v p+1 L p+1 (R) .
The functional S is often called action. We are interested in the following minimization problems on the Nehari manifold:

min{S(u) : u ∈ H 1 loc ∩ P T , u = 0, I(u) = 0}, (1.4.5)
and min{S(u

) : u ∈ H 1 loc ∩ A T 2 , u = 0, I(u) = 0}, (1.4.6)
where I is defined by

I(u) = ∂ x u 2 L 2 -a u 2 L 2 -b u p+1 L p+1 .
We also consider the following minimization problem

min 1 2 - 1 p + 1 v 2 H 1 : v = 0, I(v) 0, v ∈ H 1 loc ∩ A T 2 .
(1.4.7)

We prove that in the periodic case, with b > 0 and a < 0, the minimum of (1.4.5) is finite and there exists a unique minimizer which is solution of (1.4.4). In the anti-periodic case, with b > 0 and a < 4π 2 T 2 , the minimization problems (1.4.6) and (1.4.7) share the same minimizer. Moreover when p is an odd integer the minimizer is real and it is the solution of the ordinary differential equation (1.4.4).

Outline of the thesis

The thesis is organized as follows. In Chapter 2, we focus on the double power nonlinear Schrödinger equation, where we provide a complete stability analysis of the standing waves. In Chapter 3, we investigate the onedimensional cubic nonlinear Schrödinger equation and establish a variational characterization of the minimizer of the energy with fixed mass and momentum. We also establish a connection between these minimizers and the solutions of the ordinary differential equation. Chapter 4 extends the work on the cubic case to the more general case of a nonlinearity, where we consider various minimization problems.

Introduction

Consider the one dimensional nonlinear Schrödinger equation with double power nonlinearity

i∂ t u + ∂ 2 x u + a p |u| p-1 u + a q |u| q-1 u = 0, (2.1.1)
where u : R t × R x → C, a p , a q ∈ R \ {0} and 1 < p < q < ∞. When a p < 0, a q > 0, we say that the nonlinearity is defocusing-focusing, with analogous definitions for other possible signs combinations. Nonlinear Schrödinger equations appear in many areas of physics such as nonlinear optics (see e.g. [START_REF]Nonlinear fiber optics. Optics and Photonics[END_REF]) or Bose-Einstein condensation. The double power nonlinearity is an important example of the possible nonlinearities appearing in soliton theory (see e.g. [START_REF] Nail Akhmediev | Hamiltonianversus-energy diagrams in soliton theory[END_REF]). Via gauge transformations, the double power nonlinearity is also connected with the derivative nonlinear Schrödinger equation (see e.g. [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Le | Stability of Multisolitons for the Derivative Nonlinear Schrödinger Equation[END_REF][START_REF] Van Tin | On the derivative nonlinear Schrödinger equation on the half line with Robin boundary condition[END_REF]). The double power nonlinearity is also a typical example of a nonlinearity breaking the scaling invariance of the pure power case, while still being relatively tractable, and it may be used to study phenomena in the absence of scaling symmetry (see e.g. [START_REF] Le Coz | Minimal mass blow up solutions for a double power nonlinear Schrödinger equation[END_REF] for the construction of blowing-up solutions).

The Cauchy problem for (2.1.1) is well known (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references therein) to be well-posed in the energy space H 1 (R): for any u 0 ∈ H 1 (R), there exists a unique maximal solution u ∈ C((-T * , T * ), H 1 (R)) ∩ C 1 ((-T * , T * ), H -1 (R)) of (2.1.1) such that u(t = 0) = u 0 . Moreover, the energy E and the mass M , defined by

E(u) = 1 2 u x 2 L 2 - a p p + 1 u p+1 L p+1 - a q q + 1 u q+1 L q+1 , M (u) = 1 2 u 2 L 2 ,
are conserved along the flow and the blow-up alternative holds (i.e. if

T * < ∞ (resp. T * < ∞), then lim t→T * (resp -T * ) u(t) H 1 = ∞).
A standing wave is a solution of (2.1.1) of the form u(t, x) = e iωt φ(x) for some ω ∈ R and a profile φ ∈ C 2 (R), which then satisfies

-φ + ωφ -a p |φ| p-1 φ -a q |φ| q-1 φ = 0. (2.1.2)
We only consider real-valued φ in this paper. Define ω * by

ω * = sup ω 0 : ∃s > 0 such that ω 2 s 2 - a p p + 1 s p+1 - a q q + 1 s q+1 < 0 .
It is well known (see [START_REF] Berestycki | Nonlinear scalar field equations. I. Existence of a ground state[END_REF]) that existence of non-trivial solutions of (2.1.2) with lim |x|→∞ φ(x) = 0 holds if and only if 0 ω < ω * when a p < 0, a q > 0, 0 < ω < ω * otherwise.

In that case, the solution is positive (up to phase shift), even (up to translation) and unique. We denote it by φ ω , or simply φ when there is no ambiguity.

Solitary waves are the building blocks for the nonlinear dynamics of (2.1.1), as it is expected that, generically, a solution of (2.1.1) will decompose into a dispersive linear part and a combination of nonlinear structures as solitary waves. This vague statement is usually referred to as the Soliton Resolution Conjecture.

Therefore, understanding the dynamical properties of standing waves, in particular their stability, is a key step in the analysis of the dynamics of (2.1.1). Several stability concepts are available for standing waves. The most commonly used is orbital stability, which is defined as follows. The standing wave e iωt φ(x) solution of (2.1.1) is said to be orbitally stable if the following holds. For any ε > 0 there exists δ > 0 such that if

u 0 ∈ H 1 (R) verifies u 0 -φ H 1 < δ,
then the associated solution u of (2.1.1) exists globally and verifies

sup t∈R inf y∈R,θ∈R u(t) -e iθ φ(• -y) H 1 < ε.
In the rest of this paper, when we talk about stability/instability, we always mean orbital stability/instability.

The groundwork for orbital stability studies was laid down by Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF], Cazenave and Lions [25] and Weinstein [START_REF] Michael | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Michael | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. Two approaches lead to stability or instability results: the variational approach of [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]25], which exploits global variational characterizations combined with conservation laws or the virial identity, and the spectral approach of [START_REF] Michael | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Michael | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF], which exploits spectral and coercivity properties of linearized operators to construct a suitable Lyapunov functional. Later on, Grillakis, Shatah and Strauss [43,44] developed an abstract theory which, under certain assumptions, boils down the stability study of a branch of standing waves ω → φ ω to the study of the sign of the quantity (usually called slope) ∂ ∂ω M (φ ω ). Note that the theory of Grillakis, Shatah and Strauss has known recently a considerable revamping in the works of De Bièvre, Genoud and Rota-Nodari [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF][START_REF] De | Orbital Stability via the Energy-Momentum Method: The Case of Higher Dimensional Symmetry Groups[END_REF].

With the above mentioned techniques, the orbital stability of positive standing waves has been completely determined in the single power case (i.e. a q = 0) in any dimension d 1 in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]25,[START_REF] Michael | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Michael | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. If a q = 0, positive standing waves exist if and only if a p > 0 and ω > 0. In this case, they are stable if 1 < p < 1 + 4 d (i.e. 1 < p < 5 in dimension d = 1), and they are unstable if 1+ 4 d p < 1+ 4 (d-2) + (i.e. 5 p < ∞ in dimension d = 1). Scaling properties of the single power nonlinearity play an important role in the proof and ensure in particular that stability and instability are independent of the value of the frequency ω. It turns out that there is no scaling invariance for double power nonlinearities, which makes the stability study more delicate. As a matter of fact, only very partial results are available so far in higher dimensions. In dimension 1, the situation is a bit more favorable, as one might exploit the ODE structure of the profile equation (2.1.2) in the analysis.

Preliminary investigations for the stability of standing waves in dimension 1 were conducted by Iliev and Kirchev [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF] in the case of a generic nonlinearity. In particular, a formula for the slope condition was obtained in [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF]. The earliest work devoted to the stability of standing waves for nonlinear Schrödinger equations with double power nonlinearity in dimension 1 is the work of Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. In this work, using the integral expression for the slope condition derived by Iliev and Kirchev [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF], Ohta established the stability/instability of standing waves in a number of cases. Later on, Maeda [START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF] further refined the approach of Ohta and established the stability/instability in most of the situations not covered in [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. However, the stability picture was still not complete, as the Chapter 2. Standing waves of the double power 1d NLS following case was left partially open: a p < 0, a q > 0, 1 < p < q < 5, p + q 6 or p 7 3 .

In the above case, Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] established the stability of standing waves for ω large enough. The instability for small ω was obtained by Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] for p + q > 6, a condition which was later improved to (p + 3)(q + 3) > 32 by Fukaya and Hayashi [36]. What happens in the intermediate range of ω when (p + 3)(q + 3) > 32 and p + q 6 or p 7 3 ,

was not elucidated in [36,[START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF], nor what happens for small ω when (p + 3)(q + 3) 32, (except for the notable case p = 2, q = 3, where explicit calculations are possible and show that the wave is stable for any ω > 0).

For convenience, we adopt the following convention. When a standing wave is stable for any ω ∈ (0, ω * ), we say that it is of type S. When there exists ω 1 ∈ (0, ω * ) such that the standing wave is unstable for ω ∈ (0, ω 1 ) and stable for ω ∈ (ω 1 , ω * ), we say that it is of type US. Other types are defined in a similar manner.

Note that when instability holds the endpoint ω 1 could be included in the instability range under regularity assumptions on the nonlinearity (thanks to the criterion of Comech and Pelinovsky [START_REF] Comech | Purely nonlinear instability of standing waves with minimal energy[END_REF], see (2.2.4) and Remark 2.1.2).

Our goal in this paper is to fill out the gaps left open by the previous works [36,[START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] and to provide a complete stability picture for the standing waves of the Schrödinger equations with double power nonlinearity. Our main result is the following. Theorem 2.1.1. Let (φ ω ) ω∈(0,ω * ) be the family of standing waves of (2.1.1). The following gives the stability type of the family of standing waves.

1. Assume that a p > 0 and a q > 0.

(a) If q 5, then it is of type S.

(b) If p 5, then it is of type U . (c) If p < 5 < q, then it is of type SU .
2. Assume that a p > 0 and a q < 0. 3. Assume that a p < 0 and a q > 0.

(a) If q 7 -2p, then it is of type S.

(b) If 7 -2p < q < 5, then it is of type U S.

(c) If q 5, then it is of type U . This theorem implies in particular that stability change occurs at most once, which is conjectured in [59, p. 265], and is in contrast to NLS with triple power nonlinearity considered in [START_REF] Fei | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF].

Remark 2.1.2. The critical frequency at which the stability change occurs is not included in our statement. Indeed, it was established by Comech and Pelinovsky under a higher regularity assumption on the nonlinearity that the standing wave at the critical frequency is unstable if (2.2.4) holds. Alternative proofs with improved assumptions on the regularity are provided in [START_REF] Maeda | Stability of bound states of Hamiltonian PDEs in the degenerate cases[END_REF][START_REF] Ohta | Instability of bound states for abstract nonlinear Schrödinger equations[END_REF]. However, the regularity required is still too high to cover the whole range of double-power nonlinearities. Strategies to treat the critical frequency without regularity assumptions have been developed for other types of equations, see e.g. [START_REF] Guo | Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case[END_REF][START_REF] Wu | Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension[END_REF]. In this paper, we will not try to cover the critical frequency and we simply verify that the criterion (2.2.4) holds.

In Theorem 2.1.1, cases (1), ( 2) and (3)(c) were already covered in [START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. For the sake of completeness, and as the proofs are not very long, we will also cover them in our work. Cases (3)(a) and (b) were only partially solved. We provide a definitive result for these cases. Our approach relies on several ingredients. First of all, we express the slope condition in a concise, while easily tractable integral, factoring out terms which are in any case positive. Instead of working with the parameter ω, we manipulate the slope condition with the parameter φ 0 = φ ω (0) (which is in a bijective relation with ω). We are left with an integral expression F (φ 0 ) (see (2.3.2)), of which we need to determine the sign. A refactorisation allows us to introduce an auxiliary parameter γ, and differentiation with respect to φ 0 gives us an expression which we can prove to have sign, provided we have suitably chosen the parameter γ. This gives the information that F (φ 0 ) changes sign at most once. The sign for large ω (or equivalently large φ 0 ) had already been established in [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. On the other hand, the sign for ω close to 0 had not been computed before. Here, an astute rewriting of the slope in terms of Beta functions allows us to determine the sign for ω close to 0.

Observe that our results are not covering the zero-frequency case ω = 0. Stability or instability of the corresponding (algebraic) standing waves (when existing) can be conjectured to be the same as the one for small ω > 0 (which is consistent with the results obtained by Fukaya and Hayashi [36]).

In the case a p < 0, a q > 0 and 1 < p < q < 5, we complement our theoretical results with numerical experiments. We first represent the critical surface {ω c (p, q)} at which the stability change occurs and discuss the different possible shapes of the surface depending on the ratio a p /a q . We then simulate the dynamics of (2.1.1) around a standing wave with the Crank-Nicolson scheme with relaxation of Besse [START_REF] Besse | A relaxation scheme for the nonlinear Schrödinger equation[END_REF]. Three types of behaviors are observed depending on the type of initial data : stability, growth followed by oscillations, and scattering.

To end this introduction, we point out that many works are devoted to standing waves of the double power nonlinear Schrödinger in higher dimension (for which our approach does not apply), and just give a small sample of the existing literature. The cubic quintic case in higher dimension was investigated in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF]. Stability of standing waves in higher dimension for generic nonlinearities was considered in [37]. Strong instability was studied in [START_REF] Ohta | Strong instability of standing waves for nonlinear Schrö-dinger equations with double power nonlinearity[END_REF]. Stability results for algebraic standing waves were obtained in [36]. Uniqueness and non-degeneracy was considered in [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF]. Existence or non-existence of minimizers of the energy at fixed mass was obtained in [START_REF] Bellazzini | Ground state energy threshold and blow-up for NLS with competing nonlinearities[END_REF]. Let us also mention in dimension 1 the work [41], which is devoted to the stability of standing waves for cubic-quintic nonlinearities in the presence of a δ potential (see [START_REF] Angulo Pava | On stability properties of the cubic-quintic Schrödinger equation with δ-point interaction[END_REF][START_REF] Angulo Pava | Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity[END_REF] for further developments).

This paper is organized as follows. We start by some preliminaries in Section 2.2, recalling in particular the properties of the standing wave profiles and the stability criterion. In Section 2.3, we reformulate the slope condition for stability, using the profile equation. In Section 2.4, we analyze the limit of the slope at the endpoints of the interval of admissible frequencies ω and in particular determine the sign of the slope at the endpoints. The sign of the slope on the full interval of admissible frequencies is recovered in Section 2.5, which shows Theorem 2.1.1. Finally, numerical experiments are presented in Section 2.6.

After the first version of this paper was posted to arXiv, Professor Hayashi kindly informed us he had an independent similar result and posted it as [47]. His Theorem 1.3 is similar to our Theorem 1.1 although it does not include the case 1 < p < 9/5.

We would like to thank the anonymous referee for his valuable comments which helped in the improvement of our paper.

Preliminaries

The profile equation

We start by some analysis around the ordinary differential equation (2.1.2) and its solutions (φ ω ). Apart in a few specific cases (e.g. when q = 2p -1, see e.g. [START_REF] Fei | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF]), there does not exist an explicit formula for the full standing waves profile. Note that ω * = ∞ when a q > 0, 0 < ω * < ∞ when a p > 0 and a q < 0, and ω * = -∞ (i.e. there is no solution of (2.1.2) in H 1 (R)) when a p , a q < 0.

All along this paper, we assume that 0 < ω < ω * (excluding in particular the possibility that a p , a q < 0). Under this assumption, there exists φ 0 > 0 (depending implicitly on ω) such that

φ 0 = inf{φ > 0 : ω 2 φ 2 - a p p + 1 |φ| p+1 - a q q + 1 |φ| q+1 = 0},
and we have φ ω (0) = φ 0 .

Observe that ω may be expressed in terms of φ 0 as follows

ω = 2a p p + 1 φ p-1 0 + 2a q q + 1 φ q-1 0 . (2.2.1)
Moreover, as ω < ω * , we have

ω -a p φ p-1 0 -a q φ q-1 0 < 0.
This implies in particular φ 0 is a C 1 -function of ω. Moreover, we always have

∂φ 0 ∂ω = 2a p (p -1) p + 1 φ p-2 0 + 2a q (q -1) q + 1 φ q-2 0 -1 > 0. (2.2.2)
As a consequence, the following result holds.

Lemma 2.2.1. The function ω → φ 0 is a strictly increasing bijection from (0, ω * ) to (φ * , φ * ) where

φ * =    -ap aq q+1 p+1 1 q-p if a p < 0, 0 if a p > 0, φ * =    ∞ if a q > 0, -ap aq p-1 q-1 q+1 p+1 1 q-p if a q < 0.
(2.2.3)

The stability criterion

As we already mentioned, stability criteria have been derived in the general case in [43,[START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF]. For the double power nonlinearity, the stability of the standing wave is determined by a slope condition (the spectral condition of [43] being always verified in this case when ω > 0). The standing wave e iωt φ ω (x) will be stable if

∂ ∂ω M (φ ω ) > 0,
and it will be unstable if

∂ ∂ω M (φ ω ) < 0.
When ∂ ω M (φ ω ) = 0, under regularity assumption on the nonlinearity, the stability can be decided by looking at the second derivative, as was established by Comech and Pelinovsky [START_REF] Comech | Purely nonlinear instability of standing waves with minimal energy[END_REF]:

If ∂ ω M (φ ω ) = 0 and ∂ 2 ∂ω 2 M (φ ω ) = 0, (2.2.4)
then the standing wave e iωt φ ω (x) is unstable.

Reformulation of the slope

For notational convenience, we introduce the function J defined by

J(ω, p, q) = ∂ ∂ω M (φ ω ).
Hence the sign of J determines the stability of the corresponding standing wave.

The main idea in this section is to express J in terms of φ 0 instead of ω. Before doing that, we introduce some convenient notation. Let Φ p and Φ q be defined by

Φ p = 2a p p + 1 φ p+1 0 (1 -s p-1 ), Φ q = 2a q q + 1 φ q+1 0 (1 -s q-1 ), (2.3.1)
where a p , a q = 0, 1 < p < q < ∞ and 0 < s < 1.

Lemma 2.3.1. The function J may be expressed in terms of φ 0 as follows

J(ω, p, q) = C(φ 0 )F (φ 0 ),
where

F (φ 0 ) = 1 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 sds, (2.3.2)
and C(φ 0 ) is positive and explicitly known (see (2.3.4)).

Proof. We multiply the equation (2.1.2) of the profile by φ x and we integrate to obtain -

1 2 |φ x | 2 + ω 2 |φ| 2 - a p p + 1 |φ| p+1 - a q q + 1 |φ| q+1 = c.
When x → ∞, we know that φ(x) → 0 and φ (x) → 0. Therefore c = 0, and

- 1 2 |φ x | 2 + ω 2 |φ| 2 - a p p + 1 |φ| p+1 - a q q + 1 |φ| q+1 = 0. (2.3.3)
For x > 0, as φ is decreasing, from (2.3.3) we have

φ x = -ωφ 2 - 2a p p + 1 φ p+1 - 2a q q + 1 φ q+1 .
Still for x > 0, let t = φ(x), then

dx = dt φ x = - dt ωφ 2 -2ap p+1 φ p+1 -2aq q+1 φ q+1
.

Therefore we may perform the following change of variable:

M (φ) = 1 2 R |φ(x)| 2 dx = ∞ 0 |φ(x)| 2 dx = φ 0 0 t 2 ωt 2 -2ap p+1 t p+1 -2aq q+1 t q+1 dt.
Changing again variable by setting t = φ 0 s, we have

M (φ) = 1 0 φ 3 0 s 2 s ωφ 2 0 -2ap p+1 φ p+1 0 s p-1 -2aq q+1 φ q+1 0 s q-1
ds.

Replacing ω by its value (2.2.1) in terms of φ 0 , we have

M (φ) = 1 0 φ 3 0 s 2ap p+1 φ p+1 0 + 2aq q+1 φ q+1 0 -2ap p+1 φ p+1 0 s p-1 -2aq q+1 φ q+1 0 s q-1
ds, which, using the notation (2.3.1) for Φ p and Φ q , gives

M (φ) = 1 0 φ 3 0 s Φ p + Φ q ds.
Differentiating with respect to ω, we have

∂ ω Φ p = (p + 1)Φ p φ -1 0 ∂ ω φ 0 , ∂ ω Φ q = (q + 1)Φ q φ -1 0 ∂ ω φ 0 .
Therefore we obtain

J(ω, p, q) = ∂ ω M (φ) = 1 0 3φ 2 0 ∂ ω φ 0 s(Φ p + Φ q ) -1 2 φ 3 0 s (p + 1)Φ p φ -1 0 + (q + 1)Φ q φ -1 0 ) ∂ ω φ 0 (Φ p + Φ q ) 3 2 ds, = ∂ ω φ 0 2 φ 2 0 1 0 6(Φ p + Φ q ) -((p + 1)Φ p + (q + 1)Φ q )) (Φ p + Φ q ) 3 2 sds, = ∂ ω φ 0 2 φ 2 0 1 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 sds, = C(φ 0 )F (φ 0 ),
where F (φ 0 ) is defined in (2.3.2) and

C(φ 0 ) = ∂ ω φ 0 2 φ 2 0 . (2.3.4)
This concludes the proof.

We will now analyze the variations of J(ω, p, q) in terms of φ 0 . For future convenience (the reason for such a choice will appear clearly later), we introduce an auxiliary parameter γ in the following way

J(ω, p, q) = C(φ 0 )φ -γ 0 F γ (φ 0 ),
where

F γ (φ 0 ) = 1 0 φ γ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2
sds.

Denote the integrand of F γ by

I γ (φ 0 ) = φ γ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2
.

(2.3.5)

Observe that there is a implicit dependency in s. In the following lemma we differentiate I γ (φ 0 ) with respect to φ 0 .

Lemma 2.3.2. For any 0 < s < 1, the following holds:

∂I γ ∂ φ 0 = φ γ-1 0 ((5 -p)(2γ -(p + 1))Φ p + (5 -q)(2γ -(q + 1))Φ q ) (Φ p + Φ q ) 2 (Φ p + Φ q ) 5 2 -φ γ-1 0 3(q -p) 2 Φ p Φ q 2 (Φ p + Φ q ) 5 2
.

Proof. We start by differentiating the term in parenthesis in I γ (φ 0 ). We have

∂ φ 0 Φ p = (p + 1)Φ p φ -1 0 , ∂ φ 0 Φ q = (q + 1)Φ q φ -1 0 .
Therefore, we have

∂ φ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 = φ -1 0 ((5 -p)(p + 1)Φ p + (5 -q)(q + 1)Φ q ) (Φ p + Φ q ) (Φ p + Φ q ) 5 2 -φ -1 0 3 2 ((5 -p)Φ p + (5 -q)Φ q ) ((p + 1)Φ p + (q + 1)Φ q ) (Φ p + Φ q ) 5 2 = φ -1 0 -(5 -p)(p + 1)Φ 2 p -(5 -q)(q + 1)Φ 2 q 2 (Φ p + Φ q ) 5 2 + φ -1 0 ((5 -p)(2p -3q -1) + (5 -q)(2q -3p -1))Φ p Φ q 2 (Φ p + Φ q ) 5 2 = φ -1 0 -((5 -p)(p + 1)Φ p + (5 -q)(q + 1)Φ q ) (Φ p + Φ q ) -3(q -p) 2 Φ p Φ q 2 (Φ p + Φ q ) 5 2
.

Before going on, observe that we may rewrite the term in parentheses in

I γ (φ 0 ) as (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 = 2 ((5 -p)Φ p + (5 -q)Φ q ) (Φ p + Φ q ) 2 (Φ p + Φ q ) 5 2
.

Finally, the full derivative of I γ (φ 0 ) is given by

∂I γ ∂φ 0 = ∂ φ 0 φ γ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 , = φ γ-1 0 ((5 -p)(2γ -(p + 1))Φ p + (5 -q)(2γ -(q + 1))Φ q ) (Φ p + Φ q ) 2 (Φ p + Φ q ) 5 2 -φ γ-1 0 3(q -p) 2 Φ p Φ q 2 (Φ p + Φ q ) 5 2
.

This concludes the proof.

For future reference, we establish here the following technical lemma which we will use at several occasions.

Lemma 2.3.3. The function s → 1-s q-1 1-s p-1 is an increasing bijection from (0, 1) to (1, q-1 p-1 ). Proof. Let h(s) = 1-s q-1 1-s p-1 . We have h (s) = s p-2 (1 -s p-1 ) 2 l(s), where l(s) = (q -p)s q-1 + p -1 -(q -1)s q-p .
Note that l(1) = 0 and for 0 < s < 1, l (s) = (qp)(q -1)(s q-2s q-p-1 ) < 0.

Hence l (s) < 0 and l(s) > 0 for 0 < s < 1. We conclude that h (s) > 0 for 0 < s < 1. As a consequence, h is increasing on the interval (0, 1). Moreover, we have h(0) = 1 and, by L'Hospital's rule,

lim s→1 h(s) = q -1 p -1 .
This concludes the proof.

The slope at the endpoints

Our goal in this section is to investigate what happens for J(ω, p, q) when ω is close to 0 and ω * .

The zero frequency case

In this section, we determine the limit of J(ω, p, q) when ω tends to zero. Let J 0 be defined by J 0 (p, q) = lim ω→0 J(ω, p, q).

We first consider the case where a p > 0.

Proposition 2.4.1. Let a p > 0. The following holds.

1. If 1 < p < 7 3 , then J 0 (p, q) = 0 + . 2. If p = 7 3 , then 0 < J 0 (p, q) < ∞. 3. If 7 3 < p < 5, then J 0 (p, q) = ∞. 4. If p = 5, then three cases have to be distinguished. (a) If q < 9, then J 0 (p, q) = -sign(a q )∞. (b) If q = 9, then 0 < -sign(a q )J 0 (p, q) < ∞. (c) If q > 9, then J 0 (p, q) = 0 -sign(aq) . 5. If 5 < p, then J 0 (p, q) = -∞.
Proof. When a p > 0, we have

lim ω→0 φ 0 = φ * = 0.
Recall that we have shown in Lemma 2.3.1 that J may be written as J(ω, p, q) = C(φ 0 )F (φ 0 ). We have (recalling the definition (2.3.4) of C and the expression

(2.2.2) of ∂ ω φ 0 ) C(φ 0 ) = 1 2 ∂ ω φ 0 φ 2 0 = 1 4 a p (p -1) p + 1 φ p-4 0 + a q (q -1) q + 1 φ q-4 0 -1 = φ 4-p 0 1 4 a p (p -1) p + 1 + a q (q -1) q + 1 φ q-p 0 -1 = φ 4-p 0 p + 1 4a p (p -1) + o(1) .
The function F (defined in (2.3.2)) can be written, substituting Φ p and Φ q by their expressions (2.3.1), as

F (φ 0 ) = 1 0 2ap(5-p) p+1 (1 -s p-1 )φ p+1 0 + 2aq(5-q) q+1 (1 -s q-1 )φ q+1 0 2ap p+1 (1 -s p-1 )φ p+1 0 + 2aq q+1 (1 -s q-1 )φ q+1 0 3 2
sds.

As we are interested in the limit φ 0 → 0, we factor out the terms in φ p+1 0 to get

F (φ 0 ) = φ -p+1 2 0 1 0 2ap(5-p) p+1 (1 -s p-1 ) + 2aq(5-q) q+1 (1 -s q-1 )φ q-p 0 2ap p+1 (1 -s p-1 ) + 2aq q+1 (1 -s q-1 )φ q-p 0 3 2 sds = (5 -p)φ -p+1 2 0 1 0 2a p p + 1 (1 -s p-1 ) -1 2 sds + o(1) .
In the particular case p = 5, we instead write

F (φ 0 ) = a q (5 -q)φ q-8 0   1 0 2 q+1 (1 -s q-1 ) ap 3 (1 -s 4 ) 3 2 sds + o(1)   .
In summary, when φ 0 → 0 (i.e. ω → 0), we have established that there exists C = C(p, q) > 0 such that when p = 5 we have

J(ω, p, q) = (5 -p)φ 7-3p 2 0 C(1 + o(1)),
and when p = 5 we have

J(ω, p, q) = (5 -q)a q φ q-9 0 C(1 + o(1)).
This gives the desired result.

We now discuss the case a p < 0 and a q > 0.

Chapter 2. Standing waves of the double power 1d NLS Proposition 2.4.2. Let a p < 0 and a q > 0.

1. Assume that p < 7 3 . Then J 0 (p, q) ∈ R and the following holds. (a) If 2p + q < 7, then J 0 (p, q) > 0.

(b) If 2p + q = 7, then J 0 (p, q) = 0.

(c) If 2p + q > 7, then J 0 (p, q) < 0.

Assume that p 7

3 . Then J 0 (p, q) = -∞. We start with some preliminaries. To establish the first part of Proposition 2.4.2, we will calculate J 0 in terms of the Beta function. Recall that the Beta function, also called Euler integral of the first kind, is a special function closely related to the Gamma function. It is defined for x > 0 and y > 0 by the integral

B(x, y) = 1 0 t x-1 (1 -t) y-1 dt. (2.4.1)
The relation between the Beta function and the Gamma function is given by (see e.g [1])

B(x, y) = Γ(x)Γ(y) Γ(x + y) .
We introduce the function H defined for x > 0 and y > 0 by

H(x, y) = 1 0 t x-1 (1 -t y ) (1 -t) 3 2 dt. (2.4.2)
The relation between H and B is given in the following lemma.

Lemma 2.4.3. For x > 0 and y > 0, we have

H(x, y) = -(2x -1)B x, 1 2 + (2x + 2y -1)B x + y, 1 2 . (2.4.3) Proof. Let u(t) = t x-1 (1 -t y ). Rewrite 1 (1 -t) 3 2 = v (t) - 1 (1 -t) 1 2
, where

v(t) = 2 (1 -t) 1 2 -2(1 -t) 1 2 = 2t (1 -t) 1 2
.

We have

H(x, y) = 1 0 u(t) v (t) - 1 (1 -t) 1 2 dt, =u(1 -)v(1 -) -u(0 + )v(0 + ) - 1 0 u (t)v(t)dt - 1 0 u(t) (1 -t) 1 2 dt, =0 - 1 0 2tu (t) + u(t) (1 -t) 1 2
dt.

Above we have used u(1) = 0 of order 1 to cancel the singularity of v(1 -) of order 1 2 , and v(0) = 0 with order 1 to cancel the singularity of u(0 + ) of order x -1. Note that

2tu (t) + u(t) = (2x -1)t x-1 -(2x + 2y -1)t x+y-1 .
Therefore, using the definition of B given in (2.4.1) with y = 1 2 , we have

H(x, y) = -(2x -1)B x, 1 2 + (2x + 2y -1)B x + y, 1 2 .
This concludes the proof.

The value J 0 (p, q) may be expressed using B as follows.

Lemma 2.4.4. Let a p < 0 and a q > 0. Assume that 1 < p < 7/3. Then

J 0 (p, q) = (7 -2p -q)C 0 B 7 -3p 2(q -p) , 1 2
,
where C 0 is a positive constant explicitly known (given by (2.4.5)).

The first part of Proposition 2.4.2 is a direct consequence of Lemma 2.4.4.

Proof of Lemma 2.4.4. Let 1 < p < 7/3. Recall that J(ω, p, q) = C(φ 0 )F (φ 0 ), with C(φ 0 ) > 0 and F given by (2.3.2). Observe that, using the value of φ * given in (2.2.3), we may introduce the constant

C * = 2a q q + 1 φ q+1 * = - 2a p (p + 1) φ p+1 * .
Using the definition (2.3.4) of C(φ 0 ) and the expression (2.2.2) of ∂ ω φ 0 , we have lim

φ 0 →φ * C(φ 0 ) = C(φ * ) = φ 5 * 2C * (q -p) > 0, lim φ 0 →φ * Φ p = 2a p (p + 1) φ p+1 * (1 -s p-1 ) = -C * (1 -s p-1 ), lim φ 0 →φ * Φ q = 2a q (q + 1) φ q+1 * (1 -s q-1 ) = C * (1 -s q-1 ).
As a consequence, we get

lim φ 0 →φ * (Φ p + Φ q ) = C * (s p-1 -s q-1 ), lim φ 0 →φ * ((5 -p)Φ p + (5 -q)Φ q ) = C * -(5 -p)(1 -s p-1 ) + (5 -q)(1 -s q-1 ) .
As a consequence,

F (φ * ) = C -1 2 * 1 0 -(5 -p)(1 -s p-1 ) + (5 -q)(1 -s q-1 ) (s p-1 -s q-1 ) 3 2 sds = C -1 2 * 1 0 -(q -p)(1 -s p-1 ) + (5 -q)(s p-1 -s q-1 ) (1 -s q-p ) 3 2 s 5-3p 2 ds = C -1 2 * -(q -p) 1 0 (1 -s p-1 )s 5-3p 2 (1 -s q-p ) 3 2 ds + (5 -q) 1 0 s 3-p 2
1s q-p -1 2 ds .

(2.4.4) Changing variable t = s q-p , we obtain

F (φ * ) = C -1 2 * - 1 0 (1 -t p-1 q-p )t 7-p-2q 2(q-p) (1 -t) 3 2 ds + 5 -q q -p 1 0 t 5+p-2q 2(q-p) (1 -t) -1 2 ds .
We now use B and H to express the above quantity. Setting

(x 1 , y 1 ) = 7 -p -2q 2(q -p) + 1, p -1 q -p = 7 -3p 2(q -p) , p -1 q -p , (x 2 , y 2 ) = 5 + p -2q 2(q -p) + 1, 1 2 = 5 -p 2(q -p) , 1 2 ,
we get

F (φ * ) = C -1 2 * -H(x 1 , y 1 ) + 5 -q q -p B(x 2 , y 2 ) .
Observe that we have assumed p < 7 3 , p < q, which ensures that x 1 , x 2 , y 1 , y 2 are positive. This a posteriori justifies the fact that J 0 (p, q) is finite. The formula (2.4.3) allows us to express H(x 1 , y 1 ) in the following way (using

y 2 = 1/2): H(x 1 , y 1 ) = -(2x 1 -1)B (x 1 , y 2 ) + (2x 1 + 2y 1 -1)B (x 1 + y 1 , y 2 ) . It turns out that -(2x 1 -1) = - 7 -2p -q q -p , (2x 1 +2y 1 -1) = 5 -q q -p , x 1 +y 1 = 5 -p 2(q -p) = x 2 .
As a consequence, there is a simplification in the expression of F (φ * ), which becomes

F (φ * ) = C -1 2 * 7 -2p -q q -p B (x 1 , y 2 ) .

The slope at the endpoints 33

Setting

C 0 = C -1 2 * q -p C(φ * ) > 0 (2.4.5)
gives the desired result.

Lemma 2.4.5. Assume that a p < 0 and a q > 0. For p 7/3 and 1 < p < q, we have

lim ω→0 J(ω, p, q) = -∞.
The second part of Proposition 2.4.2 is a direct consequence of Lemma 2.4.5.

Proof. Coming back to the expression (2.4.4) of F (φ * ) in the proof of Lemma 2.4.4, we observe that if 5-3p 2 -1, i.e. p 7 3 , then F (φ * ) = -∞, and, since lim φ 0 →φ * C(φ 0 ) = C(φ * ) > 0, we also have J 0 (p, q) = -∞ when p 7 3 .

The large frequency case

In this section, we determine the limit of J(ω, p, q) when ω tends to ω * . Let J * be defined by J * (p, q) = lim ω→ω * J(ω, p, q). We first consider the case where a q > 0.

Proposition 2.4.6. Let a q > 0. The following holds.

1. If 1 < q < 7 3 , then J * (p, q) = 0 + . 2. If q = 7 3 , then 0 < J * (p, q) < ∞. 3. If 7 3 < q < 5, then J * (p, q) = ∞. 4. If q = 5, then J * (p, q) = 0 sign(ap) . 5. If 5 < q, then J * (p, q) = -∞.
Proof. Since a q > 0, we have ω * = ∞ and therefore φ * = ∞. Following similar arguments as in the proof of Proposition 2.4.1, as φ 0 → ∞, for q = 5, we have

C(φ 0 ) = φ 4-q 0 q + 1 4a q (q -1) + o(1) , F (φ 0 ) = (5 -q)φ -q+1 2 0 1 0 2a q q + 1 (1 -s q-1 ) -1 2 sds + o(1) .
As a consequence, for q = 5, when φ 0 → ∞ (i.e. ω → ∞), there exists C = C(a q , q) > 0 such that

J(ω, p, q) = (5 -q)φ 7-3q 2 0 C(1 + o(1)).
Chapter 2. Standing waves of the double power 1d NLS

In the particular case q = 5, we instead write

F (φ 0 ) = a p (5 -p)φ p-8 0   1 0 2 p+1 (1 -s p-1 ) aq 3 (1 -s 4 ) 3 2 sds + o(1)   .
and therefore we get

J(ω, p, q) = a p (5 -p)φ p-9 0 C(1 + o(1)
). The two estimates on J lead to the desired result.

Then we consider the case where a q < 0 (and thus a p > 0 to ensure existence of standing waves). Proposition 2.4.7. Let a p > 0, a q < 0 and 5 p. Then

J * (p, q) = ∞.
Proposition 2.4.7 does not cover the whole possible range of p and q. As it was not necessary in our analysis, we did not try to cover the remaining cases.

Proof of Proposition 2.4.7. By construction, ω * = ω(φ * ) is the value of ω at which ∂ φ 0 ω(φ * ) = 0. As a consequence, we have

lim φ 0 →φ * ∂φ 0 ∂ω = ∞,
which, given the value (2.3.4) of C(φ 0 ), readily implies

lim φ 0 →φ * C(φ 0 ) = ∞.
Using the expressions of (2.3.1) of Φ p and Φ q and the expression (2.3.2) of F we have

F (φ * ) = s 0 2a p (5-p)(φ * ) p+1 (1-s p-1 ) p+1 + 2a q (5-q)(φ * ) q+1 (1-s q-1 ) q+1 2a p (φ * ) p+1 (1-s p-1 ) p+1 + 2a q (φ * ) q+1 (1-s q-1 ) q+1 3 2
sds.

If p = 5, then we have F (φ * ) > 0 and the conclusion follows. From now on, assume that p > 5. Recalling the value of φ * given in (2.2.3), we infer that

2a p (5 -p)(φ * ) p+1 (1 -s p-1 ) p + 1 + 2a q (5 -q)(φ * ) q+1 (1 -s q-1 ) q + 1 = 2a q (φ * ) p+1 (1 -s q-1 ) (5 -q) q + 1 a p (5 -p)(q + 1) a q (5 -q)(p + 1) (1 -s p-1 ) (1 -s q-1 ) + (φ * ) q-p = 2a p (φ * ) p+1 (1 -s q-1 ) (5 -q) q + 1 (5 -p)(q + 1) (5 -q)(p + 1) (1 -s p-1 ) (1 -s q-1 ) - p -1 q -1 q + 1 p + 1 = 2a p (φ * ) p+1 (1 -s q-1 ) (5 -q) p + 1 (5 -p) (5 -q) (1 -s p-1 ) (1 -s q-1 ) - p -1 q -1 > 0,
where we have used in particular Lemma 2.3.3 for the last inequality. This implies that F (φ * ) > 0 which, since J(ω, p, q) = C(φ 0 )F (φ 0 ), finishes the proof.

Determination of the sign of the slope

In this section, we determine for each possible values of a p , a q , p and q the sign of J(ω, p, q). Combined with the stability criteria of Section 2.2.2, this will prove Theorem 2.1.1. The general strategy of our proofs is the following. Recall from Lemma 2.3.1 that J(ω, p, q) = C(φ 0 )F (φ 0 ), where

F (φ 0 ) = 1 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2

sds,

and C(φ 0 ) > 0. Moreover, ω and φ 0 are in an increasing one to one correspondence. Hence, to determine the sign of J, it is sufficient to determine the sign of F (φ 0 ). To do this, we have two ingredients at our disposal. First, it is usually not difficult to establish that F has a constant sign on intervals of the type (φ * , φ 0,1 ) or (φ 0,1 , φ * ). On the other hand, the expression for ∂ φ 0 F (φ 0 ) given in Lemma 2.3.2 allows us to show that ∂ φ 0 F (φ 0 ) has a constant sign on intervals of the type (φ * , φ 0,2 ) or (φ 0,2 , φ * ). If the intervals of the two ingredients overlap and if the signs are matching, the conclusion will follow. For example, if F (φ 0 ) > 0 on (φ * , φ 0,1 ), and ∂ φ 0 F (φ 0 ) > 0 on (φ 0,2 , φ * ) and φ 0,1 > φ 0,2 , then F (φ 0 ) > 0 on (φ * , φ * ). The detail of each case is given in the following sections.

The focusing-focusing case

In this section, we consider the case a p > 0, a q > 0. In this case we have Φ p > 0 and Φ q > 0.

Lemma 2.5.1. Let a p > 0, a q > 0 and q 5. Then for all ω ∈ (0, ∞) we have

J(ω, p, q) > 0,
and the family of standing waves is of type S.

Proof. If q 5, then 5p > 0 and 5q 0. Therefore for any φ 0 ∈ (0, ∞) we have

F (φ 0 ) > 0,
which gives the desired conclusion.

Chapter 2. Standing waves of the double power 1d NLS Lemma 2.5.2. Let a p > 0, a q > 0 and p 5. Then for all ω ∈ (0, ∞) we have

J(ω, p, q) < 0,
and the family of standing waves is of type U.

Proof. If p 5, then 5p 0 and 5q < 0. Therefore for any φ 0 ∈ (0, ∞) we have

F (φ 0 ) < 0,
which gives the desired conclusion.

The remaining case p < 5 < q is a bit more involved to consider.

Lemma 2.5.3. Let a p > 0, a q > 0 and p < 5 < q. There exists φ 0,1 (explicitly given in (2.5.1)) such that if φ q-p 0 > φ q-p 0,1 then

F (φ 0 ) < 0.
Proof. Using the formula (2.3.2) of F (φ 0 ) and replacing in the numerator of the integrand Φ p and Φ q by their expressions (2.3.1), we obtain

F (φ 0 ) = 1 0 (5 -p) 2ap p+1 φ p+1 0 (s -s p ) + (5 -q) 2aq q+1 φ q+1 0 (s -s q ) (Φ p + Φ q ) 3 2 ds. Let l(s) = (5 -p) 2a p p + 1 φ p+1 0 (s -s p ) + (5 -q) 2a q q + 1 φ q+1 0 (s -s q ), and
k(s) = (Φ p + Φ q ) 3 2 .
We may reformulate l(s) in the following way:

l(s) = (5 -p) 2a p p + 1 φ p+1 0 + (5 -q) 2a q q + 1 φ q+1 0 1 -s q-1 1 -s p-1 (s -s p ).
From Lemma 2.3.3, we know that the function s → 1-s q-1 1-s p-1 is increasing from 1 to q-1 p-1 when s goes from 0 to 1. Let φ 0,0 be given by φ q-p 0,0 = -a p (5p)(q + 1)(p -1) a q (5q)(p + 1)(q -1) ,

and assume from now on that φ 0 > φ 0,0 . Then

lim s→1 l(s) s -s p < 0,
and there exists s * ∈ [0, 1) such that l(s) > 0 for s ∈ (0, s * ) and l(s) < 0 for s ∈ (s * , 1).

Define

k by k(s) = k(s) k(s * ) . Then k(s * ) = 1.
As k and therefore k is a positive decreasing function of s, for all s ∈ (0, 1) we have

l(s) k(s) < l(s).
Integrating over (0, 1), we obtain

F (φ 0 ) < 1 k(s * ) 1 0 l(s)ds,
and F (φ 0 ) will be negative if the integral in the right member is. Define φ 0,1 > φ 0,0 by

φ q-p 0,1 = - a p (5 -p)(p -1)(q + 1) 2 a q (5 -q)(q -1)(p + 1) 2 . (2.5.1) If φ 0 > φ 0,1 , then 1 0 l(s)ds = (5 -p) a p p + 1 φ p+1 0 p -1 p + 1 + (5 -q) a q q + 1 φ q+1 0 q -1 q + 1 < 0.
Hence for any φ 0 > φ 0,1 we have F (φ 0 ) < 0. This concludes the proof.

Lemma 2.5.4. Let a p > 0, a q > 0 and p < 5 < q. Let γ = p+1 2 . There exists φ 0,2 (explicitly given in (2.5.4)) such that the integrand I γ of F γ defined in (2.3.5) verifies ∂I γ ∂φ 0 < 0 for all φ 0 ∈ (0, φ 0,2 ).

Proof. As γ = p+1 2 , from Lemma 2.3.2 we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -q)(p -q)Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ γ-1 0 Φ q (p -q) (5 + 2q -3p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 5 2
.

As a consequence ∂ φ 0 I γ < 0 if

(5 + 2q -3p)Φ p + (5 -q)Φ q > 0.
Chapter 2. Standing waves of the double power 1d NLS Replacing Φ p and Φ q by their expressions (2.3.1), this is equivalent to

(5 + 2q -3p) a p p + 1 φ p+1 0 (1 -s p-1 ) + (5 -q) a q q + 1 φ q+1 0 (1 -s q-1 ) > 0. (2.5.2)
Since p < 5 < q, we have 5 + 2q -3p > 0, and therefore (2.5.2) becomes

φ q-p 0 < - a p a q (5 + 2q -3p) (5 -q) (q + 1) (p + 1) (1 -s p-1 ) (1 -s q-1 ) . (2.5.3)
We know from Lemma 2.3.3 that

p -1 q -1 < 1 -s p-1 1 -s q-1 . Define φ q-p 0,2 = - a p a q (5 + 2q -3p) (5 -q) (q + 1) (p + 1) (p -1) (q -1) . (2.5.4) If φ 0 < φ 0,2 then (2.5.3
) is verified, which concludes the proof.

Lemma 2.5.5. Let a p > 0, a q > 0 and p < 5 < q. The function F γ (φ 0 ) has at most one zero in (0, ∞).

Proof. As p < q, we have

3(p -1)(p -q) < 0, hence (5 -p)(q + 1) < (p + 1)(5 + 2q -3p).
It implies

- a p a q (5 -p) (5 -q) (q + 1) 2 (p + 1) 2 (p -1) (q -1) < - a p a q (5 + 2q -3p) (5 -q) (q + 1) (p + 1) (p -1) (q -1)
.

Therefore, we have φ q-p 0,1 < φ q-p 0,2 . We know from Lemma 2.5.3 that F γ (φ 0 ) < 0 if φ 0 ∈ (φ 0,1 , ∞), and from Lemma 2.5.4 that F γ (φ 0 ) is decreasing for all φ 0 ∈ (0, φ 0,2 ). As φ 0,1 < φ 0,2 , this implies that F γ (φ 0 ) has at most one zero. Lemma 2.5.6. Let a p > 0, a q > 0 and p < 5 < q. There exists ω 1 ∈ (0, ∞) such that J(ω, p, q) > 0 for ω < ω 1 , J(ω 1 , p, q) = 0, J(ω, p, q) < 0 for ω > ω 1 , and the family of standing waves is of type SU.

Proof. From Proposition 2.4.1, we know that J(ω, p, q) > 0 for ω close to 0. Combined with Lemmas 2.5.3 and 2.5.5, this implies the desired result.

The focusing-defocusing case

In this section, we consider the case a p > 0, a q < 0. In this case Φ p > 0 and Φ q < 0.

Lemma 2.5.7. Let a p > 0, a q < 0 and p 5 < q. For any ω ∈ (0, ω * ), we have J(ω, p, q) > 0, and the family of standing waves is of type S.

Proof. We have 5p 0, and 5q < 0. Therefore F (φ 0 ) > 0 for any φ 0 ∈ (0, φ * ), which gives the desired result.

Lemma 2.5.8. Let a p > 0, a q < 0 and p < q 5. Let γ = p+1 2 . Then the integrand I γ of F γ defined in (2.3.5) verifies

∂I γ ∂φ 0 > 0 for all φ 0 ∈ (0, φ * ).
Proof. From Lemma 2.3.2 with γ = p+1 2 , we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -q)(p -q)Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

Since 5-q 0, p-q < 0 and Φ q < 0, we have ∂Iγ ∂φ 0 > 0 for any φ 0 ∈ (0, φ * ).

Lemma 2.5.9. Let a p > 0, a q < 0 and 5 < p < q. Let γ = pq + 3. Then the integrand I γ of F γ defined in (2.3.5) verifies

∂I γ ∂φ 0 > 0 for all φ 0 ∈ (0, φ * ).
Proof. Let γ = pq + 3.

∂I γ ∂φ 0 = φ γ-1 0 ((5 -p)(p -2q + 5)Φ p + (5 -q)(2p -3q + 5)Φ q ) (Φ p + Φ q ) 2 (Φ p + Φ q ) 5 2 -φ γ-1 0 3(p -q) 2 Φ p Φ q 2 (Φ p + Φ q ) 5 2
.

The sign of ∂Iγ ∂φ 0 is the same as the sign of the numerator of the fraction. Factoring out Φ 2 p , the sign is the same as the one of the second order polynomial in Φq Φp given by

(5 -q)(2p -3q + 5) Φ q Φ p 2 + 2(5 -p)(2p -3q + 5) Φ q Φ p + (5 -p)(p -2q + 5).
As 2p -3q + 5 < 0 and 5q < 0, the coefficient of the term of order 2 is positive. Therefore to show that the polynomial is positive, it is sufficient to show that the discriminant ∆, given by

∆ = 4(5 -p)(2p -3q + 5)((5 -p)(2p -3q + 5) -(5 -q)(p -2q + 5)), = -8(5 -p)(2p -3q + 5)(p -q) 2 ,
is negative. We have 2p -3q + 5 < 0 and 5p < 0, therefore ∆ < 0. This concludes the proof.

Lemma 2.5.10. Let a p > 0, a q < 0.

-Let p < q 5. Then for any ω ∈ (0, ω * ), we have

J(ω, p, q) > 0,
and the family of standing waves is of type S. -Let 5 < p < q. Then there exist ω 1 ∈ (0, ∞) such that J(ω, p, q) < 0 for ω < ω 1 , J(ω 1 , p, q) = 0, J(ω, p, q) > 0 for ω > ω 1 , and the family of standing waves is of type US.

Proof. In both cases, we infer from Lemmas 2.5.8 and 2.5.9 that for any ω ∈ (0, ω * ), the function ω → J(ω, p, q) changes sign (from negative to positive) at most once on ω ∈ (0, ω * ).

To establish the desired conclusion, we consider the values of J close to the endpoints. As ω → 0, we have established in Proposition 2.4.1 that for ω close to 0, we have J(ω, p, q) > 0 for p 5, J(ω, p, q) < 0 for p > 5.

As J is increasing, this gives the conclusion for the first part of the Lemma.

For the second part of the Lemma, we look at the limit ω → ω * (i.e. φ 0 → φ * ). From Proposition 2.4.7, for 5 < p < q and for ω close to ω * we have J(ω, p, q) > 0, which gives the second part of the Lemma.

The defocusing-focusing case

In this section, we consider the case a p < 0, a q > 0. In this case Φ p < 0 and Φ q > 0.

Lemma 2.5.11. Let a p < 0, a q > 0 and p < q < 5. Let γ = q+1

2 . If 3q 2p + 5, then the integrand I γ of F γ defined in (2.3.5) verifies

∂I γ ∂φ 0 > 0 for all φ 0 ∈ (φ * , ∞).
Proof. As γ = q+1 2 , from Lemma 2.3.2 we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -p)(q -p)Φ p ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ γ-1 0 Φ p (q -p) (5 -p)Φ p + (5 -p)Φ q + 3(p -q)Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ γ-1 0 Φ p (q -p) (5 -p)Φ p + (5 + 2p -3q)Φ q (Φ p + Φ q ) 5 2
.

As 0 < 5p and 2p + 5 -3q 0 we have

(5 -p)Φ p + (5 + 2p -3q)Φ q < 0.
As a consequence ∂Iγ φ 0 > 0 for all φ 0 ∈ (φ * , ∞) when 5 + 2p -3q 0, which is the desired conclusion. Lemma 2.5.12. Let a p < 0, a q > 0 and p < q < 5. Let γ = pq + 3. If 3q < 2p + 5 then the integrand I γ of F γ defined in (2.3.5) verifies

∂I γ ∂φ 0 > 0 for all φ 0 ∈ (φ * , ∞).
Proof. As γ = pq + 3, from Lemma 2.3.2 we have

∂I γ ∂φ 0 = φ γ-1 0 ((5 -p)(p -2q + 5)Φ p + (5 -q)(2p -3q + 5)Φ q ) (Φ p + Φ q ) 2 (Φ p + Φ q ) 5 2 + φ γ-1 0 -3(p -q) 2 Φ p Φ q 2 (Φ p + Φ q ) 5 2
.

If the numerator of the fraction is positive then the derivative is positive. Factorizing out Φ 2 p , the sign of the numerator is the same as the one of the quadratic polynomial in Φq Φp given by

(5 -q)(2p -3q + 5) Φ q Φ p 2 + 2(5 -p)(2p -3q + 5) Φ q Φ p + (5 -p)(p -2q + 5).
As 2p -3q + 5 > 0 and 5q > 0, the coefficient of the term of order 2 is positive. Therefore to show that the polynomial is positive, it is sufficient to show that the discriminant ∆, given by

∆ = 4(5 -p)(2p -3q + 5)((5 -p)(2p -3q + 5) -(5 -q)(p -2q + 5)), = -8(5 -p)(2p -3q + 5)(p -q) 2 ,
is negative. We have 2p -3q + 5 > 0 and 5p > 0, therefore ∆ < 0. This concludes the proof.

Lemma 2.5.13. Let a p < 0, a q > 0 and p < q < 5.

-If q 7 -2p, then for any ω ∈ (0, ∞), we have

J(ω, p, q) > 0,
and the family of standing waves is of type S. -If q > 7 -2p, then there exist ω 1 ∈ (0, ∞) such that J(ω, p, q) < 0 for ω < ω 1 , J(ω 1 , p, q) = 0, J(ω, p, q) > 0 for ω > ω 1 , and the family of standing waves is of type US.

Proof. Lemmas 2.5.11 and 2.5.12 implies F γ changes sign only once on (0, ω * ). From Proposition 2.4.2, we know that as ω → 0, we have J(ω, p, q) > 0 when q 7 -2p, which gives the conclusion for the first part of the Lemma. When q > 7 -2p, from Proposition 2.4.2, we know that as ω → 0, we have J(ω, p, q) < 0. Since, from Proposition 2.4.6, we know that J(ω, p, q) > 0 for large ω, the conclusion follows for the second part of the Lemma. Lemma 2.5.14. Let a p < 0, a q > 0 and p < 5 q. For any ω ∈ (0, ∞), we have J(ω, p, q) < 0, and the family of standing waves is of type U .

Proof. We have 5p > 0, 5q 0, Φ p < 0 and Φ q > 0. Therefore we directly see on the expression (2.3.2) of F (φ 0 ) that F (φ 0 ) < 0, which gives the desired result. Lemma 2.5.15. Let a p < 0, a q > 0 and 5 p < q. For any ω ∈ (0, ∞), we have J(ω, p, q) < 0, and the family of standing waves is of type U .

Proof. We know that φ * < φ 0 , therefore -ap aq q+1 p+1 < φ q-p 0 . As 5-p 5-q < 1, we have ap aq q+1 p+1

5-p 5-q < φ q-p 0 . From Lemma 2.3.3 we know that 1-s p-1 1-s q-1 < 1, hence - a p a q (q + 1) (p + 1) (5 -p) (5 -q) 1 -s p-1 1 -s q-1 < φ q-p 0 ,
which is equivalent to

(5 -p) (5 -q) < - Φ q Φ p , which implies (5 -p)Φ p + (5 -q)Φ q < 0.
This implies that F (φ 0 ) < 0 which gives the desired result.

The critical frequency

Observe that, as a by-product of the analysis of the previous sections, we should have instability at the critical frequency (under regularity assumptions on the nonlinearity) when there is a stability change. Indeed, we have

∂ 2 ω M (φ ω ) = ∂ ω C(φ 0 )φ -γ 0 F γ (φ 0 ) = ∂ ω φ 0 ∂ φ 0 C(φ 0 )φ -γ 0 F γ (φ 0 ) + C(φ 0 )φ -γ 0 ∂ φ 0 F γ (φ 0 ) .
At the stability change, we have F (φ 0 ) = 0. Therefore, at the stability change,

∂ 2 ω M (φ ω ) = (∂ ω φ 0 )C(φ 0 )φ -γ 0 ∂ φ 0 F γ (φ 0 ).
As we have shown that in this case ∂ φ 0 F γ (φ 0 ) = 0, the criterion (2.2.4) holds.

Numerical experiments

To explore further the stability/instability of standing waves, we have performed a series of numerical experiments in the case a p < 0, a q > 0, 1 < p < q < 5.

The Python language and the specific libraries Numpy, Scipy and Matplotlib have been used to perform the experiments. The code is made available in [53].

The critical surface for stability/instability

We first analyzed the critical surface in (ω, p, q) separating instability from stability. To this aim, we first have implemented the calculation of J(ω, p, q). The function integrate.quad has been used to perform the integration. While the results are overall satisfactory, in some cases the function returned incorrect results, with problems increasing as ω was taken closer to 0.

To estimate the critical ω at given (p, q), we have used the classical bisection method, which has the advantage of being very robust. The algorithm is divided into two parts.

First, we find an initial interval [ω 0 , ω 1 ] in which we are sure that ω → J(ω, p, q) changes sign. A natural choice for ω 0 is 0. To find a suitable ω 1 , we simply start with ω 1 = 1 and test if J(ω 1 , p, q) > 0. If not, we replace ω 1 by 2ω 1 and repeat until J(ω 1 , p, q) > 0. To avoid running an infinite loop, we break it when ω 1 > 10 10 and do not search for ω c in these cases. Second, we apply the bisection method to search for a root of J(ω, p, q) inside [ω 0 , ω 1 ]. As this approach, while being efficient, is also relatively slow, we took advantage of the computer power of our department to run computations in parallel on the (p, q) ∈ [1, 5] × [1,[START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF] grid with dp = dq = 0.01.

We have represented the critical surface {(p, q, ω c (p, q))}

for a p = -1 and three different values of a q = 1/2, 1, 2 in Figure 2.1. {(p,q,ω c (p,q))} for a p = -1 and a q = 1/2, 1, 2. The white lines represent q = 7 -2p and q = p, where the transition from ω c = 0 to ω c > 0 occurs.

Several observations can be made on the critical surface. As (p, q) approaches the line q = 5, we have ω c (p, q) → ∞, which is consistent with the fact that standing waves are all unstable on this line.

It can be observed that on the line q = 7 -2p the transition is continuous, no matter the value of a q . To the contrary, the transition is continuous on the line p = q when a q 1, whereas it becomes discontinuous when a q < 1, in which case ω c (p, q) → ∞ as q → p.

To investigate more the transition close to the lines q = 7 -2p and q = p, we plot slices of the critical surface for a fixed value of q in Figure 2.2. We chose to present the results when q = 4, but similar results are obtained with other values of q. On Figure 2.2, we observe that when a q = |a p | = 1, the transition between ω c (p, q) = 0 and ω c (p, q) > 0 at q = p and q = 7 -2p is Lipschitz. When a q = 2 > |a p | = 1, the transition seems smoother (but closer observations will reveal otherwise) when q = p, whereas it remains Lipchitz when q = 7 -2p. To the contrary, when a q = 1/2 < |a p | = 1, the transition is discontinuous when q = p, whereas it seems smoother at q = 7 -2p. transition from ω c > 0 to ω c = 0 on fixed q slices of Figure 2.3, we realize that the transition on the left (q = 7 -2p) seems to be always only Lipschitz, contrary to what could be inferred from the previous observation. On the other hand, the previous observation when p = q is confirmed: the transition seems smooth when a q = 2, Lipschitz when a q = 1, and discontinuous when a q = 1/2. This is reflecting the fact that when p → q, the family of soliton profiles has a different behavior for different values of a q . When a q > |a p |, soliton profiles for p = q exist and are stable (hence ω c (p, q = p) = 0), whereas for a q = |a p | the two nonlinearities exactly compensate and for |a p | > a q the defocusing nonlinearity becomes the dominant one (and solitary waves do not even exist).

From the previous observations, we know that at fixed q the map p → ω c (p, q) has a unique maximum if a q = 1 or a q = 2 (if a q = 1/2, we have seen that the map increases towards infinity as p approaches q). Denote by p max (q) the value realizing this maximum, i.e.

ω c (p max (q), q) = max 1<p<5 ω c (p, q).
The line {(p max (q), q), q > 7/3} is represented in Picture 2.4. When a q = 1, when a p = -1 and a q = 1 p = 7/3 p = q q = 7 -2p pmax(q) when a p = -1 and a q = 2 p = 7/3 p = q q = 7 -2p pmax(q) Figure 2.4 -Curve of the argument of max p ω c (p, q) in terms of q we observe that the line is tangent to the line p = q when q is close to 7/3 or close to 5. On the other hand, when a q = 2, the line seems to be tangent to the line p = 7/3 when q is close to 7/3. It approaches the point (5, 5) as q goes to 5, but does not seem to be tangent to the line p = q (it was however not possible to obtain numerically a relevant picture closer to q = 5, which leaves open the question of the behavior when q is close to 5).

Evolution for initial data close to standing waves

We now turn to numerical experiments for the stability/instability of solitary waves for the flow of (2.1.1). For the experiments, we have used the Crank-Nicolson scheme with relaxation presented in [START_REF] Besse | A relaxation scheme for the nonlinear Schrödinger equation[END_REF] which has been proved to be efficient for the numerical simulation of the Schrödinger flow (see e.g. [START_REF] Antoine | Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations[END_REF] for the comparison of various schemes used for the dynamical simulations of the nonlinear Schrödinger flow).

For a time discretization step δ t (typically δ t = 10 -3 ), denote by u n the approximation of u at time t n = nδ t . The semi-discrete (in time) relaxation scheme is then given by   

φ n+ 1 2 +φ n-1 2 2 = a p |u n | p + a q |u n | q , i u n+1 -u n δt + ∂ xx u n+1 +u n 2 = -u n+1 +u n 2 φ n+ 1 2 ,
with the understanding that u 0 = u 0 and φ -1 2 = a p |u 0 | p + a q |u 0 | q . For the implementation, the scheme is further discretized in space with second order finite differences for the second derivative operator, with Dirichlet boundary conditions.

We have performed simulations for (p, q) on the line q = 2p -1, as for this range of exponents explicit formulas are available for solitary wave profiles (see e.g. [START_REF] Fei | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF]) and can be used easily to construct initial data. Considering other ranges of (p, q) would have been possible, to the extend of additional computations to first obtain numerically solitary waves. As we do not expect different behavior to occur for other values of (p, q), the restriction to the line q = 2p -1 is harmless.

The initial data that we construct are all based on a solitary wave profile φ ω . They are of the form u 0 = φ ω + εψ, where 0 < ε 1 is used to adjust the size of the perturbation and ψ is the direction of perturbation, which can be for example

ψ = φ ω , ψ = φ ω cos, ψ = φ ω tanh, ψ = φ ω (• -3).
As our numerical scheme uses Dirichlet conditions at the bounds of the space interval, we have chosen to work with well-localized perturbation in order to avoid possible numerical reflections due to the boundary conditions. Our experiments consisted in taking one of the previous possibility as initial data, running the simulation of the nonlinear Schrödinger flow, and observe the pattern of the outcome. It turns out that after running numerous simulations, we have observed only three possible types of behavior: Chapter 2. Standing waves of the double power 1d NLS -Stability; -Growth followed by slightly decreasing oscillations; -Dispersion. Observe that our numerical results are in part similar to the ones obtained and discussed in further details in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF]Section 4] in the case of the 2d cubic-quintic (focusing-defocusing) nonlinear Schrödinger equation.

Stability means that the solution does not leave the neighborhood of φ ω (up to phase shift and translations). We obviously expect to see this behavior in the cases where the values of the parameters p, q, and ω ensure that the solitary wave will be stable. However, one thing which is not easily decided by the theory is the size of the basin of stability of the solitary wave. In other words, finding a perturbation of the solitary wave sufficiently large to be visible, but small enough so that the corresponding solution remains in the vicinity of the solitary wave requires delicate adjustments.

An example of a stable behavior is provided in Figure 2.5. Observe that while on the global scale the solution seems to be behave exactly as a solitary wave (left picture), when getting a closer look at the maximum value (right picture) we observe small oscillations (with an amplitude of order 0.03).

0 = (1 + ε)φ ω , ε = 10 -2 .
The second behavior consists in a first phase of focusing growth of the profile, which is similar to what can be observed when instability of solitons is by blow-up (e.g. for power-type supercritical nonlinearities. However, after a certain time, the focusing phase stops and is followed by a phase in which the solution seems to oscillate around another profile. The size of the oscillation is decaying, but at a slow pace, and we have not run the simulation long enough to observe convergence toward a final state. An example of such a behavior is presented in Figure 2.6.

Finally, the third behavior that we have observed could be characterized as scattering, as the profile of the solution is simultaneously decreasing in height

0 = (1 + ε)φ ω , ε = 10 -2 .
while spreading over the whole line. As before, the decay is rather slow and we have not run the simulation long enough for the solution to converge to 0. An example of such a behavior is presented in Figure 2 a p = -1, a q = 1, p = 2.67, q = 4.33, ω = 1.00, ω c = 2.14, init=2

u 0 = (1 -ε)φ ω , ε = 10 -2 .

Notebooks

This section contains the Python code that we have developed in order to illustrate our results. The first notebook provided is dedicated to the identification of the critical ω value, at which stability changes occur. The critical surface has been represented in (ω, p, q) to provide a clear visual representation of the transition between instability and stability. In addition, we have also included a second notebook in this section. This notebook centers on numerical experiments and specifically explores the evolution of initial data close to standing waves. print(f'Houston, we have a problem with {key}') pass my_shelf.close() [START_REF] Berestycki | Équations de champs scalaires euclidiens non linéaires dans le plan[END_REF]: print(f"a1={a1}") print(f"a2={a2}") ax1.plot(p1[p1>7/3],p1[p1>7/3],zorder=3,color='w') ax1.plot(p1[p1<7/3],7-2*p1[p1<7/3],zorder=3,color='w') ax1.azim = -120 ax1.set_title(rf"Critical surface when $a_p={a1}$ and $a_q={a2}$") # choose colormap from https://matplotlib.org/stable/tutorials/colors/ →colormaps.html # e.g. gist_gray or binary or inferno or viridis ax1.set_xlabel(r"$p$",fontsize=FontSize) ax1.set_ylabel(r"$q$",fontsize=FontSize) ax1.set_zlabel(r"$\omega_c$",fontsize=FontSize) ax1.xaxis.set_rotate_label(False) # disable automatic rotation ax1.yaxis.set_rotate_label(False) # disable automatic rotation ax1.zaxis.set_rotate_label(False) # disable automatic rotation fig1.savefig(f'surface_w_c_a1_is_{a1}_a2_is_{a2}.

print(f"p1={p1[1]} {p1[2]} ... {p1[-2]} {p1[-1]}") print(f"p2={p2[1]} {p2[2]} ... {p2[-2]} {p2[-1]}") a1=
→pdf',transparent=True,orientation='landscape',format='pdf', bbox_inches='tight')

Notebooks

Observing closer the transition from ω c > 0 to ω c = 0 on fixed q slices, we realize that the transition on the left (q = 7 -2p) is never smooth, whereas it is smooth when a q = 2 not smooth but continuous when a q = 1, and discontinuous when a q = 1/2. This is reflecting the fact that when p → q, the family of soliton profiles has a different behavior for different values of a q . When a q > a p , soliton profiles for p = q exist and are stable (hence ω c (p, q = p) = 0), whereas for a p = a q the two nonlinearities exactly compensate and for a p > a q the defocusing nonlinearity becomes the dominant one. →savefig(f'slice_close_up_w_c_a1_is_{a1}_a2_is_{a2}_p2_is_{value_p2}.

→pdf',transparent=True,orientation='landscape',format='pdf', bbox_inches='tight')

Notebooks

From the previous observations, we know that at fixed q the map p → ω c (p, q) has a unique maximum if a q = 1 or a q = 2 (if a q = 1/2, we have seen that the map increases towards infinity as p approaches q. Denote by p max (q) the value realizing this maximum, i.e.

ω c (p max (q), q) = max 1<p<5 ω c (p, q).
The line {(p max (q), p2), q > 7/3} is represented in Picture ??. When a q = 1, we observe that the line is tangent to the line p = q when q is close to 7/3 or close to 5. On the other hand, when a q = 2, the line seems to be tangent to the line p = 7/3 when q is close to 7/3. It approaches the point [START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF][START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF] as q goes to 5, but does not seem to be tangent to the line p = q (it was however not possible to obtain numerically a relevant picture closer to q = 5, which leaves open the question of the behavior when q is close to 5).

[12]: fig4 = plt.figure(figsize= [START_REF] Andrzej | The method of Nehari manifold[END_REF][START_REF] Andrzej | The method of Nehari manifold[END_REF]) ax4 = plt.axes() ax4.plot(np.ones_like(p2)*7/3,p2,'k-.',label='$p=7/ →3$')#,linestyle='dashdot')

ax4.plot(p1,p2,'k:',label='$p=q$') ax4.plot(p1,7-2*p1,'k--',label='$q=7-2p$')#,linestyle='dashed') ax4.set_title(f"when $a_p={a1}$ and $a_q={a2}$") ax4.set_xlabel(r"$p$", rotation=0, fontsize=FontSize) ax4.set_ylabel(r"$q$", rotation=0, fontsize=FontSize) ax4.set_ylim(7/3+0. a=np.abs(a) if a==0: print("error in a , a is 0") return a def J(w,p1,p2,a1=- To perform the numerical simulations, it might be helpful to restrict ourselves to the case q = 2p -1, for which we have explicit formulas for the standing waves (as given in the next cell).

[5]: #%%file --append 2021_07_23_script_nls_evolution.py def profile_explicit(x,beta,w=1,a1=- This function numerically solves a first order system of ODEs subject to two-point boundary conditions.

Our system is

y 1 = y 2 y 2 = ωy 1 -a p |y 1 | p-1 y 1 -a q |y 1 | q-1 y 1
with the boundary conditions φ(0) = initialvalue(φ), φ(L) = 0. def bc(ya, yb,p1,p2,w=1,a1=-1,a2=1): partial_initialvalue = partial(initialvalue,␣ →$a_q={a2}$, $p={p1:.2f}$, $q={p2:.2f}$, $\omega={w:.2f}$,␣ →$\omega_c={w_c:.2f}$, init=${init}$") ax = fig.add_subplot(111) ax.plot(Tt,max_u_t_x) ax.set_xlabel(r"$t$",fontsize=FontSize) ax.set_ylabel(r"$\max|u(t,x)|$",fontsize=FontSize) fig.savefig(f"nls_max_a1_is_{a1}_a2_is_{a2}_p1_is_{p1:. →2f}_p2_is_{p2:.2f}_w_is_{w:.2f}_w_c_is_{w_c:.2f}_init_is_{init}.

→pdf",transparent=True,orientation='landscape',format='pdf', bbox_inches='tight') [START_REF] Ethuel | Orbital stability of the black soliton to the Gross-Pitaevskii equation[END_REF]: #%%file --append 2021_07_23_script_nls_evolution.py def variables_backup(u_t_x,max_u_t_x,x,p1,p2,T=1,delta_t =␣ →1e-3,w=1,w_c="nc",c=0,a1=-1,a2=1,init=0):

# Save all variables (with in particular w_critical) to file with␣ →date in file name filename=f"nls_var_bckup_a1_is_{a1}_a2_is_{a2}_p1_is_{p1:. →2f}_p2_is_{p2:.2f}_w_is_{w:.2f}_w_c_is_{w_c:.2f}_init_is_{init}. Analysis of the quasi-periodic waves for the cubic nonlinear Schrödinger equation

Introduction

In this chapter, we consider the one dimensional cubic nonlinear Schrödinger equation

iψ t + ψ xx + b|ψ| 2 ψ = 0, (3.1.1)
where ψ : R t × R x → C and b ∈ R \ {0}. Equation (3.1.1) appears in many areas of physics such as quantum mechanics, optics, water waves and serves as a model for nonlinear dispersive wave phenomena, more generally see [START_REF] Fibich | The nonlinear Schrödinger equation[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation[END_REF]. It is said to be focusing if b > 0, where the nonlinearity is attractive and defocusing if b < 0, where the nonlinearity is repulsive. Note that (3.1.1) is invariant under the following transformations:

-spatial translation: ψ(t, x) → ψ(t, x + ξ), for ξ ∈ R, -phase multiplication: ψ(t, x) → e iα ψ(t, x), for α ∈ R. We are particularly interested in the spatially quasi-periodic solutions to (3.1.1). At least formally, they conserve the mass M , the momentum P , and the energy E, which are defined by

M (ψ) = 1 2 T 0 |ψ| 2 dx, P (ψ) = 1 2 Im T 0 ψψ x dx, E(ψ) = 1 2 T 0 |ψ x | 2 dx - b 4 T 0 |ψ| 4 dx.
The simplest non-trivial solutions of (3.1.1) are the standing waves, which have the form ψ(t, x) = e -iat u(x), a ∈ R.

The profile function u(x) satisfies the ordinary differential equation

u xx + au + b|u| 2 u = 0. (3.1.2)
The conserved quantities of (3.1.2) on C are the angular momentum J and the ordinary differential equation energy E, defined by

J = Im(u x u), E = 1 2 |u x | 2 + a 2 |u| 2 + b 4 |u| 4 .
To find the angular momentum J, we start by multiplying (3.1.2) by u, then we take the imaginary part to obtain Im(u xx u) = 0. On the other side ∂

x (u x u) = u xx u + |u x | 2 , then ∂ x (Im(u x u)) = 0
, therefore there exists J ∈ R such that Im(u x u) ≡ J. To find the energy E we multiply (3.1.2) by u x and then we take the real part. Define the space

H θ T = {f ∈ H 1 loc (R) : f (x + T ) = e iθ f (x), ∀x ∈ R},
where θ the Floquet multiplier, i.e. the increment of the phase over a period. When the Floquet multiplier is 0 we are talking about periodic waves and when it is π we are in the case of anti-periodic solutions. Non-constant, realvalued, periodic solutions of (3.1.2) are well known to be given by the Jacobi elliptic functions: dnoidal (dn), cnoidal (cn) and snoidal (sn), see Section 3.2.1 for details. We are interested in the following minimization problem:

min{E(u) : u ∈ H θ T , M (u) = m, P (u) = p}. (3.1.3)
It is known that there exists u ∞ minimizer of (3.1.3), and there exist Lagrangemultipliers ω and σ such that

-E (u ∞ ) + ωM (u ∞ ) + σP (u ∞ ) = 0, that is -∂ xx u ∞ + ωu ∞ -b|u ∞ | 2 u ∞ ± iσ∂ x u ∞ = 0.
Our main goal is to characterize variationally the solutions of the ordinary differential equation (3.1.2) by identifying them as the minimizers of the minimization problem (3.1.3). This will be achieved using a mixture of analytical and numerical methods. In Section 3.2 we start from the ordinary differential equation (3.1.2) and we consider the links between the solution of this equation and the solution of the minimization problem (3.1.3). We establish a diffeomorphic correspondence between the conserved quantities of the ordinary differential equation (3.1.2) and the conserved quantities of the nonlinear Schrödinger equation (3.1.1). To do so we start in Section 3.2.1, by reviewing the well-known properties of Jacobi elliptic functions. Next, in Section 3.2.2 we study the analysis of the profile of the ordinary differential equation (3.1.2) for the two different cases depending whether or not J = 0. For each case we describe the solutions whether we know them explicitly or not. Then we give some monotonicity results on the period T . Moreover, in Section 3.2.3 we find the expressions of the mass M and the momentum P inside the domain and on its boundaries. Afterwards, in Section 3.2.4 we prove the diffeomorphism between (J, E) and ( M (J, E), P (J, E)) where we used a part of the results and methods of Gallay and Haragus [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF].

In Section 3.3 we start from the solution of the minimization problem (3.1.3) and consider the links between the minimizer of this problem and the solution of the ordinary differential equation (3.1.2). By that means, we start in Section 3.3.1 by recalling the results concerning global variational characterizations of elliptic function periodic waves as constrained-mass energy minimizers among periodic functions and subspaces of periodic functions. After, in Section 3.3.2 we explain how to proceed to compute numerically the solution of the ordinary differential equation (3.1.2). When we know the exact solution we compare it to the numerical solution. In Section 3.3.3, we present the method of the gradient flow with discrete normalization in order to find the minimizer of the energy E with fixed mass m > 0 and fixed momentum p: at each step of time, we evolve in the direction of the gradient of the energy and renormalize the mass and the momentum of the outcome. Such scheme is popular in the physics literature under the name "imaginary time method". When there is no momentum, and only real-valued functions are considered, such approach to compute the minimizers was developed by Bao and Du [START_REF] Bao | Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow[END_REF]. In the case of the nonlinear Schrödinger equation on the line R with focusing cubic non linearity, Faou and Jezequel [START_REF] Faou | Convergence of a normalized gradient algorithm for computing ground states[END_REF] performed a theoretical analysis of the various level of the discretization of the method, from the continuous one to the fully discrete scheme. To characterize variationally the Jacobi elliptic functions, Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] developed a numerical method to obtain the minimizer of the energy with fixed mass m > 0 and fixed momentum p = 0. Another study is presented in the work of Besse, Duboscq and Le Coz [START_REF] Besse | Gradient flow approach to the calculation of stationary states on nonlinear quantum graphs[END_REF] for the gradient flow on nonlinear quantum graphs. They implemented a method based on normalized gradient flow of the energy to compute ground states of nonlinear Schrödinger equations on metric graphs. Accordingly, in Section 3.3.5 we performed different tests using the scheme described above. In every test we were comparing the minimizer that we obtained from the normalized gradient flow with the numerical solution of the ordinary differential equation. Finally, in Section 3.3.6, we evaluate the order of the scheme by making various simulations, using the numerical solution of the ODE and the solution of minimization problem.

From the ordinary differential equation to the minimization problem

Jacobi elliptic functions

The Jacobi elliptic functions are standard forms of elliptic functions. The three basic functions are denoted cn(x, k), dn(x, k), and sn(x, k), where k ∈ (0, 1) is known as the elliptic modulus.

The incomplete elliptic integral of the first kind is defined by

x = F(φ, k) := φ 0 dθ 1 -k 2 sin 2 (θ) ,
where φ is called the Jacobi amplitude, and the Jacobi elliptic functions are defined through the inverse of F (., k): sn(x, k) := sin(φ), cn(x, k) := cos(φ), dn(x, k) := 1k 2 sin 2 (φ).

The relations

1 = sn 2 + cn 2 = k 2 sn 2 + dn 2
follow. The period of the elliptic functions can be expressed in terms of the complete elliptic integral of the first kind

K(k) := F π 2 , k , K(k) → π 2 , k → 0 ∞, k → 1.
Moreover, for k ∈ (0, 1), the incomplete elliptic integral of the second kind is defined by:

E(φ, k) := φ 0 1 -k 2 sin 2 (θ)dθ.
The complete elliptic integral of the second kind is defined as

E(k) := E π 2 , k , E(0) = π 2 , E(1) = 1.
The derivatives of elliptic functions (with respect to x) are expressed in terms of elliptic functions. For fixed k ∈ (0, 1), we have:

∂ x sn = cn.dn, ∂ x cn = -sn.dn, ∂ x dn = -k 2 cn.sn.
We can easily verify that sn, cn and dn are solutions of (3.1.2):

u xx + au + b|u| 2 u = 0,
with the coefficients a, b ∈ R given by:

a = 1 + k 2 , b = -2k 2 , for u = sn, a = 1 -2k 2 , b = 2k 2 , for u = cn, a = -(2 -k 2 ), b = 2, for u = dn.

From the ordinary differential equation to the minimization problem 75

Analysis of the profile ODE

In this section, we study the bounded solutions of the ordinary differential equation (3.1.2). We distinguish between two different cases depending whether or not J = 0 and in each case we study the defocusing and focusing cases. The analysis presented here gathers elements already presented in earlier works such as [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF][START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF].

The case J = 0

We start with the case J = 0. By definition of J, this implies in particular that u(x) = 0 for all x ∈ R. Hence we can introduce the polar coordinates u(x) = r(x)e iφ(x) , with r > 0 and r, φ ∈ C2 (R). Moreover J = r 2 φ x = 0 and this implies that φ x = 0 so u(x) ∈ C with a non-trivial phase. The energy becomes

E = r 2 x 2 + V J (r),
where V J (r) is given by

V J (r) = J 2 2r 2 + a r 2 2 + b r 4 4 .
We start with the defocusing case, i.e. we assume b < 0. We will distinguish between three different cases depending on the values of J. The first case is when J 2 > 4 b 2 (which implies in particular a > 0) then we can parametrize J in a unique way as

J = q (q 2 -a) b = Q (Q 2 -a) b , where 0 < q 2 < a 3 < Q 2 < a.
With this parametrization V J (r) has a unique local minimum at

r Q = Q 2 -a b
and a unique local maximum at r q = q 2 -a b . We define:

E -(J) = V J (r Q ) = 1 4b (Q 2 -a)(3Q 2 +a), E + (J) = V J (r q ) = 1 4b (q 2 -a)(3q 2 +a).
The curves E + and E -delimit the region of the (J, E) plane where there exist bounded solutions to (3.1.2). We have the following description.

1. If E = E -(J) then u is a plane wave, i.e. u(x) = r Q e iQx up to phase shift and complex conjugate.

3. If E -(J) < E < E + (J) the modulus r = |u| and the phase derivative φ x are periodic with the same period. If we denote by r 1 < r 2 < r 3 the three positive roots of E -V J (r), the (minimal) period is

T (J, E) = 2 r 2 r 1 dr 2(E -V J (r)) . (3.2.1)
Finally the last case is when

J 2 = 4 27 a 3 b 2 .
In this case, the effective potential V J (r) is strictly decreasing over R + with an inflexion point at r = -2a 3b . Since u is bounded we must have

E -(J) = E + (J) = -a 2
3b hence u(x) = q 2 -a b e iqx with q = (a/3) 1 2 sign(J). Now for the focusing case with a positive (a negative respectively): the potential V J (r) is strictly convex for any J ∈ R, and we can parametrize J in a unique way as

J = Q (Q 2 -a) b , where Q ∈ R, Q 2 a, (Q ∈ R respectively).
Then V J (r) has a unique critical point at r Q = Q 2 -a b , where V J attains its global minimum:

E -(J) = V J (r Q ) = 1 4b (Q 2 -a)(3Q 2 + a).
In the case of positive a, the equation (3.1.2) has quasi-periodic solutions for all E > E -(J). In the case of negative a, the equation (3.1.2) has quasiperiodic solutions for all E > E -(J) with E = 0. In both cases, the period T of these solutions is given by (3.2.1).

The case J = 0

In the case J = 0, u(x) ∈ R up to a constant phase, so we're in the case of real-valued solutions. The only (non-constant) real-valued, periodic solutions of (3.1.2) are the elliptic functions. We have the following result (see e.g. [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF]). Lemma 3.2.1. Fix a period T > 0, a ∈ R and u ∈ P T a nontrivial real-valued solution of (3.1.2). By invariance under translation, and negation, we may suppose u(0) = max u > 0. There exist α > 0, β > 0, and k ∈ (0, 1), uniquely determined by T, a, b and max u such that the following hold. Given u solution of (3.1.1), the function given by ū(-t, x) is also solution of (3.1.1). Therefore without loss of generality we restrict our work to the positive case (i.e. J > 0). We define the domain of existence of nontrivial quasi-periodic solutions of (3.1.2) for each case as the following: In the defocusing case b < 0 , let

D 1 = {(J, E) ∈ R 2 : 0 < J 2 < 4 27
a 3 b 2 , E -(J) < E < E + (J)}. (3.2.2)
In the focusing case b > 0 with a > 0, let

D 2 = {(J, E) ∈ R 2 : J > 0, E > E -(J)}. (3.2.3)
In the focusing case b > 0 with a < 0, let

D 3 = {(J, E) ∈ R 2 : J > 0, E > E -(J)}. (3.2.4) Lemma 3.2.3. Assume that (J, E) ∈ D 1 (D 2 , D 3
respectively) where it is understood that for each case a and b take the corresponding values. We denote by 0 y 1 < y 2 < y 3 , (y 3 < 0 y 1 < y 2 , y 3 0 y 1 < y 2 respectively) with r = √ y, the roots of the cubic polynomial P (y) = -by 3 -2ay 2 + 4Ey -

2J 2 . Then T (J, E) = √ 2 y 2 y 1 dy -b(y -y 1 )(y -y 2 )(y -y 3 ) = 2 √ 2 π 2 0 dφ b(S(φ) -y 3 ) , (3.2.5) where S(φ) = y 1 cos 2 φ + y 2 sin 2 φ.
Proof. Notice that P (r 2 ) = 4r 2 (E -V J (r)), therefore P (y) has three roots whenever (J, E) ∈ D 1 (D 2 , D 3 respectively). Hence, using the change of variables r = √ y in (3.2.1), we obtain the first expression in (3.2.5). Moreover by setting y = S(φ), we have dy = 2 (yy 1 )(y 2y)dφ, thus we obtain the last expression.

Mass and momentum inside the domain and on the boundaries

We start by finding the mass and momentum of the solutions in the domains D 1 , D 2 and D 3 , where we know that u(x) = r(x)e iφ(x) .Then

M (u) = 1 2
T (E,J)

0 |u| 2 dx = 1 2 T (E,J) 0 (r(x)) 2 dx = r 2 r 1 r 2 dr 2(E -V J ) = √ 2 π 2 0 S(φ) b(S(φ) -y 3 ) dφ = M (J, E), and
P (u) = 1 2 Im T (E,J) 0 uu x dx = - 1 2 T 0 Jdx = - 1 2 T J = P (J, E).
Note that M and P is used to distinguish P and M as functions of J and E. Next we want to find the mass and momentum of the solutions on the boundaries E = E -(J) for the focusing/defocusing cases and on E = E + (J) for the defocusing case.

Proposition 3.2.4. Let (J, E) ∈ D 1 (D 2 , D 3 respectively). When E → E -(J)
the following holds:

lim E→E -(J) T (J, E) = π √ 2 3Q 2 -a , lim E→E -(J) M (J, E) = (Q 2 -a) 2|b| π √ 2 3Q 2 -a , lim E→E -(J) P (J, E) = - 1 2 Q (Q 2 -a) b π √ 2
3Q 2a .

From the ordinary differential equation to the minimization problem 79

Proof. We know that T

(J, E) = 2 √ 2 π 2 0 dφ √ b(S(φ)-y 3 )
, where S(φ) = y 1 cos 2 φ +

y 2 sin 2 φ. When E → E -(J), y 2 → y 1 , therefore lim E→E -(J) T (J, E) = π √ 2 √ b(y 1 -y 3 )
.

On the other hand we have

y 1 = r 2 Q = Q 2 -a
b , and since y 1 , y 2 , y 3 are solutions of the cubic equation -by 3 -2ay 2 +4Ey -2J 2 = 0, we have y 1 +y 2 +y 3 = -2a b , which implies that y 3 = -2a b -2y 1 and we can conclude that lim

E→E -(J) T (J, E) = π √ 2 √ 3Q 2 -a . Recall that J = Q (Q 2 -a)
b and on the boundary E = E -(J), we have

u(x) = (Q 2 -a) b e iQx .
We can find now the mass on this boundary

lim E→E -(J) M (J, E) = lim E→E -(J) 1 2 T 0 |u| 2 dx = lim E→E -(J) 1 2 T 0 Q 2 -a |b| dx = (Q 2 -a) 2|b| π √ 2 3Q 2 -a . (3.2.6)
The momentum is obtained by a direct computation:

lim E→E -(J) P (J, E) = - 1 2 T J = - 1 2 Q (Q 2 -a) b π √ 2 3Q 2 -a . (3.2.7)
Proposition 3.2.5. For the defocusing case, when E → E + (J) the following holds:

lim E→E + (J) T (J, E) = +∞, lim E→E + (J) M (J, E) T (J, E) = |1 -q 2 | 2 , lim E→E + (J) P (J, E) T (J, E) = - 1 2 q(1 -q 2 ). Proof. Recall that T (J, E) = 2 √ 2 π 2 0 dφ √ b(S(φ)-y 3 )
. When E → E + (J), y 3 → y 2 , therefore

lim E→E + (J) T (J, E) = 2 √ 2 b(y 1 -y 2 ) π 2 0 dφ cos(φ) = +∞.
Finally the graph of P as a function of M for each case is given in Figure 3. Note that when a = -1 in the focusing case, the behavior of the graph of P as a function of M is determined by the transition of E from negative

Diffeomorphism

In this section we establish a diffeomorphic correspondence between J, E and M, P for the defocusing case only. To this aim, we use a part of the results and methods of [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF]. Without loss of generality, we set b = -1 and a = 1. We start with monotonicity results on T . Proposition 3.2.6. We have ∂T ∂E > 0 and ∂T ∂J < 0 for all (J, E) ∈ D 1 . Proof. Let (J, E) ∈ D 1 . Since y 1 , y 2 , y 3 are solutions of the cubic equation y 3 -2y 2 + 4Ey -2J 2 = 0, we have y 1 + y 2 + y 3 = 2, and

∂y i ∂E = - 4y i 3y 2 i -4y i + 4E , ∂y i ∂J = 4J 3y 2 i -4y i + 4E , i = 1, 2, 3. (3.2.8)
In particular

∂y 1 ∂E < 0, ∂y 2 ∂E > 0, ∂y 1 ∂E + ∂y 2 ∂E = - ∂y 3 ∂E > 0,
because P (y 1 ) > 0, P (y 2 ) < 0, P (y 3 ) > 0, where P (y) = y 3 -2y 2 +4Ey-2J 2 was defined in Lemma 3.2.3. Similarly, since J > 0,

∂y 1 ∂J > 0, ∂y 2 ∂J < 0, ∂y 1 ∂J + ∂y 2 ∂J = - ∂y 3 ∂J < 0.
On the other hand, differentiating (3.2.5) with respect to E and J, we find

∂T ∂E = A 1 ∂y 1 ∂E + A 2 ∂y 2 ∂E , ∂T ∂J = A 1 ∂y 1 ∂J + A 2 ∂y 2 ∂J ,

From the ordinary differential equation to the minimization problem 81

where

A 1 = √ 2 π 2 0 1 + cos 2 φ (y 3 -S(φ)) 3 2 dφ, A 2 = √ 2 π 2 0 1 + sin 2 φ (y 3 -S(φ)) 3 2
dφ.

We have that

A 2 -A 1 = √ 2 π 2 0 sin 2 φ -cos 2 φ 2(y 3 -S(φ)) 3 2 dφ > 0.
Indeed, we have the following results. Let µ be a (Borel) probability measure on some interval I ⊂ R, and let f, g : I → R be bounded and measurable functions. If both f and g are strictly increasing or strictly decreasing, and if the support of µ is not reduced to a single point, then

I f (x)g(x)dµ > I f (x)dµ I g(x)dµ . Therefore A 2 -A 1 > 0 follows from the result above, with I = [0, π 2 ], dµ = 2
π dφ, f (φ) = sin 2 φcos 2 φ and g(φ) = (y 3 -S(φ)) -3 2 (notice that f, g are strictly increasing, and that

π 2 0 f (φ)dφ = 0). Thus, A 2 > A 1 > 0. We conclude that ∂T ∂E = (A 2 -A 1 ) ∂y 2 ∂E + A 1 ∂y 1 ∂E + ∂y 2 ∂E > 0.
Similarly,

∂T ∂J = (A 2 -A 1 ) ∂y 2 ∂J + A 1 ∂y 1 ∂J + ∂y 2 ∂J < 0.
This concludes the proof.

For all (J, E) ∈ D 1 , let

∆ = ∂ P ∂E ∂ M ∂E ∂ P ∂J ∂ M ∂J . Proposition 3.2.7. For all (J, E) ∈ D 1 we have ∆ > 0.
Proof. We have

∂ P ∂E = - 1 2 J ∂T ∂E = - 1 2 J A 1 ∂y 1 ∂E + A 2 ∂y 2 ∂E , ∂ P ∂J = - 1 2 J ∂T ∂J + T = - 1 2 J A 1 ∂y 1 ∂J + A 2 ∂y 2 ∂J + T , ∂ M ∂E = ∂ M ∂y 1 ∂y 1 ∂E + ∂ M ∂y 2 ∂y 2 ∂E + ∂ M ∂y 3 ∂y 3 ∂E , ∂ M ∂J = ∂ M ∂y 1 ∂y 1 ∂J + ∂ M ∂y 2 ∂y 2 ∂J + ∂ M ∂y 3 ∂y 3 ∂J .
We start with

∂ M ∂y 1 = √ 2 π 2 0 cos 2 φ y 3 -S(φ) --cos 2 φS(φ) 2 √ y 3 -S(φ) y 3 -S(φ) dφ, = √ 2 π 2 0 2(y 3 -S(φ)) cos 2 φ + cos 2 φS(φ) 2(y 3 -S(φ)) 3 2 dφ, = √ 2 π 2 0 2y 3 cos 2 φ -cos 2 φS(φ) 2(y 3 -S(φ)) 3 2 dφ, = √ 2 π 2 0 2y 3 cos 2 φ + S(φ) sin 2 φ -S(φ) 2(y 3 -S(φ)) 3 2
dφ.

And same for

∂ M ∂y 2 = √ 2 π 2 0 2y 3 sin 2 φ + S(φ) cos 2 φ -S(φ) 2(y 3 -S(φ)) 3 2
dφ, and ∂ M

∂y 3 = √ 2 π 2 0 -S(φ) 2(y 3 -S(φ)) 3 2
dφ.

Then

∂ M ∂E = ∂ M ∂y 1 ∂y 1 ∂E + ∂ M ∂y 2 ∂y 2 ∂E + ∂ M ∂y 3 ∂y 3 ∂E , = ∂ M ∂y 1 ∂y 1 ∂E + ∂ M ∂y 2 ∂y 2 ∂E + ∂ M ∂y 3 - ∂y 1 ∂E - ∂y 2 ∂E , = ∂y 1 ∂E ∂ M ∂y 1 - ∂ M ∂y 3 + ∂y 2 ∂E ∂ M ∂y 2 - ∂ M ∂y 3 = C 1 ∂y 1 ∂E + C 2 ∂y 2 ∂E ,
with

C 1 = √ 2 π 2 0 2y 3 cos 2 φ + S(φ) sin 2 φ 2(y 3 -S(φ)) 3 2
dφ, and

C 2 = √ 2 π 2 0 2y 3 sin 2 φ + S(φ) cos 2 φ 2(y 3 -S(φ)) 3 2
dφ.

Similarly, we have

∂ M ∂J = C 1 ∂y 1 ∂J + C 2 ∂y 2 ∂J .

From the ordinary differential equation to the minimization problem 83

Therefore,

∆ = - 1 2 J A 1 ∂y 1 ∂E + A 2 ∂y 2 ∂E C 1 ∂y 1 ∂J + C 2 ∂y 2 ∂J + 1 2 J A 1 ∂y 1 ∂J + A 2 ∂y 2 ∂J + T C 1 ∂y 1 ∂E + C 2 ∂y 2 ∂E , = - 1 2 JA 1 C 1 ∂y 1 ∂E ∂y 1 ∂J + JA 1 C 2 ∂y 1 ∂E ∂y 2 ∂J + JA 2 C 1 ∂y 2 ∂E ∂y 1 ∂J + JA 2 C 2 ∂y 2 ∂E ∂y 2 ∂J + 1 2 JA 1 C 1 ∂y 1 ∂E ∂y 1 ∂J + JA 1 C 2 ∂y 1 ∂J ∂y 2 ∂E + JA 2 C 1 ∂y 2 ∂J ∂y 1 ∂E + JA 2 C 2 ∂y 2 ∂E ∂y 2 ∂J + 1 2 T ∂ M ∂E , = (A 2 C 1 -A 1 C 2 ) 1 2 J ∂y 1 ∂E ∂y 2 ∂J - ∂y 2 ∂E ∂y 1 ∂J + 1 2 T ∂ M ∂E .
Using the identity (3.2.8)

∂y i ∂E = - y i J ∂y i ∂J ,
we find

∆ = 1 2 (A 2 C 1 -A 1 C 2 )(y 2 -y 1 ) ∂y 1 ∂J ∂y 2 ∂J + 1 2 T ∂ M ∂E .
First we will prove that C 2 > C 1 > 0.

Let f (φ) = sin 2 φcos 2 φ and g(φ) = 2y 3 -S(φ)

2(y 3 -S(φ)) 3 2
. We know that f is increasing. We will prove that g is also increasing.

g (φ) = -S (φ)(y 3 -S(φ)

3 2 + 3 2 (2y 3 -S(φ))S (φ)(y 3 -S(φ)) 1 2 2(y 3 -S(φ)) 3 = S (φ) -(y 3 -S(φ)) + 3y 3 -3 2 S(φ) 2(y 3 -S(φ)) 5 2 = S (φ)(2y 3 -1 2 S(φ)) 2(y 3 -S(φ)) 5 2 = (sin φ cos φ)(y 2 -y 1 )(2y 3 -1 2 S(φ)) (y 3 -S(φ)) 5 2 > 0,
because sin φ cos φ > 0, y 2y 1 > 0, y 3 -S(φ) > 0 and 2y 3 -1 2 S(φ) > 0. Then g is strictly increasing and same as before, as we have two strictly increasing functions, we have

C 2 > C 1 > 0. Therefore ∂ M ∂E = C 1 ∂y 1 ∂E + C 2 ∂y 2 ∂E = (C 2 -C 1 ) ∂y 2 ∂E + C 1 ( ∂y 1 ∂E + ∂y 2 ∂E ) > 0.

Now we will prove that

A 1 C 2 -A 2 C 1 > 0. Let σ(φ) = (1 -y -1 3 S(φ)) 1 2 .
Then

A 1 = √ 2 π 2 0 1 + cos 2 φ (y 3 -S(φ)) 3 2 dφ = 2 √ 2y -3 2 3 A 1 , A 1 = π 2 0 1 + cos 2 φ 2σ(φ) 3 dφ, A 2 = √ 2 π 2 0 1 + sin 2 φ (y 3 -S(φ)) 3 2 dφ = 2 √ 2y -3 2 3 A 2 , A 2 = π 2 0 1 + sin 2 φ 2σ(φ) 3 dφ, C 1 = √ 2 π 2 0 2y 3 cos 2 φ + S(φ) sin 2 φ 2(y 3 -S(φ)) 3 2 dφ = √ 2y -1 2 3 C 1 , C 1 = π 2 0 2 cos 2 φ + (1 -σ(φ) 2 ) sin 2 φ 2σ(φ) 3 dφ, C 2 = √ 2 π 2 0 2y 3 sin 2 φ + S(φ) cos 2 φ 2(y 3 -S(φ)) 3 2 dφ = √ 2y -1 2 3 C 2 , C 2 = π 2 0 2 sin 2 φ + (1 -σ(φ) 2 ) cos 2 φ 2σ(φ) 3 dφ.
Therefore

A 1 C 2 -A 2 C 1 = (2 √ 2y -3 2 3 A 1 )( √ 2y -1 2 3 C 2 ) -(2 √ 2y -3 2 3 A 2 )( √ 2y -1 2 3 C 1 ) = 4y -2 3 (A 1 C 2 -A 2 C 1 ).
We want to prove that

A 1 C 2 -A 2 C 1 > 0.
In fact,

C 1 = A 1 -C1 with C1 = π 2 0 sin 2 φ 2σ(φ) dφ, C 2 = A 2 -C2 with C2 = π 2 0 cos 2 φ 2σ(φ) dφ, which implies that A 1 C 2 -A 2 C 1 = A 1 (A 2 -C2 ) -A 2 (A 1 -C1 ) = A 1 A 2 -A 1 C2 -A 2 A 1 + A 2 C1 = A 2 C1 -A 1 C2 We know that A 2 > A 1 > 0 and C1 > C2 > 0 so A 1 C 2 -A 2 C 1 > 0 and then A 2 C 1 -A 1 C 2 < 0. Finally from A 2 C 1 -A 1 C 2 < 0 , y 2 -y 1 > 0, ∂y 1 ∂J ∂y 2 ∂J < 0 and 1 2 T ∂ M ∂E > 0, we conclude that ∆ > 0,
which concludes the proof.

Let D = {( M , P ) ∈ R 2 : M > 0, | P | < 3 M 2 +π 2 - √ 9 M 4 +4 M 2 π 2 π }.
Theorem 3.2.8. The map Ψ : D 1 → D, defined by Ψ(J, E) = ( M (J, E), P (J, E)) is a diffeomorphism.

Proof. We know from Section 3.2.3 that lim

E→E -(J) T (J, E) = π √ 2 √ 3Q 2 -1
, and

lim E→E + (J) T (J, E) = +∞. Moreover we have lim E→E -(J) M (J, E) = (1 -Q 2 ) 2 π √ 2 3Q 2 -1 , lim E→E + (J) M (J, E) = +∞, and
lim E→E -(J) P (J, E) = - 1 2 Q(1 -Q 2 ) π √ 2 3Q 2 -1 , lim E→E + (J) P (J, E) = -∞.
The range of the map M : D 1 → R is exactly the interval (0, +∞) since

∂ M ∂E > 0.
We fix M0 > 0, and let

Σ = {(J, E) ∈ D 1 : M (J, E) = M0 }.
By the implicit Function Theorem, Σ is a smooth curve in D 1 which can be represented as a graph over the J-axis. Moreover, we know that Σ connects the boundary points (0, E 0 ) and (J 0 , E -(J 0 )) is determined by the relation

(1 -Q 2 0 ) 2 π √ 2 3Q 2 0 -1 = M0 , where J 0 = Q 0 (1 -Q 2 0 ), Q 0 ∈ ( 1 √ 3 , 1
) and E 0 the corresponding energy for J = 0 and M = M0 . We know that ∆ > 0 which implies that the restriction of P to the curve Σ is a strictly decreasing function of J, because

d dJ P|Σ = ∂ M ∂E -1 ∂ M ∂E ∂ P ∂J - ∂ M ∂J ∂ P ∂E < 0.
Thus P varies from 0 to -P0 on the curve Σ, where

P0 = - 1 2 Q 0 (1 -Q 2 0 ) π √ 2 3Q 2 0 -1 = 3 M 2 0 + π 2 -9 M 4 0 + 4 M 2 0 π 2 π .
This proves that Ψ is onto and therefore Ψ is a diffeomorphism.

For the focusing case we recall the results of [START_REF] Gallay | Orbital stability of periodic waves for the nonlinear Schrödinger equation[END_REF] on the monotonicity of the period T . Proposition 3.2.9. Let b = 1, a = 1. We have

∂T ∂E < 0, ∂T ∂J < 0 for all (J, E) ∈ D 2 .
Proposition 3.2.10. Let b = 1, a = -1. Unlike in the previous cases, the period T is no longer a monotone function of the energy E. In fact the period T (J, E) satisfies

∂T ∂E (0, E) > 0 for - 1 4 < E < 0, and ∂T ∂E (0, E) < 0 for E > 0.
On the other hand, sine J > 0, the period T is still a monotone function of J and it satisfies ∂T ∂J (J, E) < 0.

From the minimization problem to the ordinary differential equation

In this section, we aim to study the minimizing problem and establish the link between the minimizer and the solution of the ordinary differential equation (3.1.2). One of the difficulties in this problem is identifying the minimizer as a solution of the ODE. The existence of the minimizer can be proven by taking a sequence that minimizes the energy and showing that it is bounded in H θ T , which implies that it has a weakly convergent subsequence in H θ T . The weak convergence of the subsequence, along with the compactness of the embedding of H θ T into L p for p ∈ (1, ∞), implies the convergence of the subsequence in L p . Therefore, the minimizing sequence converges to a function that minimizes the energy. In the cubic case, it was further shown in [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] that the minimizer is a stationary wave, which is a periodic solution of the NLS equation with a special form of the profile function. Specifically, the minimizer was identified as a Jacobi elliptic function cn or dn depending 3.3. From the minimization problem to the ordinary differential equation 87

on the value of the mass parameter. Moreover it was conjectured that the minimizer of the energy with fixed mass and momentum is a snoidal periodic wave. In the complex-valued case we need to identify the minimizer as a solution of the ODE (3.1.2).

Variational problems

We start by recalling the result of Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] on the global variational characterizations of the elliptic function periodic waves as constrained-mass energy minimizers among periodic functions. Later on, we use these results to compare our numerical method with the theoretical results.

For m > 0, we consider the following minimization problem with fixed mass min{E(u) :

u ∈ H θ T , M (u) = m}, (3.3.1)
and the minimization problem with fixed mass and momentum

min{E(u) : u ∈ H θ T , M (u) = m, P (u) = 0}. (3.3.2)
Theorem 3.3.1. Assume b > 0 (focusing case). We have the following properties:

1. Let θ = 0, we are in the periodic case:

(a) If 0 < m π 2
bT then there exists a unique (up to phase shift) minimizer for (3.3.1) and (3.3.2), which is the constant function

u min = 2m T (b) If π 2
bT < m < ∞ then there exists a unique minimizer (up to translations and phase shift) for (3.3.1) and (3.3.2), which is the rescaled function dn α,β,k = 1 α dn . β , k , where the parameters α, β, k are uniquely determined. In particular given k ∈ (0, 1), if b = 2, T = 2K(k) and m = E(k), then the unique minimizer is dn(., k).

2. If θ = π, we are in the anti-periodic case, then there exists a unique minimizer (up to translations and phase shift) for (3.3.1) and (3.3.2), which is the rescaled function cn α,β,k = 1 α cn . β , k , where the parameters α, β, k are uniquely determined. In particular given k ∈ (0, 1),

if b = 2k 2 , T = 2K(k) and m = (K-E)
k 2 , then the unique minimizer is cn(., k).

Proof. First we start with the periodic case by proving that (3.3.1) and (3.3.2) share the same minimizer. Without loss of generality we can restrict the minimization problem to real-valued non negative functions. Indeed, if u ∈ H 1 loc ∩ P T , then |u| ∈ H 1 loc ∩ P T and we have

∂ x |u| L 2 ∂ x u L 2 .
This implies that (3.3.1) and (3.3.2) share the same minimizers. Next we prove that the minimal energy is finite and negative. To prove the negativity, let φ m,0 ≡ 2m T be a test function. We have M (φ m,0 ) = m, E(φ m,0 ) < 0. To prove that it is finite, we consider a minimizing sequence (u n ) ⊂ H 1 loc ∩ P T for (3.3.1). It is bounded in H 1 loc ∩ P T . We observe that by Gagliardo-Nirenberg inequality: for any u ∈ H 1 loc ∩ P T , we have

u 4 L 4 u 3 L 2 u x L 2 + u 4 L 2 .
Consequently, for any u ∈ H 1 loc ∩ P T , such that M (u) = m, we have

E(u) u x L 2 ( u x L 2 -m 3 2 ) -m 2 .
The previous inequality implies the boundedness of ∂ x u n L 2 . Indeed, by contradiction, let's suppose that

∂ x u n L 2 → ∞, therefore E(u n ) → ∞, which is a contradiction with the minimizing nature of (u n ). Hence the sequence (u n ) is bounded in H 1 loc ∩ P T , hence E(u) is bounded from below.
Therefore up to a subsequence, (u n ) converges weakly in H 1 loc ∩ P T and strongly in L 2 loc ∩ P T and L 4 loc ∩ P T towards u ∞ ∈ H 1 loc ∩ P T . By weak convergence, we have

∂ x u ∞ 2 L 2 lim n→+∞ ∂ x u n 2 L 2 .
Hence E(u ∞ ) lim n→+∞ E(u n ). On the other hand M (u ∞ ) = m, so by the

definition of the minimizer E(u ∞ ) = lim n→+∞ E(u n ). Therefore the convergence from (u n ) to u ∞ is also strong in H 1 loc ∩ P T . Since u ∞ is a minimizer of (3.3.1), there exists a Lagrange multiplier a ∈ R such that -E (u ∞ ) + aM (u ∞ ) = 0 that is ∂ xx u ∞ + au ∞ + bu 3 ∞ = 0. (3.3.3)
Multiplying by u ∞ and integrating (recall that the functions considered are assumed to be real), we find that

a = ∂ x u ∞ 2 L 2 -b u ∞ 4 L 4 u ∞ 2 L 2 . Note that ∂ x u ∞ 2 L 2 -b u ∞ 4 L 4 = 2E(u ∞ ) - b 2 u ∞ 4 L 4 , 3.3.
From the minimization problem to the ordinary differential equation 89 therefore a < 0.

We know from Lemma 3.2.1 that the solutions of (3.3.3) are either the constant function or the dn α,β,k function, where dn α,β,k (x) = 1 α dn( x β , k). We start by proving that if m > π 2 bT then u ∞ = dn α,β,k . By contradiction we suppose that u ∞ = 2m

T . Therefore a = -2bm T . Since u ∞ is a constrained minimizer, the operator

-∂ xx -a -3bu 2 ∞ = -∂ xx - 4bm
T must have at most 1 negative eigenvalue, where the eigenvalues are given by: 2πn T

-4bm

T . For n = 0, the eigenvalue is negative.

For n = 1, 2πn T 2 -4bm T 0 if and only if m π 2
bT , which gives the contradiction. Now we prove that if

u ∞ = dn α,β,k then m > π 2
bT . We find the mass of dn α,β,k which is equal to

M (dn α,β,k ) = 4 bT E(k)K(k). We have ∂ k EK(k) > 0 therefore for k ∈ (0, 1), M (dn α,β,k ) is a strictly increasing function of k from π 2
bT , ∞ , which concludes the proof. Thus u ∞ must be constant when 0 < m π 2 bT . Finally we will prove that given k ∈ (0, 1), if b = 2, T = 2K(k) and m = E(k), then the unique minimizer is dn(., k). In this case m > π 2 bT since EK > π 2 4 and therefore from before we know that u ∞ = dn α,β,s for s ∈ (0, 1). By Sturm-Liouville, the fundamental period of u ∞ is T = 2K(s)β. So M (dn α,β,s ) = E(s)K(s) = E(k)K(k) and by the monotonicity of EK we have k = s. Thus α = β = 1 and u ∞ = dn(x, k).

For the anti-periodic case, from [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] we have the following result: Let v ∈ H 1 loc ∩ A T . There exists ṽ ∈ H 1 loc ∩ A T such that:

ṽ(x) ∈ R, ṽ L 2 = v L 2 , ∂ x ṽ L 2 = ∂ x v L 2 , ṽ L 4 v L 4 .
Therefore we can restrict the minimization problem to real valued functions, and this implies the equivalence between the two minimization problems (3.3.1) and (3.3.2). Same as for the periodic case we consider a minimizing sequence

(u n ) ⊂ H 1 loc ∩ A T and the convergence from (u n ) to u ∞ is also strong in H 1 loc ∩ P T . Since u ∞ is a minimizer of (3.3.1), there exists a Lagrange multiplier a ∈ R such that -E (u ∞ ) + aM (u ∞ ) = 0 that is ∂ xx u ∞ + au ∞ + bu 3 ∞ = 0.
We assume that max u = u(0) > 0 and therefore u ∞ = cn α,β,k . Moreover with the same method as before, using the monotonicity of m with respect to k, we prove that given k ∈ (0, 1), if b = 2k 2 , T = 2K(k) and m = (K-E) k 2 , then the unique minimizer is cn(., k). This concludes the proof. Theorem 3.3.2. Assume b < 0 (defocusing case). We have the following properties:

1. If θ = 0, we are in the periodic case, then there exists a unique minimizer (up to phase shift) for (3. Proof. For the periodic case we know that for all f ∈ L 4 (0, T ), we have by Hölder's inequality:

f L 2 T 1 4 f L 4 , with equality if and only if |f | is constant. We want to prove that φ m,0 = 2m T is the minimizer. Let v ∈ H 1 loc ∩ P T such that M (v) = m, and v = e iθ φ m,0 , (θ ∈ R).
We have:

0 = ∂ x φ m,0 2
L 2 < ∂ x v 2 L 2 ,
and φ m,0

L 4 = 4T -1 M 2 (φ m,0 ) = 4T -1 M 2 (v) v 4 L 4 4
, therefore E(φ m,0 ) < E(v) which concludes the proof of the first part of the theorem.

For the anti-periodic case we denote w(x)

= 2m T e 2iπx T . Let v ∈ H 1 loc ∩ A T such that M (v) = m and v = e iθ w (θ ∈ R). We have w 4 L 4 = 4T -1 M 2 (w) = 4T -1 M 2 (v) v 4 L 4 .
Since v ∈ A T , v must have 0 mean value. Therefore by Poincaré-Wirtinger inequality we have

v 2 L T 2π v L 2 . Moreover ∂ x w 2 L 2 = 8π 2 T M (w) = 8π 2 T M (w) < ∂ x v 2 L 2 , therefore E(φ m,0 ) < E(v) which concludes the proof.
Conjecture 3.3.3. There exists a unique minimizer (up to translations and phase shift) of (3.3.2) which is the rescaled function sn α,β,k = 1 α sn . β , k , where the parameters α, β, k are uniquely determined. In particular given k ∈ (0, 1), if b = -2k 2 , T = 2K(k) and m = 8

bT K(K -E), then the unique minimizer is sn(., k).

3.3.

From the minimization problem to the ordinary differential equation 91

Numerical solutions of the ordinary differential equation

In this section we explain how we proceeded to compute numerically the solution of the ordinary differential equation (3.1.2):

u xx + au + b|u| 2 u = 0.
The Python function odeintw has been used to find the numerical solution. In order to use this function we need to find the period T and the initial data at a fixed (J, E). Recall that

V J (r) = J 2 2r 2 + a r 2 2 + b r 4 4 .
We start our numerical experiments by fixing a value of J in the domain D 1 (D 2 , D 3 respectively) for each defocusing and focusing case. Recall that when J = 0, u is of the form u(x) = r(x)e iφ(x) . If we are in the defocusing case we find the minimum r Q and the maximum r q of V J (r). We know that

E -= V J (r Q ) and E + = V J (r q ). Then we choose E such that E -E < E + . Note that we chose E < E + since lim E→E + (J)
T (J, E) = +∞. Moreover we find the three positive roots r 1 < r 2 < r 3 of E -V J (r). If we are in the focusing case we find the minimum r Q of V J (r) and we have E -= V J (r Q ). Then we choose E such that E -E. Moreover we find the two positive roots r 1 < r 2 of E -V J (r). Recall that the period T is given by (3.2.1)

T (J, E) = 2 r 2 r 1 1 2(E -V J (r)) dr.
Furthermore, the Floquet multiplier is given by

θ(J, E) = φ(x + T ) -φ(x) = T 0 φ x (x)dx = T 0 J r(x) 2 dx = 2 r 2 r 1 J r 2 2(E -V J (r))
dr, the mass is given by

M (J, E) = 1 2 T 0 |u| 2 dx = 1 2 T 0 r(x) 2 dx = r 2 r 1 r 2 2(E -V J (r)) dr,
and the momentum is given by

P (J, E) = 1 2 Im T 0 uu x dx = - 1 2 T 0 φ x r(x) 2 dx = - 1 2 T 0 Jdx = - r 2 r 1 J 2(E -V J (r)) dr.

Chapter 3. Analysis of the quasi-periodic waves

Recall that on the boundary E = E -the period T is given by the limit when E tends to E -, which is given in Proposition (3.2.4).

To use the function odeintw, we set y = u v , where v = u , therefore

y = v -au -b|u| 2 u
. For the initial data, we suppose that u(0) = r(0) = r 1 and u (0) = iφ (0)r(0) = J r 1 i, therefore we find the numerical solution. The results of the numerical experiments are presented in Section 3.3.5.

Continuous gradient flow with discrete normalization

The variational problem that we consider is the following:

min{E(u) : u ∈ H θ T , M (u) = m, P (u) = p}, (3.3.4)
where m > 0 and p ∈ R. Our goal in this section is to compute the minimizer of the energy at fixed mass m and momentum p. The approach that we use is the normalized gradient flow: at each step we evolve in the direction of the gradient of the energy and renormalize the mass and the momentum of the outcome. One of the earliest mathematical analysis was performed by Bao and Du [START_REF] Bao | Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow[END_REF]. They proved the energy diminishing property of a normalized gradient flow and compared several discretization methods. In the case of the nonlinear Schrödinger equation on the line R with focusing cubic non linearity, Faou and Jezequel [START_REF] Faou | Convergence of a normalized gradient algorithm for computing ground states[END_REF] performed a theoretical analysis of the various level of the discretization of the method, from the continuous one to the fully discrete scheme. In the periodic setting Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] developed a numerical method to obtain the minimizer of the energy with fixed mass m > 0 and fixed momentum p = 0. More recently Besse, Duboscq and Le Coz [START_REF] Besse | Gradient flow approach to the calculation of stationary states on nonlinear quantum graphs[END_REF] implemented a method based on normalized gradient flow of the energy to compute ground states of nonlinear Schrödinger equations on metric graphs. Define an increasing sequence of time 0 = t 0 < ... < t n and take an initial data u 0 . Between each time step, let u(t, x) evolve along the gradient flow

u t = -E (u) = u xx + b|u| 2 u, u(t n , x) = u n (x), (3.3.5)
where x ∈ R, t n < t < t n+1 , n 0. At each time step t n , the function is renormalized so as to have the desired mass m and momentum p. Recall that, as noted in [START_REF] Bao | Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow[END_REF], the renormalization of the mass is equivalent to solving 3.3. From the minimization problem to the ordinary differential equation 93

exactly the following ordinary differential equation:

u t = µ n u, t n < t < t n+1 , n 0, µ n = 1 t n+1 -t n ln √ 2m u(t n ) L 2 .
(3.3.6) Inspired by this remark, we consider the following numerical algorithm which gives the desired mass m and momentum p simultaneously by one flow. Suppose ũn+1 has been computed from u n using the semi-implicit Euler scheme.

To normalize ũn+1 we proceed in the following way. Let

m 0 = M (ũ n+1 ), p 0 = P (ũ n+1 ), k 0 = 1 2 T 0 |∂ x ũn+1 | 2 dx.
Consider the following flow

∂ t u = (µ + iω∂ x )u.
We have at (t = t n+1 )

d dt M (u) = Re T 0 ūu t dx = 2m 0 µ + 2p 0 ω, d dt P (u) = Im T 0 ūx u t dx = 2p 0 µ + 2k 0 ω.
We want to choose µ n = µδt and ω n = ωδt so that

2m 0 µ n + 2p 0 ω n = m -m 0 , 2p 0 µ n + 2k 0 ω n = p -p 0 .
Note that by Cauchy-Schwarz

p 2 0 = 1 2 Im T 0 uū x dx 2 1 2 T 0 |u| 2 dx 1 2 T 0 |u x | 2 dx = m 0 k 0 .
In the case m 0 k 0p 2 0 > 0, we can solve

µ n = k 0 (m -m 0 ) -p 0 (p -p 0 ) 2(m 0 k 0 -p 2 0 )
,

ω n = m 0 (p -p 0 ) -p 0 (m -m 0 ) 2(m 0 k 0 -p 2 0 )
.

In the case m 0 k 0p 2 0 = 0, we have Cauchy-Schwarz equality. To be periodic, ũn+1 is a plane wave and for some real constants r, α for the Cauchy-Schwarz equality we have ũn+1 = re iαx .

We choose

u n+1 = 2m T e -i p m x .

Discretization

Several numerical methods can be considered for discretizing (3.3.5). For example, a standard Crank-Nicolson scheme would consist in

u n+1 -u n δ t = ∂ xx u n+1 + u n 2 + b|u n | 2 u n+1 + u n 2 .
This method can be proved to be energy diminishing. However in the above discretization, we need to solve a fully nonlinear system at every time step, which is time and resource consuming in practical computation. A more efficient method is the Gradient Flow with Discrete Normalization (GFDN) which consists into one step of classical gradient flow followed by a mass and momentum normalization step. The discretized problem is of the form

       ũn+1 -u n δ t = ∂ xx ũn+1 + b|u n | 2 ũn+1 , u n+1 -ũn+1 δ t = (µ + iω∂ x )u n+1 .
Note that in the first equation, we used Euler explicit for the nonlinear term, and Euler implicit for the linear terms. Therefore we obtain a semi-implicit Euler scheme and the first equation becomes linear. Finally we present the fully discretized problem. We discretize the space interval [0, T ] by setting

x 0 = 0, x l = x 0 + lδx, δx = T N , N ∈ N.
We denote by u l n the numerical approximation of u(t n , x l ). Using the (Backward Euler) semi-implicit scheme for time discretization and second-order centered finite difference for the second spatial derivatives, and the first centered finite difference for the first spatial derivatives, we obtain the following scheme:

       ũl n+1 -u l n δ t = ũl-1 n+1 -2ũ l n+1 + ũl+1 n+1 δx 2 + b|u l n | 2 ũl n+1 , u l n+1 -ũl n+1 δ t = µu l n+1 + iω u l-1 n+1 -u l+1 n+1 2δx .
Recall that as we are in the space H θ T , we know that u(x + T ) = e iθ u(x), therefore with the discretization on the space interval [0, T ], we have

u L n+1 = e iθ u 0 n+1 Remark 3.3.4.
Since the problem is invariant under phase multiplication and spatial translation, at each step we shift u j so that the minimum of its modulus is at the boundary, and is real. We do the same for the numerical solutions of the ordinary differential equation.

3.3.

From the minimization problem to the ordinary differential equation 95

Experiments

Our main observation is the following. We performed different tests using the schemes described above. In every test we were comparing the minimizer that we obtained from the normalized gradient flow with the numerical solution of the ordinary differential equation. The choice of the initial data was arbitrary and equal to u 0 (x) = 1 + i + cos 2πx

T . Other initial data lead to similar results. We use N = 1000 grid points for the interval [0, T ]. We run the algorithm until a maximal difference of = 10 -6 between the absolute values of the moduli of u l j . For the time discretization we take δ t = 10 -3 .

Minimization among periodic functions is completely covered by theoretical results (see Theorems 3.3.2, 3.3.1). Therefore we started the experiments by testing the conjecture on these cases: the dnoidal, the cnoidal and the snoidal functions. We have chosen to fix k = 0.9. Recall that to run the algorithm we have to fix a, b, J and E. For these three cases we fix J = 0.

We start with the periodic case. Let

E = -0.095, b = 2, a = -(2 -k 2 ).
As mentioned in Section 3.3.2, after fixing J and E we can now find the period T , the Floquet multiplier θ , the mass m and the momentum p. We found the following quantities:

T = 2K(k), θ = 0, m = E(k), p = 0,
which are exactly the period, mass and momentum of the dnoidal function. We start by plotting u ode the numerical solution of the ordinary differential equation (3.1.2) and the function dn(x, k). We can see on the left of Figure 3.4 that u ode is very close to the exact solution dn. The second step is to run the algorithm and find the minimizer u min which we compare to the solution of the ODE u ode . We observe convergence towards dnoidal functions as we see on the right of Figure 3.4 with a maximum difference between the solutions of 3 × 10 -3 . The second case is the anti-periodic functions. We fix

E = 0.095, b = 2k 2 , a = 1 -2k 2 .
Same as before we find the period T , the mass m and the momentum p. For the Floquet multiplier θ, we impose θ = π and therefore we have:

T = 2K(k), θ = π, m = (K -E) k 2 , p = 0.
We start by comparing the numerical solution of the ODE with the exact solution. We can see on the left of Figure of 3.5 that the numerical solution of the ODE is the exact solution cn. Then we compare it to the solution of the minimization problem. We observe convergence towards cnoidal functions as we see on the right of Figure 3.5 with a maximum difference between the solutions of 3 × 10 -4 .

From the minimization problem to the ordinary differential equation 97

With the same method as before we have

T = 2K(k), θ = π, m = 8 bT K(K -E), p = 0.
First we can see on the left of Figure of 3.6 that the numerical solution of the ODE is the exact solution sn. Then we compare it to the solution of the minimization problem. We observe convergence towards snoidal functions as we see on the right of Figure 3.6 with a maximum difference between the solutions of 1 × 10 -2 . This case is not covered by the theoretical result. Our observation here is similar to the one of [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF]: snoidal functions minimize the energy on fixed mass and 0 momentum among anti-periodic functions. Now we will test the conjecture on the other cases where we do not know the theoretical results i.e. for complex valued solutions of (3.1.2). To do so we fix different values of J and for each of these values, we choose E such that (J, E) ∈ D 1 , ((D 2 ), (D 3 )) defined in (3.2.2), (3.2.3), (3.2.4). We plot the numerical solution of the ODE and the solution of the minimization problem and we do the comparison.

We start with the defocusing case, and we fix b = -1 and a = 1. We choose J = 0.2 arbitrarily. For other values of (J, E) ∈ D 1 that we tested, we obtain the same result. Then we fix three values of E such that: the first one corresponds to E = E -(J) = V (J, r Q ), where we know that the solution is a plane wave, the second is strictly between E -(J)and E + (J) and the third is very close to E + (J). For each value of E we plot the numerical solution of the ODE and the solution of the minimization problem. As we can see in Figure 3.7, we obtain a very good agreement with the conjecture. For the first value of E we have a maximum difference between the solutions of 5 × 10 -8 , for the second value of E a maximum difference of 1 × 10 -2 and for the third value of E a maximum difference of 3 × 10 -2 . We do the same for the focusing case, with positive a. We fix b = 1 and a = 1 and an arbitrary J = 1. We choose two values for E. The first one E = E -(J) = V (J, r Q ) and the second such that E -(J) < E = 5. We plot the solutions and we can see in Figure 3.8 that the solution of the ODE is the minimizer with a maximum difference of 1.2 × 10 -7 for the first value of E and 1 × 10 -2 in the second. Finally the focusing case with negative a. We fix b = 1 and a = -1 and an arbitrary J = 4. We choose two values for E. The first one E = E -(J) = V (J, r Q ) and the second E -(J) < E = 7. We plot the solutions and we can see in Figure 3.9 that the solution of the ODE is the minimizer. For the first value of E we have a maximum difference between the solutions of 6 × 10 -8 , for the second value of E a maximum difference of 2 × 10 -3 .

Order of the scheme

Using the numerical solution of the ODE and the solution of minimization problem, we were able to evaluate the order of the scheme which is of order 1. To confirm this we make various simulations for different mesh sizes δ x and present the results below. We tried to change the Laplacien and the matrix of the derivative to order 4 in order to obtain a better order of the scheme but the observed order of the scheme was still 1.

For the defocusing case, we choose to fix (J, E) = (0.15, 0.25) and (J, E) = (0.3, 0.29) arbitrarily. We present the results in Figure 3.10. For the focusing case with positive a = 1, we choose to fix (J, E) = (1, 2) and (J, E) = (2, 5) arbitrarily. We present the results in Figure 3.11. For a = -1 we fix (J, E) = (2, 6) and (J, E) = [START_REF]Nonlinear fiber optics. Optics and Photonics[END_REF][START_REF] Andrzej | The method of Nehari manifold[END_REF]. We can see in Figure 3.12 that the scheme is of order 1.

Remark 3.3.6. When we minimize the energy without the momentum constraint, for the dnoidal function the scheme is of order 2. For the cnoidal function it is of order 1.

Surfaces

The conserved quantities surfaces

In this section we plot the surfaces of the conserved quantities of the ordinary differential equation with respect to the period T , the Floquet multiplier θ and the conserved mass and momentum of the nonlinear Schrödinger equation (3.1.1). We start with the defocusing case. Let b = -1 and a = 1. We take 25 values of J between 0.1 and 4 27 a 3 b 2 . For each value of J we take 40 value of E such that E -(J) E < E + (J) -0.001. For each couple (J, E) we find the period T , the Floquet multiplier θ, the mass M and the momentum P . Recall that from Section 3.2.4 we know that in the defocusing case the period T is strictly increasing in function of E and strictly decreasing in function of J for all J > 0. We represent each of the surfaces in Figures 3.13 and 3.14 .

For the focusing case let b = 1. For each positive or negative a, we take 50 values of J between 0.1 and 10. For each value of J we take 40 value of E such that E -(J) E 30. For each couple (J, E) we find the period T , the Floquet multiplier θ, the mass M and the momentum P . Recall that from Section 3.2.4 we know that in the focusing case with positive a the period T is strictly decreasing in function of E and strictly decreasing in function of J for all J > 0. Unlike the other cases for the focusing case with a negative T is no longer a monotonic function of E (see Proposition 3.2.10). We represent each of the surfaces in Figures 3. [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]

The energy surfaces

In this section we will represent the surface in (M, P, E) for each of the focusing/defocusing case. We fix N = 500, T = π and we will take 2 values of θ which are 0 and π. We start with the defocusing case. Let b = -1 and a = 1. Let θ = 0. We choose the mass such that it varies from 1 to 3 with r = 17 values. Then for each value of M we let P varies in (0, 2M ) with k = 11 values. Using the gradient flow algorithm we find the solution for every fixed (M, P ) and we calculate the energy E of every solution and we represent the surface (M, P, E). As we can see in Figure 3.19, for the defocusing case with θ = 0 the energy is increasing as a function of P and the minimum of the energy is at P = 0 which is consistent with the fact that in the defocusing case with periodic solutions the minimizer of the energy with fixed mass is the constant function. Now we fix θ = π. In this case and same as before we represent the surface in (M, P, E) in Figure 3.20. We can see that the energy starts decreasing from P = 0 to P = M and then increases. This is consistent with the fact that in the anti-periodic case the minimizer is a plane wave which occurs at P = M .

Finally, for the focusing case let b = 1 and a = -1. In this case the surface (M, P, E) is represented in Figure 3.21. We can see that the energy is an increasing function of P .

Notebook

This section includes the Python code that we have developed to demonstrate our findings. Our goal is to compute the minimizer of the energy at a fixed mass m and momentum p using the normalized gradient flow approach. Furthermore, we compare the obtained minimizer with the solution of the ordinary differential equation, which we also compute. On the boundary E = E -the period T is given by the limit when E tends to E -. The period T is given by

T(J, E) = 2 r 2 r 1 1 2(E -V J (r))
dr.

The Floquet multiplier is given by

θ(J, E) = φ(x + T) -φ(x) = T 0 φ x (x)dx = T 0 J r 2 (x) dx = 2 r 2 r 1 J r 2 2(E -V J (r)) dr.
The mass is given by

M(J, E) = 1 2 T 0 |u| 2 dx = 1 2 T 0 r 2 (x)dx = r 2 r 1 r 2
2(E -V J (r)) dr.

Notebook

And the momentum is given by

P(J, E) = 1 2 T 0 u ūx dx = - 1 2 T 0 φ x r 2 dx = - 1 2 T 0 Jdx = - JT 2 = - r 2 r 1 J 2(E -V J (r)) dr.
When E → E -(J) the following holds:

lim

E→E -(J) T(J, E) = π √ 2 3Q 2 -a , lim E→E -(J) M(J, E) = |Q 2 -a| 2|b| π √ 2 3Q 2 -a , lim E→E -(J) P(J, E) = - 1 2 Q (Q 2 -a) b π √ 2 3Q 2 -a .
For the defocusing case, when E → E + (J) the following holds: t=integrate.quad(lambda r: 2/(math.

lim E→E + (J) T(J, E) = +∞, lim E→E + (J) M(J, E) = +∞, lim E→E + (J) P(J, E) = -∞.
→sqrt(2*(E[i]-V(J,r)))),R[i][0],R[i][1])[0]
theta=integrate.quad(lambda r: 2*J/(r**2*math.

→sqrt(2*(E[i]-V(J,r)))),R[i][0],R[i][1])[0]
mass=integrate.quad(lambda r: r**2/(math.

→sqrt(2*(E[i]-V(J,r)))),R[i][0],R[i][1])[0]
p=integrate.quad(lambda r: -J/(math. General nonlinearity

→sqrt(2*(E[i]-V(J,r)))),R[i][0],R[i][1])[0]

Introduction

We consider in one space dimension the nonlinear Schrödinger equation

iψ t + ψ xx + bf (ψ) = 0, (4.1.1)
where ψ : R t × R x → C, the nonlinearity f :

C → C is defined for any z ∈ C by f (z) = g(|z| 2 )z with g ∈ C 0 ([0, +∞), R) ∩ C 2 ((0, +∞), R) and b ∈ R \ {0}.
For simplicity, we denote by f the derivative of f |R . We make the following assumptions on the nonlinearity:

• The function h(s) := (sf (s) -2F (s))s -2 , (4.1.2)
is strictly increasing on (0, +∞), where F (z) = • There exist M > 0 , 1 < p < 5 and s 0 such that for all s s 0 we have |f (s)| M s p . Under these assumptions, the functions s → F (s) s 2 and s → f (s) s are increasing functions that grow to infinity. We also assume that f satisfies the inequality:

s 2 f (s) > sf (s) -f (s). (4.1.3)
Finally, to avoid linear behavior of f at infinity, we assume that:

lim s→∞ f (s) s -f (s) = -∞. (4.1.4)
Note that these assumptions hold for nonlinearities of the form of sum of powers with positive coefficients, for example. We are particularly interested in the spatially periodic solutions ψ(t, .) ∈ H 1 loc ∩ P T , and anti-periodic solutions ψ(t, .) ∈ H 1 loc ∩ A T where

P T = {f ∈ L 2 loc (R) : f (x + T ) = f (x)}, and
A T = {f ∈ L 2 loc (R) : f (x + T ) = -f (x)}.
The Cauchy problem (4.1.1) is known to be globally well posed in H 1 loc ∩ P T (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). Solutions to (4.1.1) conserve the mass M , the momentum P , and energy E:

M (ψ) = 1 2 T 0 |ψ| 2 dx, P (ψ) = 1 2 Im T 0 ψψ x dx E(ψ) = 1 2 T 0 |ψ x | 2 dx -b T 0 F (ψ)dx.
The simplest non-trivial solutions of (4.1.1) are the standing waves. They are solutions of the form

ψ(t, x) = e -iat u(x), a ∈ R.
The profile function u(x) satisfies the ordinary differential equation

u xx + au + bf (u) = 0. (4.1.5)
It is an integrable ordinary differential equation, whose conserved quantities (on C) are the momentum J and the energy E, given by

J = Im(u x u), E = 1 2 |u x | 2 + a 2 |u| 2 + bF (u).
In the work of Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] they provided a global variational characterization of the cnoidal, snoidal, and dnoidal elliptic functions for the cubic case, and proved some orbital stability results for the corresponding solutions of the nonlinear Schrödinger equation.

In this work, we aim to generalize their results to the case of a general nonlinearity. Specifically, we start with the analysis of the profile equation (4.1.5), which describes the behavior of stationary solutions to (4.1.1). We will consider the real and complex-valued solutions and represent their phase portraits for various parameter regimes. We will then study the minimization problems for the energy functional with fixed mass and momentum constraints for the periodic and anti-periodic cases, extending the previous results to this more general setting. We will analyze the existence, uniqueness, and stability of the minimizers for each case, and investigate their properties in detail using variational methods. Finally, we will consider the minimization problem on the Nehari manifold. The ground state solutions for the minimization problem lie on the Nehari manifold, and we can characterize them by minimizing the energy functional subject to the Nehari constraint. The analysis of these problems is expected to provide a better understanding of the dynamics of the nonlinear Schrödinger equation.

Preliminaries

When dealing with functions in P T , we will denote norms such L q (0, T ) by

u L q = u L q (0,T ) = T 0 |u| q 1 q
, and the complex L 2 inner product by

(f, g) = T 0 f ḡdx.

Orbital stability

Recall that we say that the standing wave ψ(t, x) = e -iat u(x) is orbitally stable for the flow of (4.1.1) in the function space X if for all > 0, there exists δ > 0 such that the following holds: if ψ 0 ∈ X verifies ψ 0u X δ, then the solution ψ of (4.1.1) with initial data ψ(0, x) = ψ 0 verifies for all t ∈ R the estimate inf θ∈R,y∈R ψ(t, .)e iθ u(.y) X < .

Analysis of the profile equation

In this section, we study the bounded solutions of the profile equation (4.1.5). We will distinguish between two different cases depending whether or not J = 0. We introduce the polar coordinates u(x) = r(x)e iφ(x) , with r > 0 and r, φ ∈ C 2 (R). The invariants become

J = r 2 φ x , E = r x 2 2 + J 2 2r 2 + a r 2 2 + bF (r).
If J = 0, then replacing r(x)e iφ(x) with r(x)e iφ for some φ ∈ [0, 2π] we can assume that u(x) ∈ R up to a constant phase. If J = 0, then u(x) = 0 for all x ∈ R, and φ x = 0. Define the effective potential by

V J (r) = J 2 2r 2 + a r 2 2 + bF (r).
We will study the variations of the function h, which will give us the graph of the potential. We have

h (r) = 4ar 3 + bf (r)r 3 + 3br 2 f (r) = r 3 4a + bf (r) + 3b f (r) r .
We start with the defocusing case where b < 0 and a > 0. In this case the graph of h(r) as a function of r is presented on the left of 1. If λ 1 , λ 2 are real numbers of the same sign the point is called a node.

2. If λ 1 , λ 2 are real numbers and non-zero of opposite sign the point is called a saddle.

3. If λ 1 , λ 2 are complex numbers, the real parts are equal and non-zero the point is called focus.

4. If λ 1 , λ 2 are purely imaginary numbers the point is called center.

Let's start with the defocusing case where b < 0 and a > 0. We know that f (r)

r is an increasing function on (0, ∞) therefore in this case there exists a unique r 0 > 0 such that ar 0 + bf (r 0 ) = 0. Thus we have three equilibrium points (0, 0), (r 0 , 0) and (-r 0 , 0). Hence

I 0 = {(y 1 , y 2 ) ∈ R 2 : y 1 ∈ {0, ±r 0 }, y 2 = 0}, and
I ∞ = {(y 1 , y 2 ) ∈ R 2 : y 1 / ∈ {0, ±r 0 }, y 2 = 0}.
The characteristic polynomial of the Jacobian matrix J F is given by P (λ) = λ 2 + a + bf (y 1 ). At the equilibrium point (0, 0) the eigenvalues are λ = ±i √ a (recall that a > 0). Since the eigenvalues are purely imaginary, the equilibrium point (0, 0) is a center. At the equilibrium points (±r 0 , 0) we have a + bf (r 0 ) < 0, therefore the eigenvalues are non-zero real numbers of opposite signs and the equilibrium point is a saddle point. The phase portrait is given in The last case is the focusing case with b > 0 and a < 0. There exists a unique r 0 > 0 such that ar 0 + bf (r 0 ) = 0 and we have three equilibrium points (0, 0), (r 0 , 0) and (-r 0 , 0). Same as before we have the iscolines I 0 and I ∞ . At the equilibrium point (0, 0) the eigenvalues are λ = ± √ -a and the equilibrium point (0, 0) is a saddle. At the equilibrium points (±r 0 , 0) the eigenvalues are purely imaginary numbers hence the equilibrium point is a center. The phase portrait is given on the right of Figure 4.5.

The second case is when J = 0. We have The Jacobian matrix of F (y) is of the form

I 0 = {(y 1 , y 2 ) ∈ R 2 : J 2 y 3 1 -ay 1 -bf (y 1 ) = 0, y 2 = 0}, and
I ∞ = {(y 1 , y 2 ) ∈ R 2 : J 2 y 3 1 -ay 1 -bf (y 1 ) = 0, y 2 = 0}.
J F = 0 1 -3 J 2 y 4 1 -a -bf (y 1 ) 0 .
We start with the defocusing case where b < 0 and a > 0. In this case the equation J 2 y 4 1 -ay 1 -bf (y 1 ) = 0 has 2 solutions r Q and r q such that 0 < r Q < r q . Thus we have two equilibrium points (r Q , 0) and (r q , 0). The characteristic polynomial of the Jacobian matrix J F is given by P (λ) = λ 2 + 3J 2 y 4 1 +a+bf (y 1 ).

On the first equilibrium point

(r Q , 0) we have λ 2 = -3J 2 r 4 Q -1 + f (r Q ) = -V J (r Q ) < 0, because V J (r
) is convex at r Q and therefore the eigenvalues are purely imaginary and the equilibrium point (r Q , 0) is a center. On the second equilibrium point (r q , 0) we have λ

2 = -3J 2 r 4 q -1 + f (r q ) = -V J (r q ) > 0,
because V J (r) is concave at r q and therefore the eigenvalues are non-zero real numbers of opposite signs hence the equilibrium point is a saddle. The phase portrait is given on the left of Figure 4.6.

For the focusing case where b > 0, with both cases a > 0 or a < 0, the equation J 2 y 3 1 ay 1bf (y 1 ) = 0 has 1 solution r Q . On the equilibrium point

(r Q , 0) we have λ 2 = -3J 2 r 4 Q -a -f (r Q ) < 0,

Triple power nonlinearity

In this section we treat a special case not included in the general case we treated before. Let f (u) = a 1 |u|u + a 2 |u| 2 u + a 3 |u| 3 u = 0, where a 1 , a 3 > 0 and a 2 < 0. After using the symmetry of (4.1.1), we may assume a 1 = a 3 = 1 and a

2 = -γ < 0. Let f (φ) = |φ|φ -γ|φ| 2 φ + |φ| 3 φ. Denote also F (φ) = 1 3 |φ| 3 - γ 4 |φ| 4 + 1 5 |φ| 5 .
We study the critical points of V . Since f is gauge-invariant, V is even in r and we may restrict the study to positive critical points. Let ω = -a, we have

V (r) = -ωr + f (r). Define f 1 (r) = f (r) r .
The difference between this case and the case we treated before is that f 1 (r) is not strictly increasing. A positive zero of V is a positive solution of

0 = -ω + f 1 (r) = -ω + r -γr 2 + r 3 . (4.3.1)
To determine the number of zeros of V , we analyze the variations of f 1 . We have f 1 (r) = 1 -2γr + 3r 2 , which has constant sign when γ < √ 3 and otherwise has two (positive) zeros given by

r ± = 1 3 γ ± γ 2 -3 .
As a consequence, when 0 < γ √ 3, the function f 1 is strictly increasing on [0, ∞) and there exists a (unique) positive solution of (4.3.1) if and only if ω > 0.

When γ > √ 3, we have f 1 (r) > 0 for r ∈ (0, r -)∪(r + , ∞) and f 1 (r) < 0 for r ∈ (r -, r + ). In this case, (4.3.1) has between 0 and 3 solutions. In particular, (4.3.1) has three positive solutions if and only if ω > 0 and

1 27 γ(-2γ 2 + 9) -2(γ 2 -3) 3 2 = f 1 (r + ) < ω < f (r -) = 1 27 γ(-2γ 2 + 9) + 2(γ 2 -3) 3 2
.

The γω regions of existence of solutions for (4.3.1) is represented in the figure below (zero solution, one solution, two solutions, three solutions). Whenever they exist, we denote the solutions of (4.3.1) by

0 < c 1 < r -< c 2 < r + < c 3 ,
with the convention that when r ± do not exist the solution is called c 1 .

Let us now distinguish the various possibles phase portraits depending on γ and ω.

4.3.2.1 Case ω < min{0, f 1 (r + )}
In this case the only critical point of V is 0, which is a center. Solutions of (4.1.5) are all of sn/cn type. The phase portrait is given in Figure 4.7.

Case

ω > 0, ω ∈ {(f 1 (r + ), f 1 (r -))}
In this case, V has two non-negative critical points: 0 and c 1 . The point 0 is a saddle point. The other critical point c 1 is a center. The phase portrait is similar to the one of the single focusing power. We have dn type solutions close to the center and cn type solutions for higher first integrals. The phase portrait is given in Figure 4.8.

Case

f 1 (r + ) < ω < 0
In this case, V has three non-negative critical points: 0 and c 2 , c 3 . The points 0 and c 3 are centers. The other critical point c 2 is a saddle point. Only one possible phase portrait with discussion about the co-existence of periodic solutions for same values of the first integral. The phase portrait is given in Figure 4.9.

4.3.2.4 Case max(0, f 1 (r + )) < ω < f 1 (r -)
In this case, V has four non-negative critical points: 0 and c 1 , c 2 , c 3 . The points 0 and c 2 are saddle points. The other critical point c 1 and c 3 are centers. Three possible phase portraits depending on the value of V (c 2 ). If V (c 2 ) > 0, then we have a homoclinic solution connecting 0 to itself without passing through c 2 and an heteroclinic solution connecting c 2 to -c 2 . If V (c 2 ) < 0, then the heteroclinic solution connecting 0 to itself passes through c 2 and c 3 (observe that this is nothing but the line of non-existence of solitons found in [START_REF] Fei | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF]). The phase portraits are given in Figure 4.10.

The minimization problems

In the work of Gustafson, Le Coz and Tsai [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF], the authors gave global variational characterizations of the cnoidal, snoidal, and dnoidal elliptic functions for the cubic nonlinear Schrödinger equation. Specifically, they showed that these functions arise as minimizers of the energy functional subject to that is

-∂ xx u ∞ + ωu ∞ -b|u ∞ | 2 u ∞ ± iσ∂ x u ∞ = 0.
In this chapter we will only focus on the case p = 0. Let us prove that the minimal energy is negative. To do so, let φ m,0 ≡ 2m T be a test function. We have

M (φ m,0 ) = m, E(φ m,0 ) = - T 0 F 2m T dx = -T F 2m T < 0,
where the last inequality holds because F (z) > 0 for any z ∈ C by the assumptions on f . Consider now a minimizing sequence (u n ) ⊂ H 1 loc ∩ P T for (4.4.1). We first prove that it is bounded in H 1 loc ∩ P T . To this aim, we rely on the Gagliardo-Nirenberg inequality: for any u ∈ H 1 loc ∩ P T , we have

u p+1 L p+1 u x α(p+1) L 2 u (1-α)(p+1) L 2 + u p+1 L 2 ,
where α = 1 2 -1 p+1 . We also know that there exists p > 1 such that

F (u) |F (u)| |u| 2 + |u| p+1 .
Consequently, for any u ∈ H 1 loc ∩ P T , such that M (u) = m, we have

E(u) = 1 2 u x 2 L 2 - T 0 F (u)dx, 1 2 u x 2 L 2 -u 2 L 2 -u p+1 L p+1 , 1 2 u x 2 L 2 -m -u x α(p+1) L 2 m (1-α)(p+1) 2 -m p+1 2 , = u x 2 L 2 1 2 -u x α(p+1)-2 m (1-α)(p+1) 2 -m p+1 2 -m.
The previous inequality implies the boundedness of ∂ x u n L 2 when 1 < p < 5. Indeed, by contradiction, we suppose that ∂ x u n L 2 → ∞. Since 1 < p < 5, we have α(p + 1) -2 < 0, and this implies that

∂ x u n α(p+1)-2 L 2
→ 0, and therefore E(u n ) → ∞, which is a contradiction with the minimizing nature of (u n ). Moreover, the same arguments show that if 1 < p < 5, then the minimal energy is finite. Hence the sequence (u n ) is bounded in H 1 loc ∩P T . Therefore up to a subsequence, (u n ) converges weakly in H 1 loc ∩ P T and strongly in L 2 loc ∩ P T and L p+1 loc ∩ P T towards u ∞ ∈ H 1 loc ∩ P T . We now show that (u n ) converges strongly towards u ∞ in H 1 loc ∩ P T . By weak convergence, we have

∂ x u ∞ 2 L 2 lim n→+∞ ∂ x u n 2 L 2 .
Up to a subsequence, we also have F (u n ) → F (u ∞ ) almost everywhere. Moreover, we have

|F (u n )| |u n | 2 + |u n | p+1 u n 2 L ∞ + u n p+1 L ∞ u n 2 H 1 + u n p+1 H 1 max n∈N { u n 2 H 1 + u n p+1 H 1 } < ∞.
then by the dominated convergence theorem we have

lim n→+∞ T 0 F (u n )dx = T 0 F (u)dx.
Combining the previous arguments, we obtain

E(u ∞ ) lim n→+∞ E(u n ), M (u n ) = m, which in turn implies ∂ x u ∞ 2 L 2 = lim n→+∞ ∂ x u n 2 L 2 .
Therefore the convergence from (u n ) to u ∞ is also strong in H 1 loc ∩ P T .

Proposition 4.4.2. If m > m, then the minimizer is not a constant, the associated Lagrange multiplier verifies a < 0, the minimizer is positive, it is a solution of the ordinary differential equation (4.1.5).

Remark 4.4.3. In the cubic case, it is known that for small enough values of m, the minimizer of the energy functional in this case is the constant function. However, in our work, we do not provide a rigorous proof of this conjecture, but we believe it to be true based on the known result in the cubic case.

Proof. Since u ∞ is a minimizer of (4.4.1), there exists a Lagrange multiplier

a ∈ R such that -E (u ∞ ) + aM (u ∞ ) = 0 that is ∂ xx u ∞ + au ∞ + bf (u ∞ ) = 0.
Multiplying by u ∞ and integrating (recall that the functions considered are assumed to be real), we find that

a = ∂ x u ∞ 2 L 2 -b T 0 f (u ∞ )u ∞ dx u ∞ 2 L 2 . Note that ∂ x u ∞ 2 L 2 -b T 0 f (u ∞ )u ∞ dx = 2E(u ∞ ) + 2b T 0 F (u ∞ )dx - T 0 bf (u ∞ )u ∞ dx = 2E(u ∞ ) + b T 0 (2F (u ∞ ) -f (u ∞ )u ∞ )dx, where E(u ∞ ) < 0 and 2F (u ∞ ) -f (u ∞ )u ∞ < 0,
by the assumption on (4.1.2). Therefore, we have a < 0.

We introduce an auxiliary function

A(s) = 4π 2 T 2 + b f (s) s -f (s) .
By assumption (4.1.3), we have

A (s) = b f (s)s -f (s) s 2 -f (s) < 0.
Therefore A is a decreasing function, from 4π 2 T 2 to -∞ from assumption (4.1.4). Let m * be such that A(m * ) = 0 and define

m = T m * 2 2
We want to prove that if m > m, then u ∞ is not constant. By contradiction, we assume that u ∞ is constant for m > m. Then we necessarily have u ∞ ≡ 2m T . The Lagrange multiplier can also be computed and we find

a = -b T 0 f 2m T 2m T dx 2m = -bf 2m T T 2m .
Since u ∞ is supposed to be a constrained minimizer for (4.4.1), the operator

-∂ xx -a -bf (u ∞ ) = -∂ xx + b T 2m f 2m T -bf 2m T ,
must have Morse Index at most 1, i.e, at most 1 negative eigenvalue. The eigenvalues are given for n ∈ Z by the following formula:

2πn T 2 + b T 2m f 2m T -bf 2m T , n ∈ Z.
If n = 0, the eigenvalue is negative:

b T 2m f 2m T -bf 2m T < 0.
Indeed as f (s) s is an increasing function we have that for all s > 0, f (s)

s = f (s)s-f (s) s 2 > 0.
If n = 1 the eigenvalue is of the form:

4π 2 T 2 + b T 2m f 2m T -bf 2m T = A 2m T .
Recall that A Proof. Consider a minimizing sequence (u n ) ⊂ H 1 loc ∩ P T for (4.4.1). We first prove that it is bounded in H 1 loc ∩ P T . We have

E(u n ) = 1 2 ∂ x u n 2 L 2 + T 0 F (u n )dx.
By contradiction, we suppose that ∂ x u n L 2 → ∞. Therefore E(u n ) → ∞, which is a contradiction with the minimizing nature of (u n ). Moreover, the same argument show that the minimal energy is finite. Hence the sequence (u n ) is bounded in H 1 loc ∩ P T . Therefore up to a subsequence, (u n ) converges weakly in H 1 loc ∩ P T and strongly in L 2 loc ∩ P T and L p+1 loc ∩ P T towards u ∞ ∈ H 1 loc ∩ P T . As in the proof of Proposition 4.4.1 we have that (u n ) converges strongly towards u ∞ in H 1 loc ∩ P T . As in the proof of Proposition 4.4.2, we know that there exists a Lagrange multiplier a such that

-E (u ∞ ) + aM (u ∞ ) = 0, (4.4.3)
i.e. u ∞ satisfies the ordinary differential equation (4.1.5). Hence a might be explicitly expressed in the following way:

a = ∂ x u ∞ 2 L 2 -b T 0 f (u ∞ )u ∞ dx u ∞ 2 L 2 .
Since b < 0, we have a > 0. In this case we know that the phase portrait for real valued solutions of (4.1.5) is given in Figure 4.4.

As for the focusing case, for any v ∈ H 1 loc ∩ P T , we have M (|v|) = M (v), E(|v|) E(v), therefore we may assume that u ∞ 0. The only solutions of (4. Therefore, the eigenvalues of the associated linearized operator

-∂ xx -a -bf (u ∞ ) = -∂ xx + b T 2m f 2m T -bf 2m T
are given for n ∈ Z by the following formula:

2πn T 2 + b T 2m f 2m T -f 2m T .
Since b < 0 and f (s)s -1f (s) < 0 (see assumption (4.1.2)), we remark that the eigenvalues are all positive.

Minimization among anti-periodic functions

We will also consider variational problems restricted to anti-symmetric functions: In this section, we restrict ourselves to the sum of two powers to use the Hölder inequality that cannot be used in the general case. Moreover we can generalize to the sum of several powers. Proposition 4.4.6. Let f (u) = |u| p-1 u + |u| q-1 u, with p, q > 1. There exists a unique (up to phase shift and complex conjugate) minimizer of (4.

v L 2 T 2π v L 2 ,
and that the optimizers of the Poincaré-Wirtinger inequality are of the form ce ± iπx T , c ∈ C. This implies that

∂ x w 2 L 2 = 8π 2 T 2 M (w) = 8π 2 T 2 M (v) < ∂ x v 2 L 2 .
We will prove now that T 0 F (w)dx T 0 F (v)dx. We have

T 0 1 p + 1 |w| p+1 + 1 q + 1 |w| q+1 dx, = T 0   1 p + 1 2m T p+1 + 1 q + 1 2m T q+1   dx, = T 1 p + 1 T -p+1 2 2 p+1 2 m p+1 2 + 1 q + 1 T -q+1 2 2 q+1 2 m q+1 2 , = T 1 p + 1 T -p+1 2 v p+1 L 2 + 1 q + 1 T -q+1 2 v q+1 L 2 , T 1 p + 1 T -p+1 2 T p-1 2 v p+1 L p+1 + 1 q + 1 T -q+1 2 T q-1 2 v q+1 L q+1
, where the last inequality came from Hölder inequality:

v p+1 L 2 T p-1 2 v p+1 L p+1 , 1 p+1 2 + 1 p+1 p-1 = 1.
Therefore we have

1 p + 1 w p+1 L p+1 + 1 q + 1 w q+1 L q+1 1 p + 1 v p+1 L p+1 + 1 q + 1 v q+1 L q+1
which implies that E(w) < E(v), which concludes the proof. and p an odd integer. Then there exists ṽ ∈ H 1 loc ∩ A T 2 such that:

A Fourier rearrangement inequality

ṽ(x) ∈ R, ṽ L 2 = v L 2 , ∂ x ṽ L 2 = ∂ x v L 2 , ṽ L p+1 v L p+1 . Proof. Since v ∈ H 1 loc ∩ A T 2
, its Fourier series expansion contains only terms indexed by odd integers:

v(x) = j∈Z jodd v j e ij 2π T x .
We define ṽ by its Fourier series expansion ṽ(x) = j∈Z jodd ṽj e ij 2π T x , ṽj :=

|v j | 2 + |v -j | 2 2 .
It is clear that ṽ(x) ∈ R, and by Plancherel formula, we have

ṽ L 2 = v L 2 , ∂ x ṽ L 2 = ∂ x v L 2
, so all we have to prove is that ṽ L p+1 v L p+1 . We have where by wn , we denote the quantity defined similarly as in (4.5.1) for (ṽ j ).

Minimizing problem on the Nehari manifold

In this section we restrict ourselves to the nonlinearity of the form f (u) = |u| p-1 u, with p > 1. We define the functional S : H 1 loc → R by setting for u ∈ H 1 loc S(u) :=

1 2 ∂ x u 2 L 2 - a 2 u 2 L 2 - b p + 1 u p+1 L p+1 .
It is standard that S is of class C 2 . The Fréchet derivative of S at u is given by S (u) = -u xxau -b|u| p-1 u.

Therefore, u is a solution of the ordinary differential equation The minimization problem on the Nehari manifold has been studied in numerous works. In this regard, we mention the work of Szulkin and Weth [START_REF] Andrzej | The method of Nehari manifold[END_REF], the work of Pankov [START_REF] Pankov | Gap solitons in periodic discrete nonlinear Schrodinger equations II: A generalized Nehari manifold approach[END_REF] and Pankov and Zhang [START_REF] Zhang | Standing waves of the discrete nonlinear Schrödinger equations with growing potentials[END_REF] for the discrete nonlinear Schrödinger equation. We also mention the work of Hayashi [47] on the nonlinear Schrödinger equation of derivative type and the work of Colin and Watanabe [28] on the nonlinear Klein-Gordon-Maxwell type system. Lemma 4.6.1. The minimum of (4.6.1) is finite and there exists a minimizer solution of (4.1.5).

Proof. Consider a minimizing sequence (u n ) ⊂ H 1 loc ∩ P T for (4.6.1). We have I(u n ) = 0, therefore

S(u n ) = S(u n ) - 1 p + 1 I(u n ) = 1 2 - 1 p + 1 ∂ x u n 2 L 2 -a u n 2 L 2 . (4.6.3)
We have the boundedness of the sequence (u n ) in H 1 loc ∩ P T . Indeed, by contradiction we suppose that u n 2 L 2 → ∞, or ∂ x u n 2 L 2 → ∞, therefore S(u n ) → ∞, which is a contradiction with the minimizing nature of (u n ). Therefore up to a subsequence, (u n ) converges weakly in H 1 loc ∩P T and strongly in L 2 loc ∩ P T and L p+1 loc ∩ P T towards u ∞ ∈ H 1 loc ∩ P T . By the weak convergence we have

u ∞ H 1 lim n→∞ inf u n H 1 , then ∂ x u ∞ 2 L 2 -a u ∞ 2 L 2 lim n→∞ inf ∂ x u n 2 L 2 -a u n 2 L 2 .
Therefore S(u ∞ ) -

1 p + 1 I(u ∞ ) lim n→∞ inf S(u n ).
On the other hand we have The graph of I(tu) is given in the figure 4.11. We know that I(t 1 u ∞ ) 0, there exists t 0 < 1 such that I(t 0 u ∞ ) = 0. We have In the general case the nonlinearity f : C → C is defined for any z ∈ C by f (z) = g(|z| 2 )z with g ∈ C 0 ([0, +∞), R) ∩ C 1 ((0, +∞), R). Let G : t → g(|φ + th| 2 )(φ + th).

I(u ∞ ) = ∂ x u ∞ 2 L 2 -a u ∞ 2 L 2 -b u ∞ p+1 L p+1
S(t 0 u ∞ ) = S(t 0 u ∞ ) - 1 p + 1 I(t 0 u ∞ ), = 1 2 - 1 p + 1 t 0 u ∞ 2 H 1 , 1 2 - 1 p + 1 u ∞ 2 H 1 , = S(u ∞ ) - 1 p + 1 I(u ∞ ) lim n→∞ inf S(u n ).
We suppose that φ ∈ R, we have

∂G ∂t |t=0

= hg(φ 2 ) + 2φ 2 Re(h)g (φ 2 ) = (g(φ 2 ) + 2φ 2 g (φ 2 ))Re(h) + ig(φ 2 )Im(h) = f (φ)Re(h) + i f (φ) φ Im(h).

We know that u is a solution of (4.1.1), therefore h verifies the following equation

h t = ih xx -iωh + if (φ)Re(h) - f (φ) φ Im(h) + N (h),
where N (h) holds in all of the nonlinear terms. We separate h into real and imaginary parts and let h = h R + ih I , hence we have:

h R h I t = L h R h I + N (h)
where

L = 0 -∂ xx (.) + ω -f (φ) φ ∂ xx (.) -ω + f (φ) 0 .
For the triple power case where f (u) = |u|u -γ|u| 2 u + |u| 3 u we have the following lemma. T . For a given value of θ, there exists a constant γ * such that if γ > γ * , then the determinant of the linearization of the nonlinear Schrödinger equation around the constant solution is positive. As a consequence, the eigenvalues of the linearization are purely imaginary, and the constant solution is spectrally stable. The value of γ * depends on θ and the specific form of the nonlinear term in the equation.

Proof. We have |u|u = (φ 0 + h)

Study of periodic and standing waves for the nonlinear Schrödinger equation

Abstract:

The thesis focuses on the study of periodic and standing waves for the one dimensional nonlinear Schrödinger equation. The first part of the thesis focuses on the orbital stability of standing waves for the double power nonlinear Schrödinger equation. In this thesis we provide a complete stability picture for the problem using the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. We provided numerical experiments to complement our theoretical results. In the second part of the thesis, we study the quasi-periodic solutions for the cubic nonlinear Schrödinger equation. We use a variational approach to study the quasi-periodic solutions by minimizing the energy at fixed mass and momentum, using the method of the gradient flow with discrete normalization:at each step of time, we evolve in the direction of the gradient of the energy and renormalize the mass and the momentum of the outcome. We establish a connection between the minimizers of the energy and the solutions of the ordinary differential equation. The final part extends the work on the cubic case to a more general nonlinearity and provides variational characterizations for this case. Overall, the thesis contributes to the understanding of the dynamics of nonlinear Schrödinger equations, with a focus on the study of periodic and standing waves. We use a combination of analytical, numerical, and variational methods to provide a deeper understanding of the behavior of the equations and their solutions.

Theorem 1 . 3 . 1 .

 131 Let b > 0.

(a) If p 5 ,

 5 then it is of type S. (b) If p > 5, then it is of type U S.

101010

  Critical surface when ap = -1 and aq = 0.5 Critical surface when ap = -1 and aq = 1 Critical surface when ap = -1 and aq = 2

Figure 2 . 1 -

 21 Figure 2.1 -Critical surface {(p, q, ω c (p, q))} for a p = -1 and a q = 1/2, 1, 2. The white lines represent q = 7 -2p and q = p, where the transition from ω c = 0 to ω c > 0 occurs.

Figure 2 . 2 -whenwhenwhen

 22 Figure 2.2 -Slices of the critical surface for fixed value of q = 4

Figure 2 . 3 -

 23 Figure 2.3 -Zoom on slices of the critical surface for fixed value of q = 4

a

  p = -1, a q = 1, p = 1.25, q = 1.50, ω = 1.00, ω c = 0.00, init=1 Plot of t → max |u(t, x)| when ap = -1, aq = 1, p = 1.25, q = 1.50, ω = 1.00, ωc = 0.00, init=1

Figure 2 . 5 -

 25 Figure 2.5 -Example of a stable numerical solution. The initial data is u 0 = (1 + ε)φ ω , ε = 10 -2 .

aFigure 2 . 6 -

 26 Figure 2.6 -Example of a growing/oscillating numerical solution. The initial data is u 0 = (1 + ε)φ ω , ε = 10 -2 .

  .7. Observe that the domain of calculation is [-50, 50], but the solution is represented only on [-20, 20], which explains the non-zero values observed at the boundaries on the left figure.

Figure 2 . 7 -

 27 Figure 2.7 -Example of a scattering numerical solution. The initial data isu 0 = (1ε)φ ω , ε = 10 -2 .

[ 13 ]

 13 : %%file --append script_critical_omega.py # computation of the critical omega map with multi-processing p1 = np.linspace(2, 5, 5, endpoint=False) p2 = np.linspace(2, 5, 5, endpoint=False) P1,P2 = np.meshgrid(p1,p2) #P1P2=np.array([P1.flatten(),P2.flatten()]) # construct a new w_critical function for fixed values of a1 and a2 a1=-1 a2=0.5 partial_w_critical = partial(w_critical,a1=a1,a2=a2) w_critical_P1P2=[] if __name__ == "__main__": PPE = ProcessPoolExecutor() w_critical_P1P2 = list(PPE.map(partial_w_critical,P1.flatten(),P2. →flatten())) Appending to script_critical_omega.py [14]: %%file --append script_critical_omega.py # Save all variables (with in particular w_critical) to file with date␣ →in file name date = datetime.now().strftime("%Y_%m_%d-%I_%M_%S_%p") filename=f'w_critical_{date}.out' my_shelf = shelve.open(filename,'n') # 'n' for new for key in dir(): try: my_shelf[key] = globals()[key] except: # 2.7. Notebooks # __builtins__, my_shelf, and imported modules can not be␣ →shelved. # pass my_shelf.close() Appending to script_critical_omega.py [15]: # To restore all needed variables (with in particular p1,p2, and␣ →w_critical) import shelve #filename='w_critical_2021_07_10-07_59_17_PM.out' # a2=2 filename='w_critical_2021_07_10-08_00_51_PM.out' # a2=1 #filename='w_critical_a1_is_-1_a2_is_0.5_2021_07_20-01_03_33_PM.out'#␣ →a2=1/2 my_shelf = shelve.open(filename) list_of_variables = ['a1','a2','p1','p2','P1','P2','w_critical_P1P2'] for key in list_of_variables:#my_shelf: try: globals()[key]=my_shelf[key] except:

[ 11 ]

 11 : # close up view to check the regularity on the left and on the right fig3 = plt.figure(figsize=[START_REF] Andrzej | The method of Nehari manifold[END_REF][START_REF] Andrzej | The method of Nehari manifold[END_REF]) ax3 = plt.axes() # Slice of the critical surface for $q={value_p2}$\n ax3.set_title(f"when $a_p={a1}$ and $a_q={a2}$") ax3.plot(p1,w_critical_P1P2[index,:],label=f'$\omega_c(p,q={p2[index]:. →2f})$') ax3.set_xlabel(r"$p$", rotation=0, fontsize=FontSize) y_max=1 ax3.set_ylim(-0.3,y_max) legend = ax3.legend() fig3.

  →pdf',transparent=True,orientation='landscape',format='pdf', bbox_inches='tight') elif a2==1: ax4.plot(p1_of_p2,p2,'k',label='$p_{\max}(q)$') legend=ax4.legend(loc="upper left",prop={'size': 12}) fig4.savefig(f'p1_line_of_max_w_c_a1_is_{a1}_a2_is_{a2}. →pdf',transparent=True,orientation='landscape',format='pdf', bbox_inches='tight') #p$ which realizes $\max_{p}\omega_c(p,q) [3]: #%%file --append 2021_07_23_script_nls_evolution.py L=50 # half length of the interval on which we discretize N=10000+1 # number of discretization points #x,dx=np.linspace(-L, L, num=N, endpoint=False,retstep=True) # without␣ →right end point for periodic x,dx=np.linspace(-L, L, num=N,retstep=True) # with right end point for␣ →Dirichlet # construction of the Laplacian matrix with Dirichlet conditions def Laplace(N,dx): main_diag = [-2 for i in range(N)] upper_diag= [1 for i in range(N-1)] lower_diag= upper_diag diagonals=[lower_diag,main_diag,upper_diag] Lap=scs.diags(diagonals,[-1,0,1])/dx**2 return Lap Lap=Laplace(N,dx)Before simulating the evolution flow, we compute the critical omega associated with our parameters.

[ 4 ]

 4 : #%%file --append 2021_07_23_script_nls_evolution.py def f1(z,p1,p2,a1=-1,a2=1): s=np.abs(z) return -a1*s**((p1-1)/2)-a2*s**((p2-1)/2) def F1(z,p1,p2,a1=-1,a2=1): s=np.abs(z) return -(a1*s**((p1+1)/2))*2/(p1+1)-(a2*s**((p2+1)/2))*(2/(p2+1)) def U(s,w,p1,p2,a1=-1,a2=1): s=np.abs(s) return w*s+F1(s,p1,p2,a1,a2) def U_der(s,w,p1,p2,a1=-1,a2=1): s=np.abs(s) return w+f1(s,p1,p2,a1,a2) def U_div_s(s,w,p1,p2,a1=-1,a2=1): s=np.abs(s) return (w-(a1*s**((p1-1)/2))*2/(p1+1)-(a2*s**((p2-1)/2))*(2/(p2+1))) def U_div_s_prime(s,w,p1,p2,a1=-1,a2=1): s=np.abs(s) return -(a1*s**((p1-3)/2))*2/(p1+1)*(p1-1)/2-(a2*s**((p2-3)/2))*(2/ →(p2+1))*(p2-1)/2 def a(w,p1,p2,a1=-1,a2=1): a=sco.fsolve(U_div_s,␣ →10,args=(w,p1,p2,a1,a2),fprime=U_div_s_prime,xtol=10**(-12))

[ 6 ]

 6 : #%%file --append 2021_07_23_script_nls_evolution.py def initialvalue(phi,p1,p2,w=1,a1=-1,a2=1): return (a1/(p1+1)*phi**(p1-1)+a2/(p2+1)*phi**(p2-1)-w/2) def fun(x, y,p1,p2,w=1,a1=-1,a2=1): return np.vstack((y[1], w*y[0]-a1*np.abs(y[0])**(p1-1)*y[0]-a2*np. →abs(y[0])**(p2-1)*y[0]))

  →2f}_p2_is_{p2:.2f}_w_is_{w:.2f}_w_c_is_{w_c:.2f}_init_is_{init}.→html")#https://plotly.com/python/3d-surface-plots/ [11]: #%%file --append 2021_07_23_script_nls_evolution.py def make_max_figure(max_u_t_x,p1,p2,T=1,delta_t =␣ →1e-3,w=1,w_c=float("nan"),c=0,a1=-1,a2=1,init=0): N_t=len(max_u_t_x) Tt=np.linspace(0,T,N_t) fig = plt.figure(figsize=(8,6)) fig.suptitle(f"Plot of $t\mapsto\max|u(t,x)|$ when\n$a_p={a1}$,␣

27 a 3 b 2 .

 2732 Then V J (r) < 0 for all r > 0, hence (3.1.2) has no bounded solution in this case. If J 2 < 4 27 a 3

1 . 2 . 3 . 2 .

 1232 If b < 0 (defocusing case) then 0 < |b|u(0) 2 < a, and u(x)= 1 α sn(K(k)+ x β , k) or u(0) 2 = |a| b and u(x) = -a b . If b > 0 (focusing case), we have 3 possible cases:(a) if 0 min u < u(0), then a < 0, |a| < bu(0) 2 < 2|a|, and u(x)= 1 α dn( x β , k), From the ordinary differential equation to the minimization problem 77(b) if min u = u(0), then a < 0, u(0) 2 = |a| b , and u(x) = -a b , (c) if min u < 0, then max(0, -2a) < bu(0) 2 , and u(x) = 1 α cn( x β , k). The parameters satisfy the a-independent relation: bβ 2 = -2k 2 α 2 for (1), bβ 2 = 2α 2 for (2)-(a) and bβ 2 = 2k 2 α 2 for (2)-(b). In addition, there exists n ∈ N such that 4K(k)βn = T for (1) and (2)-(b) and 2K(k)βn = T for (2)-(a). Remark 3.2.2. In the case of non periodic solution, in the defocusing case if E = -a 2 4b then the solution can be u(x) = -a b tanh(x a 2 ) and in the focusing case with negative a if E = 0 the equation (3.1.2) can have the pulse-like solution u(x) = -2a b sech(x √ -a). The graph of E as a function of J for each case is given in Figure 3.1. From left to right we chose (b = -1, a = 1), (b = 1, a = 1) and (b = 1, a = -1).

Figure 3 . 1 -

 31 Figure 3.1 -Graphs of E as a function of J.

2 .

 2 From left to right we chose (b = -1, a = 1), (b = 1, a = 1) and (b = 1, a = -1).

Figure 3 . 2 -

 32 Figure 3.2 -Graphs of P as a function of M .

3. 2 .Figure 3 . 3 -

 233 Figure 3.3 -Domain D.

3 . 1 ) 2 .

 312 and (3.3.2), which is the constant function u min = 2mT . If θ = π, we are in the anti-periodic case: There exists a unique minimizer (up to phase shift) for (3.3.1) which is the plane wave u min =

Observation 3 . 3 . 5 .

 335 Let (J, E) ∈ D 1 (D 2 , D 3 respectively) for each defocusing and focusing case, u ode be the associated solution of the ordinary differential equation (3.1.2). Then (up to phase shift and translation) the minimizer of the problemmin{E(u) : u ∈ H θ T , M (u) = M (u ode ), P (u) = P (u ode ), u ∈ H θ T },is given by u ode .

Figure 3 . 4 -

 34 Figure 3.4 -Comparison of the dn function and the solutions of the ODE and the minimization problem.

Figure 3 . 5 -

 35 Figure 3.5 -Comparison of the cn function and the solutions of the ODE and the minimization problem.

Figure 3 . 6 -

 36 Figure 3.6 -Comparison of the sn function and the solutions of the ODE and the minimization problem.

Figure 3 . 7 -

 37 Figure 3.7 -Comparison between u min and u ode for the defocusing case.

Figure 3 . 8 -

 38 Figure 3.8 -Comparison between u min and u ode for the focusing case with a = 1.

3. 3 .Figure 3 . 9 -

 339 Figure 3.9 -Comparison between u min and u ode for the focusing case with a = -1.

Figure 3 . 10 -

 310 Figure 3.10 -Defocusing case.

Figure 3 . 11 -

 311 Figure 3.11 -Focusing case with a = 1.

Figure 3 . 12 -

 312 Figure 3.12 -Focusing case with a = -1.

  and 3.16 for a = 1 and in Figures 3.17

  and 3.18 for a = -1.

Figure 3 .

 3 Figure 3.13 -Surfaces (J, E, T ) and (J, E, θ) for the defocusing case.

Figure 3 .

 3 Figure 3.14 -Surfaces (J, E, M ) and (J, E, P ) for the defocusing case.

Figure 3 .

 3 Figure 3.15 -Surfaces (J, E, T ) and (J, E, θ) for the focusing case with a = 1.

Figure 3 .

 3 Figure 3.16 -Surfaces (J, E, M ) and (J, E, P ) for the focusing case with a = 1.

Figure 3 .

 3 Figure 3.17 -Surface (J, E, T ) for the focusing case with a = -1.

Figure 3 .

 3 Figure 3.18 -Surfaces (J, E, θ), (J, E, M ) and (J, E, P ) for the focusing case with a = -1.

Figure 3 . 19 -

 319 Figure 3.19 -Defocusing case with θ = 0.

Figure 3 .

 3 Figure 3.20 -Defocusing case with θ = π.

Figure 3 . 21 -

 321 Figure 3.21 -Focusing case with a = -1 and θ = π.

  print(i) c=[-b,0,-2*a,0,4*E[i],0,-2*J**2] l=np.roots(c) real_valued = l.real[abs(l.imag)<1e-5] #where I chose 1-e5 as a␣ →threshold roots= real_valued[real_valued>0] roots.sort() print(roots) R+=[roots]

[ 32 ]

 32 : # Warning: does not take J=0 into account T=[] Theta=[] Mass=[] P=[] for i in range(n): if E[i]==V(J,rQ): #plane wave on the boundary E=E_ t=(math.pi* math.sqrt(2))/math.sqrt(abs((3*Q**2-a))) theta=Q*t mass=abs((Q**2-a)/b)*t/2 p=-Q*abs((Q**2-a)/b)*t/2 else:

3 .

 3 Analysis of the quasi-periodic wavesChapter 4

|z| 0 f

 0 (s)ds = 1 2 G(|z| 2 ), with G = g and G(0) = 0. Also lim s→0 h(s) = 0 and lim s→0 f (s) s = 0.

Figure 4 .

 4 2. Hence V J (r) = 0 has 2 solutions for J 2 < r c where h (r c ) = 0 and the maximum occurs. Then the graph of V J as function of r is presented on the left of Figure 4.3. The second case is the focusing case where b > 0 and a > 0. We know that h(r) is a strictly increasing function presented on the center of Figure 4.2. Hence V J (r) = 0 has a unique solution. Then the graph of V J as function of r is given on the center of Figure 4.3. The last case is the focusing case where b > 0 and a < 0. The graph of h as a function of r is represented on the right of Figure 4.2. Hence V J (r) = 0 has a unique solution. Then the graph of V J as function of r is represented on the right of Figure 4.3.

Figure 4 . 2 -

 42 Figure 4.2 -h(r) as a function of r.

Figure 4 . 3 -

 43 Figure 4.3 -V J (r) as a function of r with J = 0.

Figure 4 . 4 .

 44 For the focusing case where b > 0 with a > 0 we have one equilibrium point (0, 0) satisfying the equation ar 0 + bf (r 0 ) = 0. The eigenvalues are given at the equilibrium point (0, 0) by λ = ±i √ a and the equilibrium point (0, 0) is a center. The phase portrait is given on the left of Figure of 4.5.

Figure 4 . 4 -

 44 Figure 4.4 -Phase portraits of the solutions for the defocusing when J = 0.

Figure 4 . 5 -

 45 Figure 4.5 -Phase portraits of the solutions for the focusing case when J = 0.

  therefore the eigenvalues are purely imaginary and the equilibrium point (r Q , 0) is a center. The phase portrait for these two cases is given on the right of Figure 4.6.

Figure 4 . 6 -

 46 Figure 4.6 -Phase portraits of the solutions when J = 0.

Figure 4 . 7 -

 47 Figure 4.7 -Phase portrait 0 solution.

Figure 4 . 8 -

 48 Figure 4.8 -Phase portrait 1 solution.

Figure 4 . 9 -

 49 Figure 4.9 -Phase portrait 3 solution.

  2m T is non-negative if and only if 2m T m * which is equivalent to m m which gives the contradiction. Therefore when m > m * the minimizer u ∞ is not constant, which concludes the proof. 4.4.1.2 The defocusing case in P T Assume that b < 0. Proposition 4.4.4. For all m ∈ (0, ∞) the constrained minimization problems (4.4.1) and (4.4.2) have the same unique (up to phase shift) minimizer, which is the constant function u ∞ ≡ 2mT .

  1.5) that do not change sign are the constant functions ± 2m T . As a consequence, there exists θ ∈ R such that u ∞ = e iθ 2m T , which concludes the proof. Remark 4.4.5. Under the assumptions of Proposition 4.4.4, the minimizer is u ∞ ≡ 2m T (up to phase shift), and therefore the associated Lagrange multiplier is given by a = -bf 2m T T 2m .

2 , 2 ,

 22 min{E(u) : u ∈ H 1 loc ∩ A T M (u) = m}, (4.4.4) min{E(u) : u ∈ H 1 loc ∩ A T M (u) = m, P (u) = 0}. (4.4.5) 4.4.2.1 The defocusing case for anti-periodic functions Assume b < 0.

AT 2 ,

 2 4.4). It is the plane waveu ∞ ≡ 2m T e iπx T .Proof. Denote the supposed minimizer by w(x) = 2m T e ± iπx T . Let v ∈ H 1 loc ∩ A T 2 such that: M (v) = m and v ≡ e iθ w (θ ∈ R). Since v ∈ H 1loc ∩ v must have 0 mean value. Recall that in this case v verifies the Poincaré-Wirtinger inequality

Lemma 4 . 5 . 1 .

 451 Let v ∈ H 1 loc ∩ AT

2

 2

|v(x)| 2 = j∈Z jodd |v j | 2 + n∈2N n 2 w

 2 n e in 2π T x + wn e in 2π T x ,where we have definedw n = j>k,j+k=n j,kodd v j v-k + v k v-j .

4. 5 . A Fourier rearrangement inequality 139 Let N = p+1 2 . 2 w 2 w 2 wwp 1 w p l e ip l 2π T x , where 0 l=1 2 w

 51392222102 We start with|v| p+1 = |v| 2 p+1 n e in 2π T x + wn e in 2π n e in 2π T x + wn e in 2π n e in 2π T x + wn e in 2π • • • wps • w ps w p s+1 • • • w p k e i(-p 1 -•••-ps+p s+1 +•••+p k ) 2π T x , wp l e -ip l 2π T x = 1, k l=k+1w p l e ip l 2π T x = 1. n e in 2π T x + wn e in 2π

w 0 e in 2π T x dx = 0 .

 00 p l , where σ = {(p 1 , ..., p n ) : ∃α ∈ {0, 1} n : j (-1) α j p j = 0}, and where we have used the fact that for n ∈ N, n = 0, we have T On the other hand, we observe thatw n =

0 |ṽ|

 0 p+1 dx, which concludes the proof.

2 L 2 -a u 2 L 2 - 2 , u = 0 ,

 222220 (4.1.5) if and only if S (u) = 0. Let I(u) = ∂ x u b u p+1 L p+1 . The set {u ∈ H 1 loc : u = 0, I(u) = 0} is called Nehari manifold. We are interested in the minimization problem on the Nehari manifold: min{S(u) : u ∈ H 1 loc ∩ P T , u = 0, I(u) = 0}, (4.6.1) and min{S(u) : u ∈ H 1 loc ∩ A T I(u) = 0}. (4.6.2)

4. 6 . 1

 61 Periodic case 4.6.1.1 Focusing case Let b > 0 and a < 0. We have the following lemma.

lim n→∞ inf ∂ x u n 2 L 2 -a u n 2 L 2

 2222 n ) = 0. Then I(u ∞ ) 0,and this implies thatS(u ∞ ) S(u ∞ ) -1 p + 1 I(u ∞ ) lim n→∞ inf S(u n ).

Therefore S(t 0

 0 u ∞ ) lim n→∞ inf S(u n ), I(t 0 u ∞ ) = 0,which implies the existence of the minimizer.

Figure 4 .

 4 Figure 4.11 -I(tu) in function of t.We haveI(u) = ∂ x u 2 L 2a v 2 L 2b v p+1 L p+1 . Therefore I (u) = -2u xx -2aub(p + 1)|u| p-1 u, = -2u xx -2au -2b|u| p-1 u + b(2 -(p + 1))|u| p-1 u, = 2S (u)b(p -1)|u| p-1 u. Moreover I(u) =< S (u), u >, therefore < I (u), u > = 2 < S (u), u > -b(p -1) u p+1 L p+1 , = -b(p -1) u p+1 L p+1 = 0.

Lemma 4 . 7 . 1 .

 471 Let θ = 2kπ

3 2 (φ 0 + h) 1 2 5 2

 315 e iωt , |u| 2 u = (φ 0 + h) 2 (φ 0 + h)e iωt , |u| 3 u = (φ 0 + h) (φ 0 + h)

  rapid calculations. Outcomes of the script are saved under w_critical_{date}.out Exploitation of datas obtained with the script is given at the end of␣

	52	Chapter 2. Standing waves of the double power 1d NLS
	Notebook 1	w1=w_tmp
	def U_div_s_prime(s,w,p1,p2,a1=-1,a2=1): precision=(w2-w1)/w2 [1]: %%file script_critical_omega.py s=np.abs(s) if w2<10**(-12):
	return -(a1*s**((p1-3)/2))*2/(p1+1)*(p1-1)/2-(a2*s**((p2-3)/2))*(2/ break Script for the surface w_critical(p1,p2). """ except: →(p2+1))*(p2-1)/2 pass Here with small values of p1 p2. def a(w,p1,p2,a1=-1,a2=1): return w1 True script is runned on a dedicated server where multiprocessing will␣ a=sco.fsolve(U_div_s,␣ a=np.abs(a) →10,args=(w,p1,p2,a1,a2),fprime=U_div_s_prime,xtol=10**(-12)) Appending to script_critical_omega.py
	if a==0: →the notebook. print("error in a , a is 0") """ return a
	import numpy as np def J(w,p1,p2,a1=-1,a2=1): import scipy as sp A=a(w,p1,p2,a1,a2) import scipy.optimize as sco def integrand(s): from scipy import integrate C=-A**(1.5)/(2*U_der(A,w,p1,p2,a1,a2)) import time from datetime import datetime I=(3+A*s*(U_der(A,w,p1,p2,a1,a2)-U_der(A*s,w,p1,p2,a1,a2))/
	from concurrent.futures import ProcessPoolExecutor
	from functools import partial return C*I import shelve I=integrate.quad(integrand, 0, 1,epsabs=10**(-100),limit=100)
	Overwriting script_critical_omega.py return I[0]
	[2]: %%file --append script_critical_omega.py # construction of a function returning the critical omega
	def w_critical(p1,p2,a1=-1,a2=1):
	dp1=0.0001 def f1(z,p1,p2,a1=-1,a2=1): dp2=0.0001 s=np.abs(z) if ((p2<=p1+dp1/2) or (p2-dp2/2<(-2*p1+7))): return -a1*s**((p1-1)/2)-a2*s**((p2-1)/2) return 0
	else: def F1(z,p1,p2,a1=-1,a2=1): try: s=np.abs(z) precision=1 return -(a1*s**((p1+1)/2))*2/(p1+1)-(a2*s**((p2+1)/2))*(2/(p2+1)) tol=0.00001
	w1,w2=0,1 def U(s,w,p1,p2,a1=-1,a2=1): while J(w2,p1,p2,a1,a2)<0: s=np.abs(s) w2=2*w2 return w*s+F1(s,p1,p2,a1,a2) if w2>10**(10):
	w1=10**(10) def U_der(s,w,p1,p2,a1=-1,a2=1): break s=np.abs(s) while precision>tol: return w+f1(s,p1,p2,a1,a2) w_tmp=w1+(w2-w1)/2
	if J(w_tmp,p1,p2,a1,a2)>0: def U_div_s(s,w,p1,p2,a1=-1,a2=1): w2=w_tmp s=np.abs(s) else:

→allow return (w-(a1*s**((p1-1)/2))*2/(p1+1)-(a2*s**((p2-1)/2))*(2/(p2+1))) →U(A*s,w,p1,p2,a1,a2))*np.sqrt(s)/np.sqrt(np. →abs(U(A*s,w,p1,p2,a1,a2)))

  Proof. Without loss of generality, we can restrict the minimization to real valued non-negative functions. Indeed, if u ∈ H 1 loc ∩ P T , then |u| ∈ H 1 loc ∩ P T and we have ∂ x |u| L 2 ∂ x u L 2 . This implies that (4.4.1) and (4.4.2) share the same minimizers.

	4.4.1.1 The focusing case in P T
	Assume that b > 0.
	Proposition 4.4.1. For all m > 0, the minimization problem (4.4.1) admits a
	minimizer which is also a minimizer of the minimization problem (4.4.2). The
	minimal energy is finite and negative.

  Etude des ondes périodiques et stationnaires pour l'équation de Schrödinger non linéaire Résumé: La thèse s'intéresse à l'étude des ondes périodiques et stationnaires pour l'équation de Schrödinger non linéaire unidimensionnelle. La première partie de la thèse se concentre sur la stabilité orbitale des ondes stationnaires pour l'équation de Schrödinger non linéaire à double puissance. Dans cette thèse, nous fournissons une image de stabilité complète pour le problème en utilisant le critère de pente de Grillakis, Shatah et Strauss. Les principaux nouveaux ingrédients de notre approche sont une reformulation de la pente et le calcul explicite de la valeur de la pente dans le cas de la fréquence zéro. Nous avons fourni des expériences numériques pour compléter nos résultats théoriques. Dans la deuxième partie de la thèse, nous étudions les solutions quasi-périodiques pour l'équation de Schrödinger cubique non-linéaire. Nous utilisons une approche variationnelle pour étudier les solutions quasi-périodiques, en minimisant l'énergie à masse et moment fixes en utilisant la méthode du flux de gradient avec normalisation discrète : à chaque pas de temps, nous évoluons dans la direction du gradient de l'énergie et renormalisons la masse et le moment du résultat. Nous établissons un lien entre les minimiseurs de l'énergie et les solutions de l'équation différentielle ordinaire. La dernière partie étend le travail sur le cas cubique à une non-linéarité plus générale et fournit des caractérisations variationnelles pour ce cas. La thèse dans son ensemble, contribue à la compréhension de la dynamique des équations de Schrödinger non linéaires, en mettant l'accent sur l'étude des ondes périodiques et stationnaires. Nous utilisons une combinaison de méthodes analytiques, numériques et variationnelles pour fournir une compréhension plus profonde du comportement des équations et de leurs solutions.

				3 2 e iωt .
	Hence we have
	h R h I t	= L	h R h I	+ N (h)

2.7. Notebooks

If E = E + (J) then u is a plane wave (i.e. u(x) = r q e iqx up to phase shift and complex conjugate) or |u| is homoclinic to r q as x → +∞.

Legend: Critical surface {(p, q, ω c (p, q))} for a p = -1 and a q = 1/2, 1, 2. The white lines represent q = 7 -2p and q = p, where the transition from ω c = 0 to ω c = 0 occurs.

Several observations can be made on the critical surface. As (p, q) approaches the line q = 5, we have ω c (p, q) → ∞, which is consistent with the fact that standing waves are all unstable on this line, or in other word ω 2 (p, q = 5) = ∞.

Different behaviors appear close to the line p = q depending on the value of a p and a q . When a q > a p , the transition is smooth, when a q = a p , the transition is sharp, whereas when a q < a p we have ω c (p, q) → ∞ as q → p.

To investigate more the transition close the lines q = 7 -2p and q = p, we plot slices of the critical surface for fixed values of q.

[10]: value_p2=4 index=np.argmax(p2>=value_p2) fig2 = plt.figure(figsize= [START_REF] Andrzej | The method of Nehari manifold[END_REF][START_REF] Andrzej | The method of Nehari manifold[END_REF]) ax2 = plt.axes() # Slice of the critical surface for $q={value_p2}$\n ax2.set_title(f"when $a_p={a1}$ and $a_q={a2}$") if np.isclose(a2,0. In the following cell, we introduce the discretization of the real line R as a finite number N of points equally distributed between -L and L and we construct the discretized Laplacian matrix with Dirichlet boundary conditions.

[ The effective potential

[17]: def V(J,r): return (J**2)/(2*r**2)+a*(r**2)/2+b*(r**4)/4

We start our numerical experiments by fixing a value of J in the domain D for each focusing/defocusing case. If we are in the defocusing case we find r Q and r q where the minimum and the maximum of V J (r) occurs. We know that E -= V J (r Q ) and E + = V J (r q ). If we are in the focusing case we find r Q where the minimum of V J (r) occurs and we have E -= V J (r Q ).

[23]: def v(J,r): #derivative of V(J,r) to find the minimum and the maximum return -(J**2)/(r**3)+a*r+b*r**3 [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]: coeff= [b,0,a,0,0,0,-J**2] s=np.roots(coeff) real_valued = s.real[abs(s.imag)<1e-8] # where I chose 1-e5 as a␣ →threshold roots=real_valued[real_valued>=0]

Notebook

By elementary calculations, we have

In what follows, we describe the potential V J . We start with the case J = 0. Then

We know that f (r) r is an increasing function for all r > 0 therefore there exists at most one value r 0 > 0 such that

We now discuss what happens depending on the values of a and b. We know that lim

Now we suppose that

Phase portraits

In this section we will represent the phase portraits for each focusing and defocusing cases and depending whether or not J = 0. In polar coordinates, the equation (4.1.5) becomes

We rewrite this second order differential equation in the form of a first order system by introducing new coordinates

Then the differential system is the following

We start by finding the equilibrium points y such that G(y) = 0. Then we find the isoclines I 0 and I ∞ , where

We start with the case J = 0. We have

and

These isoclines I 0 and I ∞ meet at the equilibrium points of the system and determine the regions where the trajectories are monotone:

Then we study the stability of the equilibrium points. The Jacobian matrix of F is of the form

Classification of equilibrium points is determined by the eigenvalues λ 1 and λ 2 of the Jacobian matrix J F . There are 4 different types of equilibrium points: appropriate mass and momentum constraints. Their work built upon earlier results in [25], which established the stability of the minimizers of the energy functional subject to fixed mass constraints. By combining the variational characterization of stability with the existence of minimizers for the massconstrained problem, the authors of [START_REF] Gustafson | Stability of periodic waves of 1D cubic nonlinear Schrödinger equations[END_REF] were able to prove the stability of the cnoidal, snoidal, and dnoidal elliptic functions.

In this section, we are looking to study the minimization problems where we minimize the energy to fixed mass and momentum constraints for the nonlinear Schrödinger equation (4.1.1). We shall provide some stability results on the minimizer.

Minimization among the periodic functions

Let m > 0. The basic variational problem is to minimize the energy at fixed mass:

Since the momentum is also conserved for (4.1.1), it is natural to consider the problem with a further momentum constraint:

The minimization problems (4.4.1) and (4.4.2) seek to find functions u which minimize the energy subject to the constraint that the mass is fixed and, in the case of (4.4.2), the momentum is also zero. Note that when we minimize the energy with fixed mass and fixed momentum p = 0 the problem is more complicated. It is known that there exists u ∞ minimizer, and there exist Lagrange-multipliers ω and σ such that

On the other hand S (u) = λI (u), this implies that 0 =< S (u), u >= λ < I (u), u > .

Since < I (u), u > = 0, this implies

Therefore the minimizer u ∞ verifies S (u ∞ ) = 0, so it is a solution of the ordinary differential equation (4.1.5). Proof. Consider a minimizing sequence

Anti-periodic case

for (4.6.2). We have

We will distinguish between two cases whether a < 0 or a > 0. In the first case, as in the periodic case, we can directly conclude by contradiction with the minimizing nature of (u n ) that it is bounded in

In the second case, we suppose that a > 0. Since u n ∈ A T 2 , u n must have 0 mean value. In that case u n verifies the Poincaré-Wirtinger inequality:

Replacing in (4.6.4), we obtain that

Then by the same arguments as in the first case we can prove that

. By the weak convergence we have

Minimizing problem on the Nehari manifold 145

If a < 0, by the equivalence of the norms we have

And if a > 0, by the strong convergence in L 2 loc ∩ AT 2 we also have the above inequality. Therefore

On the other hand we have

and this implies that

As in the periodic case with the Figure 4.11 we prove that there exists t 0 < 1 such that I(t 0 u ∞ ) = 0 and S(t 0 u ∞ ) lim n→∞ inf S(u n ) which implies the existence of the minimizer.

We now consider the following minimization problem:

(4.6.5)

We have the following lemma.

Lemma 4.6.3. The minimization problems (4.6.2) and (4.6.5) share the same minimizer. Moreover when p is an odd integer the minimizer is real and it is a solution of (4.1.5).

Proof.

Let

Chapter 4. General nonlinearity

We will prove that m 1 = m 2 . Let u be such that m 1 is reached. Hence I(u) = 0. We have

Let u be such that m 2 is reached. Then I(u) 0. We will prove that I(u) = 0.

By contradiction, we suppose that I(u) < 0. As we can see in Figure 4.11 there exists t 0 < 1 such that I(t 0 u) = 0. Therefore we have

which gives the contradiction. Thus I(u) = 0. That being the case, we have

Hence m 1 = m 2 . On the other hand from Lemma 4.5.1 of the Fourier rearrangement inequality, we conclude that if p is an odd integer, then there exists

Hence the minimizer is real. Moreover as in the periodic case the minimizer is a solution of (4.1.5) and this concludes the proof.

Spectral stability around the constant

This section is concerned with the spectral stability of the constant solution for the nonlinear Schrödinger equation. We begin by examining the linearization of the equation around the constant solution in the general case.

In particular, we focus on the triple power nonlinearity case and we establish conditions under which the constant solution is spectrally stable.

Given a standing wave u(t, x) = e iωt φ(x) solution to (4.1.1), we consider the linearization of (4.1.1) around this solution:

where L denotes the linear part and N the non linear part. We have

Spectral stability around the constant

where

As the perturbations are periodic, using the Fourrier variable we have

The characteristic polyniomal of L is given by

where

We are interested in whether the entire spectrum of L lies on the imaginary axis, in which case, we say the periodic wave u is spectrally stable. If there exists at least one eigenvalue λ such that Re(λ) > 0 then the solution is spectrally unstable. Therefore if det( L) is positive then the eigenvalues of L are purely imaginary and we have the spectral stability. As θ = 2kπ T , we have det( L) = θ 2 + ω -2φ 0 + 3γφ 2 0 -4φ 3 0 θ 2 + ωφ 0 + γφ 2 0φ 3 0 .

If φ 0 = 0 then det( L) = (θ 2 + ω) 2 > 0. Moreover we know that φ 0 is a solution of (4.1.5), therefore as we suppose that φ 0 is a real constant we have -ωφ 0 + φ 2 0γφ 3 0 + φ 4 0 = 0.

Hence if φ 0 = 0 we have ω = φ 0γφ 2 0 + φ 3 0 which gives det( L) = θ 2 θ 2 + ω -2φ 0 + 3γφ 2 0 -4φ 3 0 .

Replacing ω as a function of φ 0 we obtain det( L) = θ 2 2φ 0 γ + θ 2 -3φ 3 0φ 0 .

We can see that 2φ 0 γ + θ 2 -3φ 3 0φ 0 is linear and increasing in γ. If we fix θ and φ 0 , then there exists a γ * depending on θ such that det( L) > 0 if γ > γ * .