Corentin Audiard

Existence of multi-traveling waves in capillary fluids

We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler-Korteweg system in dimension one. Such solutions behaves asymptotically in time like several traveling waves far away from each other. A kink is a traveling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.

Introduction

The Euler-Korteweg model The Euler-Korteweg equations read   

∂ t ρ + ∂ x (ρv) = 0, ∂ t v + v∂ x v + g ′ (ρ)∂ x ρ = ∂ x K(ρ)∂ 2 x ρ + 1 2 K ′ (ρ)(∂ x ρ) 2 , (x, t) ∈ R × R + . (1.1) 1 INTRODUCTION 2
They are a modification of the usual Euler equations that model capillary forces in non viscous fluids. The function K(ρ) is supposed to be smooth R + * → R + * . In some relevant cases it is not bounded near 0, in particular for K = 1/ρ there exists a change of variable, the Madelung transform, that converts at least formally solutions of (1.1) into solutions of the nonlinear Schrödinger equation (for details on this interesting feature see the review article [START_REF] Carles | Madelung, Gross-Pitaevskii and Korteweg[END_REF]).

There is a formally conserved energy

H[ρ, v] = R 1 2 ρv 2 + K(ρ)(∂ x ρ) 2 + G(ρ)dx,
where G is a primitive of g, and under appropriate functional settings, denoting δH the variational derivative of H, V = ρ v (1.1) can be viewed as a hamiltonian system

∂ t V = J∂ x δH[V ], with J = 0 -1 -1 0 . ( 1.2)
(1.1) also has a formally conserved momentum P (ρ, v) = R ρv, whose conservation is related to the identity δP [V ] = -JV . Although formal these identities are used at least as notations in this article. Due to the intricate quasilinear nature of (1.1), only local well-posedness was obtained so far in dimension one and, even for data close to a constant state, global well-posedness is an open problem (on well-posedness and stability in larger dimension, see also [START_REF] Benzoni-Gavage | On the well-posedness for the Euler-Korteweg model in several space dimensions[END_REF], [START_REF] Audiard | Global Well-Posedness of the Euler-Korteweg System for Small Irrotational Data[END_REF], [START_REF] Giesselmann | Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics[END_REF]). It was proved in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] by ODE technics that (1.1) admits traveling waves as solutions, namely solutions of the form (ρ, v)(x -ct). There exists two classes of traveling waves: those such that lim

+∞ (ρ, v)(z) = lim -∞
(ρ, v)(z) are labelled as kinks, while solitons satisfy lim

+∞ (ρ, v)(z) = lim -∞ (ρ, v)(z).
No quantity is assumed to be zero at infinity.

Both types of traveling waves are physically relevant, especially kinks are supposed to model phase transition in capillary fluid (e.g. liquid to vapor). While kinks are known to be always stable, solitons are not and a (conditional) stability criterion in the spirit of [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] was derived in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF]. This article is devoted to a related, yet different issue : the existence of multi-traveling waves, i.e. solutions that decouple as t → +∞ to a sum of traveling waves.

Multi-traveling waves in the litterature The existence of multiple traveling waves is now a classical topic. While the first examples came from the field of integrable equations (for example, see the pioneering work of Zakharov-Shabat [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]), flexible and powerful methods have since been developed to tackle non-integrable equation. In particular considerable progress was achieved for the KdV and nonlinear Schrödinger equations over the last twenty years.

In the framework of the nonlinear Schrödinger equation we refer to the work of Martel, Merle and coauthors [START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF], in particular based the use of modulation parameters and a compactness argument, see also Le Coz and Tsai [START_REF] Le | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] for an approach based on dispersive estimates. To the best of our knowledge, the inclusion of a kink in the asymptotic profile is a rather rare feature in the field of multiple traveling waves, to the noticeable exception of the work of Le Coz-Tsai [START_REF] Le | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], see also [START_REF] Lin | Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations[END_REF] in higher dimension. All those results share the fact that they are more conveniently applied to equations that have a "good" well-posedness theory available (existence of global solutions in not too restrictive spaces). As such, their adaptation to quasilinear systems like the Euler-Korteweg system raises some difficulties. To explain it roughly, a key step of the compactness argument of Martel-Merle requires the existence of solutions for t ∈ R + , while the well-posedness theory for the Euler-Korteweg system only allows existence in finite time.

In the context of the water waves, this difficulty was overcome by Ming-Rousset-Tzvetkov [START_REF] Ming | Multi-solitons and related solutions for the water-waves system[END_REF] with the construction of global approximate solutions with some fast decay in time on the approximation error. This is also, to some extent, the approach that we follow here.

The traveling waves A short description of the construction of traveling waves is provided in the appendix, for more details we refer to [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF]. Their main features are the following:

• A traveling wave is a solution of (1.1) of the form V (x -ct), c is its speed.

• All the traveling waves that we consider are smooth and bounded. Their derivatives are exponentially decreasing at ±∞. Consequently all traveling waves have limits at ±∞.

• Kinks are travelling waves that have different endstates at ±∞, say (ρ ± , u ± ). The function ρ is monotonous.

• Solitons are traveling waves with same endstates at ±∞. ρ changes monotony only once, when it reaches its unique extremum. In the appendix we only deal with the case where this extremum is a minimum. By analogy with the Schrödinger equation (see e.g. [START_REF] Barashenkov | Soliton-like "bubbles" in a system of interacting bosons[END_REF]), we label such solutions bubbles (note that for fluids such solutions correspond to a negative bump in the density, therefore the word bubble is consistent).

• For a fixed endstate solitons can be smoothly parametrized by their speed.

Main result Let (V c j ) 1≤j≤n be travelling waves with ordered speeds c j < c j+1 . We assume that all V c j are stable, and we consider one of the following two cases:

• V c 1 is a kink, and V c j are solitons with ∀ j ≥ 2, lim

±∞ V c j = lim +∞ V c 1 .
• V c j are all solitons with the same endstate.

For V c a soliton, we denote

U c = V c - ρ + v +
, and define the rescaled momentum

P (V c ) = P (ρ c , v c ) = R (ρ c -ρ + )(v c -v + )dx. (1.3)
We assume that the solitons are stable in the following sense Stability condition:

d dc (ρ c -ρ + )(v c -v + )dx < 0.
We refer to the appendix A for the proof that our conditions can be met, where we also show that this stability criterion coincides with the one derived in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF].

We define the multi-soliton

S(x, t) = V c 1 (x -c 1 t) + n k=2 U c k (x -c k t - k j=2 A j ),
A j ≥ A a large constant to choose later. Our main result is that there exists a solution which converges to S as t → ∞ (see section 2 for the definition of H n ).

Theorem 1.1. For A 0 large enough, A ≥ A 0 , and n ≥ 3, there exists a global solution of (1.1)

such that V -S ∈ C(R + , H 2n ) and lim t→∞ V (t) -S(t) H 2n → 0.
Remark 1. It may be tempting to think that theorem 1.1 hints towards the stability of multisolitons. This is not correct as the solution constructed is quite peculiar: it is a pure soliton solution with no dispersive part. For NLS multi-solitons have been constructed in cases where each soliton is unstable [START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF][START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF]. Note however that in the case of the Gross-Pitaevskii equation, whose hydrodynamics formulation is a special case of (1.1) with K = 1/ρ, g = ρ -1, nonlinear stability of multi-solitons was obtained by Béthuel-Gravejat-Smets [START_REF] Béthuel | Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation[END_REF]. It is expectable that a similar result holds (at least in some regime) for (1.1), however, due to the lack of global well-posedness, going beyond conditional stability, that is stability until blow-up, requires significant new ideas. Remark 2. It is apparent from the proof that multiple traveling waves can be constructed in more complicated configurations, such as kink-soliton-kink, soliton-kink-kink etc. We chose not to aim at such results to keep a reasonably simple proof, and because configurations with multiple kinks and stable solitons might require very exotic presssure laws to exist.

Scheme of proof

The key is to construct an approximate solution V a to (1.1) that satisfies

∂ t V a -J∂ x δH[V a ] = f a ,
which is defined globally, converges as t → ∞ to the multi-soliton, and such that the error term f a decays rapidly in time. Once V a is constructed, we use the local well-posedness theory with some improved energy estimates to construct a sequence of exact solutions V k close to V a , defined on [0, k] with V k (k) = U a (k). A compactness argument then provides a global solution of the Euler-Korteweg system which converges at t → ∞ to the multi-soliton. The construction of V a is quite intricate, it requires fine estimates on the flow generated by the linearized operator J∂ x δ 2 H[S], building upon a spectral decomposition of δ 2 H[V c j ] -c j δ P . Once these estimates are proved, the approximate solution is constructed by a Newton iteration method applied to the problem

∂ t V -J∂ x δH[V ] = 0.
Plan of the article In section 2 we define some notations and functional settings. The energy estimate for (1.1) are proved in section 3. Section 4 is the core of the article. We first give a convenient spectral decomposition of the operator δ 2 E[V c j ] -c j δ 2 P . We deduce some estimates on the flow of J∂ x δ 2 H that are not useful for this paper, but contain most of the ideas for the much more technical estimates on the flow of

J∂ x δ 2 H[S].
With these estimates at hand we construct in section 5 an approximate solution by following Newton's iteration method. The compactness argument that provides the multi-soliton solution is detailed in section 6. Finally, as the existence of a "kink-stable solitons" configuration is not obvious, we prove it in the appendix. The appendix is also used to recall how kinks and solitons for (1.1) are constructed.

Acknowledgement The author was supported by the French National Research Agency project NABUCO, grant ANR-17-CE40-0025. The author thanks Stefan Le Coz for pointing out some useful references.

Notations, functional spaces

Reference state of a solution Any solution V of (1.1) that we consider is of the form

V = V ref + U, (2.1)
where U vanishes at infinity, V ref is a reference state which is a smooth function with finite limit at ±∞, and for any k + j ≥ 1, ∂ k x ∂ j t V ref decays exponentially at ±∞. The notation V = V ref + U will be used without explanation when the context is clear, in particular for a soliton of endstate (ρ + , v + ) we always take

V ref = (ρ + , v + ) t . If any sub/superscript is present we denote V a = V a ref + U a , V j = V ref,j + U j etc. We always denote V = ρ v , U = r u
, and similarly for V a , U a ...

Symbols and conventions of computation

The constant C in inequalities A ≤ CB changes from line to line. Depending on the context, they are allowed to depend on some quantities, but for conciseness this dependency is not explicited. For example when proving A ≤ C( u ∞ )B, we write freely |uv| ≤ C|v|.

The inequality A B means A ≤ CB for some constant C > 0, where the previous rule applies to C. The L 2 scalar product for real vector valued functions is denoted •, • .

Sobolev spaces

Even for functions of one variable, we use the notation u ′ = ∂ x u. H n is the usual L 2 based Sobolev space

H n = {u ∈ S ′ : ∀ 0 ≤ k ≤ n, ∂ k x u ∈ L 2 }, u 2 H n = n 0 ∂ k x u 2 L 2 .
We denote C n b the set of n times differentiable functions that are bounded as well as their derivatives. For a vector valued distribution U = r u , we also define

U 2 H n = r 2 H n+1 + u 2 H n , and
U X n = U H n+1 + ∂ t U H n .
We have the interpolation property

∀ 0 ≤ k ≤ n, u H k ≤ u 1-k/n L 2 u k/n H n , the continuous embedding H n ⊂ C n-1 b
. For n ≥ 1, H n is a Banach algebra. The following composition composition rules hold for a ∈ C n b + H n , u ∈ H n and F smooth on Im(a), Im(a + u):

F (a + u) -F (a) H n ≤ C( a C n b +H n + u H n ) u H n , (2.2)
in particular if

F (0) = 0, F (u) H n ≤ C( u H n ) u H n .
A similar second order rule holds

F (a + u) -F (a) -uF ′ (a) H n ≤ C( a C n b +H n + u H n ) u 2 H n . (2.3)
Both are consequences of a combination of the Faa Di Bruno formula, Sobolev's embedding, Hölder's inequality and Taylor's formula.

Energy estimates

An essential step is to bound the distance between an exact solution and a smoother approximate solution V a = (ρ a , v a ) satisfying

∂ t V a = J∂ x δH[V a ] + f a , for some remainder f a .
Due to the quasi-linear nature of the system the flow map is (probably) not Lipschitz even in high regularity Sobolev spaces, nevertheless Lipschitz bounds with harmless loss of derivatives on V a can be obtained. Energy estimates were obtained by Benzoni et al [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF] thanks to a change of variable (initially due to F. Coquel), and this section is actually more or less contained in [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF]. Let us shortly describe the argument : if (ρ, v) is a smooth solution of (1.1) without vacuum, n ≥ 2, set w = K/ρ∂ x ρ, and z = v + iw. Then z satisfies

∂ t z + v∂ x z + iw∂ x z + i∂ x a∂ x z + g ′ (ρ)∂ x ρ = 0, (3.1)
with a(ρ) = √ ρK. This equation has a nice structure : i∂ x a∂ x is antisymetric, v∂ x too up to zero order terms, g ′ ∂ x ρ is of order zero since w is a derivative of ρ. The only bad term iw∂ x z is dealt with thanks to a gauge method.

A few preliminary notations : for V = (ρ, v), solution of (1.1) and V a = (ρ a , v a ) an approximate solution, we denote z and z a the associated new variables. We assume that

V a = V ref + U a and V = V ref + U (same reference state) so that V -V a = U -U a . Generically for F a function of ρ we denote ∆F = F (ρ) -F (ρ a ), for F a function of v, ∆F = F (v) -F (v a ) etc. The gauge function of order n is ϕ n (ρ) := a n/2 √ ρ and the modified norm ∆V H 2n := ∆ρ L 2 + √ ρ∆z L 2 + ϕ n ∂ 2n x ∆z L 2 .
This notation is quite incorrect as the "norm" depends on V in a nonlinear way. Nevertheless, using the computation rules 2.2 and with constants depending continuously on

V H 2n + V a H 2n + ρ + 1 ρ L ∞ + ρ a + 1 ρ a L ∞ , we have ∆V H 2n ∼ ∆ρ L 2 + ∆z H 2n , and (∆ρ, ∆v) H 2n ∼ ∆z H 2n + ∆ρ L 2 , so ∆V H 2n ∼ ∆V H 2n . (3.2)
The main result is the following: Proposition 3.1. Let V a ref be a reference state smooth, bounded with its derivatives rapidly decaying at infinity. Let V a = (ρ a , v a ) = V a ref + U a be an approximate solution of (1.1)

∂ t V a = J∂ x δH[V a ] + f a ,
and V a solution of (1.1)

such that U = V -V a ref ∈ H 2n , n ≥ 1.
Then the estimate holds

1 2 d dt ∆V H 2n ≤ C U H 2n + U a H 2n+2 + 1/ρ + 1/ρ a L ∞ ∆V H 2n + f a H 2n , with C a continuous, positive nondecreasing function R + → R + * .
Proof. We recall the convention of section 2; the hidden constants in , ∼ are as the function

C of the statement. If f a = (f a 1 , f a 2 ), the equations on ∆ρ, ∆z are      ∂ t ∆ρ + ∂ x (∆ρv + ρ a ∆v) = f a 1 , ∂ t ∆z + v∂ x ∆z + ∆v∂ x z a + iw∂ x ∆z + i∆w∂ x z a +∂ x ∆g + i∂ x a∂ x ∆z + ∆a∂ x z a ) = i K ρ a ∂ x f a 1 + f a 2 := h a .
∆ρ L 2 is estimated by multiplying the first equation by ∆ρ and space integration 1 2

d dt ∆ρ 2 L 2 ≤ ∂ x ∆ρ L 2 ( ∆ρ L 2 v L ∞ + ρ a L ∞ ∆v L 2 ) + ∆ρ L 2 f a 1 L 2 ∆V H 2n + f a 2 H 2n ∆V H 2n . (3.3)
The main issue is thus to control ∆z. Let us first note that

h a 2 H 2n ∂ x f a 1 2 H 2n + f a 2 2 H 2n ≤ f a 2 H 2n . (3.4)
For 0 ≤ k ≤ n, we apply a k √ ρ∂ 2k

x := ϕ k ∂ 2k

x to the second equation. Denoting ∆z k = ϕ k ∂ 2k

x ∆z we find after some commutations

∂ t ∆z k + v∂ x ∆z k + i∂ x (a∂ x ∆z k ) + i(ϕ k w + 2k∂ x (a)ϕ k -2a∂ x ϕ k )∂ 2k+1 x ∆z +iϕ k ∂ 2k+1 x (∆a∂ x z a ) = R + ϕ k ∂ 2k x h a (3.5)
where R is a remainder term containing derivatives of ∆z of order at most 2k, and derivatives of z a of order at most 2k + 2

R = [z∂ x , ϕ k ∂ 2k x ]∆z -iϕ k ∂ 2k x (∆v∂ x z a + i∆w∂ x z a ) +i[∂ x (a∂ x •), ϕ k ∂ 2k x ]∆z + 2ik∂ x (a)ϕ k ∂ 2k+1 x ∆z -ϕ k ∂ 2k+1 x (∆g) + iϕ k ∂ 2k+1 x (∆a∂ x z a ) -ϕ ′ k ∂ x (ρv)∂ 2k x ∆z.
By construction,

ϕ k w + 2k∂ x (a)ϕ k -2a∂ x ϕ k = a k √ K + 2k ρK = K ρ a k √ ρ + 2ka k a ′ √ ρ -2ka k a ′ √ ρ - a k+1 √ ρ ∂ x ρ = 0.
Therefore, multiplying (3.5) by ∆z k and integrating,

d dt ∆z k 2 L 2 ( v L ∞ ∆z k L 2 + R L 2 + C z a H 2k+2 ∆z H 2k + ϕ k ∂ 2k x h a L 2 ) ∆z k L 2 . (3.6) Using section 2.2 and Faa di Bruno formula R L 2 ∆V H 2k , moreover from (3.4) ϕ k ∂ 2k x h a L 2 f a H 2k , (3.6) rewrites d dt ∆z k 2 L 2 ∆V H 2k ∆V H 2k + + f a H 2k .
Thanks to (3.2), ∆V H 2k ∆V H 2n . Adding estimates (3.3) and (3.6) with k = 0 and

k = n we conclude d dt ∆V 2 H 2n = d dt ∆ρ 2 L 2 + √ ρ∆z 2 L 2 + ϕ n ∂ 2n x ∆z 2 L 2 ) ∆V H 2n ∆V H 2n + f a H 2n .

Linear estimates

This section is devoted to estimates in H n on the flows associated to

J∂ x δ 2 H[V c ] (V c a traveling wave) and J∂ x δ 2 H[S].
We recall the notation δH

[V ] = -K∂ 2 x ρ -1 2 K ′ (∂ x ρ) 2 + g(ρ) + v 2 /2 ρv , in the same spirit δ 2 H[V ] r u = (-K ′ (ρ)∂ 2 x ρ -1 2 K ′′ (ρ)(∂ x ρ) 2 + g ′ (ρ) r -∂ x (K∂ x r) + uv ρu + rv , (4.1)
or in a matrix operator notation

δ 2 H[V ] = -K ′ (ρ)∂ 2 x ρ -1 2 K ′′ (ρ)(∂ x ρ) 2 + g ′ (ρ) -∂ x (K∂ x •) v v ρ .
As can be expected, δ 2 H is a symmetric operator. Recalling •, • is the L 2 scalar product, we shall use frequently that

δ 2 H[V ] r u , r 1 u 1 = R -K ′ (ρ)∂ 2 x ρ - 1 2 K ′′ (ρ)(∂ x ρ) 2 + g ′ (ρ) rr 1 +K∂ x r∂ x r 1 + vur 1 + vru 1 + ρuu 1 dx, (4.2)
so that δ 2 H induces a continuous bilinear form on H 0 if V is smooth enough.

Linear stability of a traveling wave

The case of a kink Let V c be a kink of speed c. The system (1.1)linearized near V c reads after the change of variables x → x -ct

∂ t U (x, t) = J∂ x δ 2 (H -cP )[V c (x)]U (x, t).
We define a modified energy functional E = H -cP . According to lemma 3 in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] (see also remark 2 in this reference) kinks are always stable in the following sense:

Lemma 4.1. For any U ∈ H there exists a unique orthogonal decomposition

U = α∂ x V c + W, ∂ x V c ∈ Ker(δ 2 E) and δ 2 E[V c ]U, U W 2 H 0 . (4.3)
For the link between linear stability and δ 2 E being definite positive, see e.g. theorem 3.1 of Pego-Weinstein [START_REF] Pego | Eigenvalues, and instabilities of solitary waves[END_REF].

The case of a soliton We consider a branch of solitons V c . As it is more convenient here to work on U c , we denote

P [U c ] = r c u c and abusively δH[U c ] = δH[V c ]. We recall (see (1.2)) that J = 0 -1 -1 0 so that δ 2 P = -J. From -c∂ x U c = J∂ x δH[U c ] we have a number of useful identities ∀ U, V, δP [U ] = δ 2 P [V ]U = -JU, (4.4)
(δH -cδP )[U c ] := δE[U c ] is constant, (4.5) δ 2 E[U c ] • ∂ x U c = 0 (differentiation of (4.5) in x), (4.6)
δ 2 E[U c ]∂ c U c -δP [U c ] = 0 (differentiation in c) (4.7) ⇔ δ 2 E[U c ]∂ c U c = -JU c . (4.8)
Stability assumption We assume that U c is stable, namely it satisfies :

dP [U s ] ds | s=c < 0.
(see the appendix for a link with the so called Boussinesq momentum of instability). This also implies that ∂ c U c is an unstable direction in the sense that

δ 2 E[U c ]∂ c U c , ∂ c U c = δP [U c ], ∂ c U c = d dc P [U c ] < 0.
Let us first recall a result from [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] (proved for the formulation of the Euler-Korteweg system in Lagrangian coordinates, see also [START_REF] Audiard | Small energy traveling waves for the Euler-Korteweg system[END_REF] appendix B for a proof in Eulerian coordinates).

Lemma 4.2. Under the stability assumption, the operator

δ 2 E[U c ] is block diagonal on the orthogonal decomposition H = vect(U -) ⊕ ⊥ vect(∂ x U c ) ⊕ ⊥ G
, where ∂ x U c spans the kernel of δ 2 E, U -is a normalized eigenvector associated to the unique negative eigenvalue, and

∀ W ∈ G, δ 2 E[V c ]W, W W 2 H 0 . Lemma 4.3.
For U ∈ H, there exists a unique orthogonal decomposition

U = αδP [U c ] + β∂ x U c + W, W ∈ (δP [U c ], ∂ x U c ) ⊥ and δ 2 EU, U W 2 H 0 -Cα 2 .
(4.9) Remark 3. To underline the unity between this decomposition and (4.3) in the case of a kink, let us point out that since the reference state is constant

∂ x V c = ∂ x U c . Proof. The momentum being invariant by translation, δP [U c ], ∂ x U c = 0 and according to (4.6), ∂ x U c ∈ Ker(δ 2 E). Therefore the only thing to prove is δ 2 EW, W W 2 H 0 . By contradiction we assume the existence of W ∈ (δP [U c ], ∂ x U c ) ⊥ \{0} such that δ 2 EW, W ≤ 0,
then for any (α, β, γ) ∈ R 3 , using identities (4.6), (4.7)

δ 2 E(α∂ c U c + β∂ x U c + γW ), α∂ c U c + β∂ x U c + γW = δ 2 E(α∂ c U c + γW ), α∂ c U c + γW = α 2 δP [U c ], ∂ c U c + γ 2 δ 2 EW, W +2αγ δP [U c ], W = α 2 δP [U c ], ∂ c U c + γ 2 δ 2 EW, W , by orthogonality. By definition, ∂ x U c , W = 0 and ∂ c U c , δP [U c ] < 0 therefore (∂ c U c , ∂ x U c , W ) is free. But δ 2 E is thus nonpositive on a dimension 3 space, which contradicts lemma 4.2. As a consequence ∀ W ∈ (δP [U c ], ∂ x U c ) ⊥ \ {0}, δ 2 EW, W > 0. (4.10)
The improved inequality δ 2 EW, W W 2 H 0 follows from a (probably standard) compactness argument : consider a sequence V n of (δP

[U c ], ∂ x U c ) ⊥ such that V n H 0 = 1 and δ 2 EV n , V n → 0. Using lemma 4.2 we write V n = α n U -+ β n ∂ x U c + W n , W n ∈ G. By assumption, β n = 0 and up to an extraction, α n → n→∞ α, W n ⇀ W ∈ G. Denoting -λ -the negative eigenvalue, 1 = V n 2 H = α 2 n + W n 2 H 0 = α 2 + lim n W n 2 H , 0 = lim n→∞ δ 2 EV n , V n = lim n→∞ -λ -α 2 n + δ 2 EW n , W n ≥ -λ -α 2 + c lim inf n W n 2 H 0 .
This implies α = 0. Let V = αU -+ W , then by weak convergence

δ 2 EV, V = -λ -α 2 + δ 2 EW, W ≤ -λ -α 2 + lim n δ 2 EW n , W n = 0, but since V is the weak limit of V n , it belongs to (δP [U c ], ∂ x U c ) ⊥
, and (4.10) implies V = 0, which contradicts α = 0.

As a consequence, we deduce the following linear stability result, whose proof will be a guideline for the computations in the multi-soliton case.

Theorem 4.4. Under the stability assumption (4.1), the solution of

∂ t U (x, t) = J∂ x δ 2 H[U c (x -ct)]U (x, t), U | t=0 = U 0 , satisfies for t ∈ R U (t) H 0 (1 + |t|) U 0 H 0 .
Proof. For conciseness we write

δ 2 H for δ 2 H[U c ]. Using δ 2 P = -J d dt (δ 2 H -cδ 2 P )U, U = [∂ t , δ 2 H]U, U + 2 δ 2 HJ∂ x δ 2 HU, U + c [∂ x , δ 2 H]U, U = [∂ t + c∂ x , δ 2 H]U, U .
Since the coefficients of the operator δ 2 H only depend on x -ct,

[∂ t + c∂ x , δ 2 H] = 0, so d dt (δ 2 H -cδ 2 P )U, U = 0. (4.11)
We use the decomposition (4.9) for the solution

U (t) = α(t)δP [U c (x -ct)] + β(t)∂ x U c + W (t). Since ∂ x U c ∈ Ker(δ 2 E) α ′ (t) = J∂ x δ 2 HU, δ 2 P U c + U, -c∂ x δ 2 P U c δP [U c ], δP [u c ] = U, (δ 2 H -cδ 2 P )∂ x U c δP [U c ], δP [u c ] = 0.
By the conservation (4.11) and the continuity of δ 2 H as a bilinear form (4.2)

δ 2 EU (0), U (0) = δ 2 EU (t), U (t) = δ 2 E(αδP [U c ] + W ), αδP [U c ] + W W 2 H 0 -Cα 2 ⇒ W (t) 2 H 0 α(0) 2 + U (0) 2 H 0 . Moreover |α(0)| = U (0), δP [U c ] / δP [U c ] 2 H U (0) H 0 .
The last term is estimated thanks to the bounds on α, V :

|β ′ (t)| = d dt U (t), ∂ x U c ∂ x U c 2 L 2 = J∂ x δ 2 E(αδP [U c ] + V ), ∂ x U c ∂ x U c 2 L 2 U (0) L 2 ,
and by integration,

|β(t)| ≤ |β(0)| + |t| U (0) H 0 .
Remark 4. The linear growth in time is unavoidable, indeed we have

J∂ x δ 2 E∂ c U c = ∂ x JδP [U c ] = ∂ x U c ∈ Ker(J∂ x δ 2 E), therefore (∂ c U c , ∂ x U c
) is associated to a Jordan block of the eigenvalue 0 of J∂ x δ 2 E.

Remark 5. Of course to study the stability of a single soliton it is much more natural to do the galilean change of variable y = x -ct and consider the autonomous linear problem

∂ t U = (δ 2 H -cδ 2 P )[U c (y)]U .
The proof of theorem 4.4 is simplified in this frame. Nevertheless, when considering multi-soliton such a change variable is not available and this first simple case is a good warm up before the more technical computations of section 4.2.

Stability near multiple traveling waves

We recall that the multi-soliton is defined as

S(x, t) = V c 1 (x -c 1 t) + n k=2 U c k (x -c k t - k j=2 A j ) = V 1 + n 2 U k , A j ≥ A,
where

V c k = V c k ref + U c k are traveling waves, c 1 < c 2 • • • < c n .
We have exponential decay

∀ p ≥ 0, 1 ≤ k ≤ n, ∃ α > 0 : |∂ p x U k (x, t)| e -α|x-c k t-k 2 A j | . (4.12)
The aim of this section is to get bounds on the flow associated to J∂ x δ 2 H[S + η] where η is a small perturbation of limited smoothness that depends on x and t. When there is no ambiguity, we write δ 2 H for δ 2 H[S + η] and

δ 2 E k := δ 2 H[V k ] -c k δ 2 P.
Lemma 4.5. For s ≥ 0, let U solve

∂ t U = J∂ x δ 2 H[S + η]U, U | t=s = U 0 , (4.13)
with η a smooth perturbation. There exist C and ε 0 such that for ε :

= 1/A + η X 1 ≤ ε 0 , ∀ t ≥ 0, U (t) H 0 ≤ C(1 + |t -s|)e Cε 1/4 |t-s| U 0 H 0 .
Remark 6. The estimate is not true for t ≤ 0 as the key argument is that the distance between the traveling waves must be (in some sense) larger than A.

The proof requires some preliminaries that will be used through the section. Let 2c 0 = inf j<k c k -c j , and for 1 ≤ k < n, c k+1/2 = (c k + c k+1 )/2. We first define localizing functions : pick a nondecreasing χ ∈ C ∞ (R), supp(χ) = [0, ∞], χ| [1/2,∞) = 1, 0 < χ < 1 on (0, 1/2), and set

ϕ 1 (x, t) = 1 -χ x -c 1+1/2 t -A 2 /2 A 2 , ∀ 2 ≤ k < n, ϕ k (x, t) = χ x -c k-1/2 t - k 2 A j -A k /2 A k -χ x -c k+1/2 t - k 2 A j + A k+1 /2 A k+1 , ϕ n (x, t) = χ x -c n-1/2 t - n 2 A j -A n /2 A n . It is easily seen that supp(ϕ 1 ) = (-∞, c 3/2 t + A 2 ], ∀ 2 ≤ k ≤ n -1, supp(ϕ k ) = [c k-1/2 t + k 2 A j - A k 2 , c k+1/2 t + k+1 2 A j ], supp(ϕ n ) = [c n-1/2 t + n 2 A j -A n /2, ∞),
and n k=1 ϕ 2 k ≥ c > 0 for some constant independent of x. The localizing functions are then defined as

χ j = ϕ j n 1 ϕ 2 j so that χ 2 j = 1. (4.14)
Note that ϕ j and χ j have same support. Thanks to (4.12) we have the following estimates, uniformly for A large

∂ j x ∂ k t χ j L ∞ x,t = O(1/A k+j ) (slow variation ), (4.15)
∀ j = k, (p, q) ∈ N 2 , r ≥ 1, ∃ α > 0 : ∂ p x ∂ q t U k L r x (supp(χ j )) = O(e -αc 0 t /A), (4.16) if (p, q) = (0, 0), r ≥ 1, ∂ p x ∂ q t V k L r x (supp(χ j )) = O(e -αc 0 t /A), (4.17)
(support decorrelation).

Proof of lemma 4.5. In the spirit of the proof of theorem 4.4 we define the modified energy

E(t) = δ 2 H[S + η]U (t), U (t) - n k=1 c k δ 2 P χ k U (t), χ k U (t) . (4.18)
Similarly to theorem 4.4, the proof has three steps : 1) Control of d E/dt, 2) control of U (t) 2 H 0 by E up to a finite number of parameters, 3) Control of these parameters.

Step 1: Control of d E/dt. From basic computations, using δ 2 P = -J, J 2 = I,

d dt E = ([∂ t , δ 2 H] + δ 2 HJ∂ x δ 2 H)U, U + n k=1 2c k χ ′ k JU, χ k U + n k=1 c k χ k ∂ x δ 2 HU, χ k U + χ k U, χ k ∂ x δ 2 HU = [∂ t , δ 2 H]U, U + n k=1 2c k χ ′ k JU, χ k U + n k=1 c k [χ 2 k , ∂ x δ 2 H]U, U + [∂ x , δ 2 H]χ 2 k U, U = n k=1 ([∂ t , δ 2 H] + c k [∂ x , δ 2 H])χ 2 k U, U + c k 2 χ ′ k JU, χ k U + c k [χ 2 k , ∂ x δ 2 H]U, U = n k=1 C 1,k (t) + C 2,k (t) + C 3,k (t).
We first point out that X 1 controls the L ∞ norm, therefore for η X 1 small enough the density of S +η remains bounded away from 0 and the computations rules in (2.2),(2.3) can be applied. C 2,k and C 3,k are not difficult to control : let us write [χ 2 k , ∂ x δ 2 H] = (L i,j ) 1≤i,j≤2 as a matrix of operators, S + η = ρ 1 v 1 and detail the estimate for L 1,1 r, r

L 1,1 r, r = χ 2 k , ∂ x (g ′ -K ′′ (∂ x ρ 1 ) 2 -K ′ ∂ 2 x ρ 1 ) -∂ x K∂ x r, r = -2χ k ∂ x (χ k )(g ′ -K ′′ (∂ x ρ 1 ) 2 -K ′ ∂ 2 x ρ 1 )r, r (4.19) -2 χ k ∂ x χ k ∂ x (K∂ x r), r -[χ 2 k , ∂ x K∂ x ]r, ∂ x r . (4.20)
Using the Sobolev estimates (2.2) and (4.15), we find (4.19) r 2 H 1 /A, the second one is estimated by an integration by part

-2 χ k ∂ x χ k ∂ x (K∂ x r), r ≤ R K∂ x r∂ x 2χ k ∂ x χ k r dx 1 A r(t) 2 H 1 .
The last term in (4.20) is estimated similarly with the explicit commutator formula

[χ 2 k , ∂ x K∂ x ]r = -2∂ x (χ 2 k )K∂ x r -∂ x (χ 2 k )∂ x (Kr)
. Similar computations eventually lead to

|C 2,k + C 3,k | 1 A U 2 H 0 . (4.21)
To bound C 1,k we introduce the (bilinear) operator δ 3 H such that

[∂ t , δ 2 H[S + η]]U = δ 3 H[S + η](U, ∂ t (S + η)). (4.22)
This is merely a convenient notation, as writing

S + η = ρ 1 v 1 , δ 3 H(•, ∂ t (S + η)) is explicitly δ 3 H[S + η](•, ∂ t (S + η)) = M t ∂ t v 1 ∂ t v 1 ∂ t ρ 1 , (4.23)
with

M t r = g ′′ ∂ t ρ 1 - K ′′′ ∂ t ρ 1 (∂ x ρ 1 ) 2 + 2K ′′ ∂ x ρ 1 ∂ xt ρ 1 2 r -(K ′′ ∂ t ρ 1 ∂ 2 x ρ 1 + K ′ ∂ xxt ρ 1 )r -∂ x (K ′ ∂ t ρ 1 ∂ x r),
and we use the same notation for [∂ x , δ 2 H] := δ 3 H[S](•, ∂ x S). We can thus rewrite using Step 2: Lower bounds for E. The key here is the decompositions (4.3) and (4.9). For 1

S = V 1 + n j=2 U j , with ∂ t V 1 = -c 1 ∂ x V 1 , ∂ t U j = -c j ∂ x U j , for k > 1 C 1,k (t) = δ 3 H(U, ∂ t S + c k ∂ x S)χ 2 k U, U + δ 3 H(U, ∂ t η + c k ∂ x η)χ 2 k U, U = δ 3 H(U, -c 1 ∂ x V 1 - j =k c j ∂ x U j )χ 2 k U, U + δ 3 H(U, ∂ t η + c k ∂ x η)χ 2 k U, U ,
and if k = 1 C 1,1 (t) = δ 3 H(U, - n j=2 c j ∂ x U j )χ 2 1 U, U + δ 3 H(U, ∂ t η + c 1 ∂ x η)χ 2 1 U, U .
≤ k ≤ n we set χ k U (t) = α k (t)δP [U k ] + β k (t)∂ x V k + W k (t)
, with the convention that if V 1 is a kink, α 1 = 0 (in this case the relevant decomposition is (4.3)). Using the translation invariance of the L 2 norm, the lower bound in (4.9) gives for some m, C > 0

∀ 1 ≤ k ≤ n, δ 2 E[U k ]χ k U, χ k U ≥ m W k 2 H 0 -Cα 2 k .
According to this, we split E as a sum of localized terms and remainders:

E = δ 2 H[S + η]U, U - n k=1 c k δ 2 P χ k U, χ k U = n k=1 δ 2 H[S + η]χ 2 k U, U -c k δ 2 P χ k U, χ k U ,
so commuting δ 2 H and χ k we obtain

E(t) = n k=1 δ 2 E k χ k U, χ k U + (δ 2 H[S + η] -δ 2 H[U k ])χ k U, χ k U + [δ 2 H[S + η], χ k ]χ k U, U ≥ n k=1 m W k (t) 2 H 0 -Cα 2 k (t) + (δ 2 H[S + η] -δ 2 H[V k ])χ k U, χ k U + [δ 2 H, χ k ]χ k U, U . (4.26)
The last term is estimated as in (4.20),

[δ 2 H, χ k ]χ k U (t), U (t) ε U (t)
(δ 2 H[S + η] -δ 2 H[V k ])χ k U, χ k U e -αc 0 t A + η H 2 U (t) 2 H 0 . (4.28)
For example the term associated to ∂ x (K∂ x r) is controlled as follows

∂ x (K(S + η) -K(V k ))∂ x (χ k r) , χ k r ≤ K(S + η) -K(V k ) L ∞ (supp(χ k )) ∂ x (χ k r) 2 L 2 e -αc 0 t A + η(t) H 0 r(t) 2 H 1 .
Note that U 2

H 0 n 1 W k 2 H 0 + α 2 k + β 2 k
, so for ε small enough, from (4.26),(4.27),(4.28), there exists constants m, C 0 , C 1 (m is not the same as in (4.26)) such that

E(t) ≥ n k=1 m W k (t) 2 H 0 -C 0 α 2 k (t) -C 1 εβ 2 k (t). ( 4

.29)

Step 3: Control of the parameters. Once more it is a matter of repeating the proof of theorem 4.4 with some commutators. Let us start with (α

k (t)) 1≤k≤n (k > 2 when V 1 is a kink): α ′ k (t) = d dt χ k U, δ 2 P U k δ 2 P U k 2 L 2 = (∂ t χ k )U, δ 2 P U k + χ k J∂ x δ 2 H[S + η]U, δ 2 P U k + χ k U, -c k δ 2 P ∂ x U k δ 2 P U k 2 L 2 = (∂ t χ k )U, δ 2 P U k + [χ k , J∂ x δ 2 H[S + η]]U, δ 2 P U k δ 2 P U k 2 L 2 + J∂ x (δ 2 H[S + η] -δ 2 H[V k ])χ k U, δ 2 P U k + χ k U, δ 2 E k ∂ x U k δ 2 P U k 2 L 2 .
There are four terms. The fourth one is actually 0, thanks to identity (4.6). From the same argument as for C 2,k , C 3,k in (4.20), the first and second ones are O(|U (t) H 0 /A). Using integration by part, the smoothness of U k and the same argument as for (4.28), the third one is O( U (t) H 0 (e -αc 0 t /A + η H 2 )). To summarize

∀ 1 ≤ k ≤ n, t ≥ 0, |α ′ k (t)| ε U (t) H 0 . (4.30)
We bound now β k (t):

∂ x V k 2 L 2 β ′ k (t) = d dt χ k U, ∂ x V k = (∂ t χ k )U, ∂ x V k + χ k J∂ x δ 2 HU, ∂ x V k + χ k U, -c k ∂ 2 x V k = (∂ t χ k )U, ∂ x U k + J∂ x δ 2 E k (χ k U ), ∂ x V k + χ k J∂ x (δ 2 H[S + η] -δ 2 H[V k ])U, ∂ x V k + [χ k , J∂ x δ 2 H[V k ]]U, ∂ x V k . Since χ k U = α k δP [U k ] + β k ∂ x V k + W k , with δ 2 E k ∂ x V k = 0, we have J∂ x δ 2 E k χ k U (t) , ∂ x V k (t) |α k (t)| + W k (t) H 0 .
The other terms are estimated as for α ′ k , leading to

|β ′ k (t)| |α k (t)| + W k (t) H 0 + ε U (t) H 0 . (4.31)
Conclusion. Let us rewrite (4.25),(4.30),(4.31) : there exists some C > 0 such that for ε small enough

E ′ (t) ≤ Cε U (t) 2 H 0 ds, |α ′ k (t)| ≤ Cε U (t) H 0 , |β ′ k (t)| ≤ C |α k (t)| + V k (t) H 0 + ε U (t) H 0 .
With the same constants as in (4.29), let E(t) := E(t)

+ n 1 (C 0 + m)α 2 k + ε 1/2 β 2 k , then for ε small enough E(t) n 1 α 2 k + W k 2 H 0 + ε 1/2 β 2 k
, and

E ′ (t) ≤ n k=1 Cε W k 2 H 0 + α 2 k + β 2 k + Cε 1/2 |β k | |α k | + W k H 0 + ε U H 0 ≤ Cε 1/2 E + Cε 1/2 α 2 k + W k 2 H 0 ε 1/4 + ε 1/4 β 2 k ≤ Cε 1/4 E(t).
With Gronwall's lemma and thanks to (4.29) we get

n k=1 m W k (t) 2 H 0 + α 2 k (t) + ε 1/2 β 2 k (t) ≤ E(t) ≤ E(s)e Cε 1/4 |t-s| .
We can assume ε ≤ 1/(4C 2 1 ) so that ε 1/2 -C 1 ε ≥ ε 1/2 /2, and

n k=1 W k (t) 2 H 0 + α k (t) 2 U (s) 2 H 0 e Cε 1/4 |t-s| = U 0 2 H 0 e Cε 1/4 |t-s| .
To bound β k independently of ε, we plug the estimate above in the differential inequality (4.31)

n k=1 |β ′ k (t)| ≤ M n k=1 |α k (t)| + W k (t) H 0 + ε U (t) H 0 ≤ M 1 U 0 H 0 e Cε
|β k (t)| ≤ n k=1 |β k (s)|e Cε 1/4 |t-s|/2 + 2M 1 U 0 H 0 Cε 1/4 e Cε 1/4 |t-s| -1 U 0 H 0 1 + |t -s| e Cε 1/4 |t-s| .
Combining this with the estimate on W k , α k , we conclude

U (t) H 0 U 0 H 0 (1 + |t -s|)e Cε 1/4 |t-s| .
It is useful to restate the result of lemma 4.5 in a slightly more abstract way:

Corollary 4.6. Let R η (t, s) the resolvent operator associated to

∂ t U = J∂ x δ 2 H[S + η]U .
There exists ε 0 and C such that if 1/A + η X 1 := ε ≤ ε 0 , t, s ≥ 0:

R η (t, s) L(H 0 ) ≤ (1 + |t -s|)Ce Cε 1/4 |t-s| .
We deduce now similar estimates at any level of regularity.

Theorem 4.7 (Higher order estimates). Let ε := 1/A + η X 2n , n ∈ N * . There exists ε n , C e,n , C n such that for any ε ≤ ε n , we have the resolvent estimate

R η (t, s) L(H 2n ) ≤ (1 + |t -s|)C n e Ce,nε 1/4 |t-s|
Proof. The hamiltonian structure is useless here, so we denote for conciseness

L := J∂ x δ 2 H[S + η].
As for the energy estimate, an important issue is that J∂ x δ 2 H[S + η] does not commute with ∂ t , this will be overcome with the same method as for the zero order estimate. The commutator

[∂ t , L] = J∂ x δ 3 H( • , ∂ t (S + η)) := δL( • , ∂ t (S + η))
is not zero (for the definition of δ 3 H, see (4.22) in the proof of lemma 4.5), thus to get higher order estimates it is more natural to use the operator

L := L + n k=1 c k χ 2 k ∂ x . Indeed using [L, L] = n k=1 [L, c k χ 2 k ∂ x ] = n k=1 c k [L, χ 2 k ]∂ x -c k χ 2 k δL(•, ∂ x (S + η)),
we find for any V

[∂ t -L, L]V = n k=1 χ 2 k δL(V, (∂ t + c k ∂ x )(S + η)) + 2c k χ k (∂ t χ k )∂ x V -[L, c k χ 2 k ]∂ x V, (4.33) therefore recalling ∂ x V j = ∂ x U j for j ≥ 2 ∂ t L n U = L L n U + n-1 q=0 L q [∂ t -L, L]L n-q-1 U = L( L n U ) + n-1 q=0 L q j =k χ 2 k δL(•, (c k -c j )∂ x V j ) + χ 2 k δL(•, (∂ t + c k ∂ x )η) L n-q-1 U + n-1 q=0 L q n k=1 2c k χ k (∂ t χ k )∂ x -[L, c k χ 2 k ]∂ x L n-q-1 U = L( L n U ) + C 1 + C 2 . (4.34) Let us recall that if S + η = ρ 1 v 1 , for any U = (r, u) t , we have LV = -∂ x   uρ 1 + rv 1 g ′ -K ′′ (∂xρ 1 ) 2 2 -K ′ ∂ 2 x ρ 1 r -∂ x (K∂ x r) + uv 1   ,
the first coefficient contains derivatives of r, u, ρ 1 , v 1 of order at most 1, the second contains derivatives of v 1 , v of order at most 1, and derivatives of ρ 1 , l of order at most 3 so using the rules of section 2.2,

∀ N ≥ 0, LV H N ≤ C N ( η H N+2 ) V H N+2 , (4.35)
⇒ LV H N ≤ M N ( η H N+2 ) V H N+2 , (4.36)
with C N , M N positive locally bounded functions (we recall that S is smooth and is unimportant in the estimates). With this observation, estimate (4.15) and computations similar to (4.20) we deduce

C 2 (t) H 0 U (t) H 2n A ≤ ε U (t) H 2n .
Similarly, C 1 (t) is estimated thanks to (4.16) as for (4.24),

C 1 (t) H 0 e -αc 0 t A + η X 2n U (t) H 2n ≤ ε U (t) H 2n .
Conversely thanks to the interpolation estimate

∂ p x ϕ L 2 ≤ ϕ 1-p/q L 2 ∂ q x ϕ p/q L 2 , and Young's inequality, ∂ p x ϕ L 2 ≤ C(ε) ϕ L 2 + ε ∂ q x ϕ L 2
for any ε > 0, p < q, therefore using once more the explicit formula for LV , we obtain

LV H 0 V H 2 -C V H 0 , and by induction L n U H 0 U H 2n -C n U H 0 . (4.37)
Equation (4.34) is of the form ∂ t V = LV + f . We apply Duhamel's formula and corollary 4.6

L n U (t) H 0 (1 + |t -s|) U 0 H 2n e Cε 1/4 |t-s| + ε t s (1 + |t -τ |)e Cε 1/4 |t-τ | U (τ ) H 2n ds (4.38) (1 + |t -s|) U 0 H 2n e Cε 1/4 |t-s| + ε 3/4 t s e 2Cε 1/4 |t-τ | U (τ ) H 2n dτ, (4.39)
where the last estimate simply follows from (1 + s)e s e 2s . We use (4.39), the bound U (t) H 0 ≤ C(1 + |t -s|)e Cε 1/4 |t-s| U 0 H 0 of lemma 4.5 and the lower bound (4.37) with Gronwall's lemma

U (t) H 2n (1 + |t -s|)e Cε 1/4 |t-s| U 0 H 2n + ε 3/4 t s e 2Cε 1/4 (t-τ ) U (s) H 2n ds, ⇒ U (t) H 2n (1 + |t -s|) U 0 H 2n e C(ε 1/4 +ε 3/4 )|t-s| . For ε small ε 3/4 + ε 1/4 = O(ε 1/4
) and the proof is complete.

Construction of an approximate solution

We construct here an approximate solution V a close to the multi-soliton and such that the error ∂ t V a -J∂ x δH[V a ] is rapidly decaying in t. It is done with Newton's algorithm, initialized with S as the first approximate solution.

Theorem 5.1. For any n ∈ N, ε > 0, C e > 0, there exists A 0 such that for A ≥ A 0 , there exists

U a ∈ L ∞ t H n , α > 0 such that ∀ t ≥ 0, ∂ t (S + U a ) -J∂ x δH[S + U a ] H n ≤ εe -Cet and U a (t) H n e -αc 0 t A .
We recall Newton's algorithm: S 0 = S, f 0 = ∂ t S 0 -J∂ x δH[S 0 ], η 0 = 0, and recursively η j+1 is the solution of

∂ t η j+1 = J∂ x δ 2 H[S j ]η j+1 -f j , lim t→∞ η j+1 = 0, S j+1 = S j + η j+1 , f j+1 = ∂ t S j+1 -J∂ x δH[S j+1 ]. Of course since ∂ t S j+1 -J∂ x δH[S j+1 ] = J∂ x δH[S j ] + J∂ x δ 2 H[S j ]η j+1 -J∂ x δH[S j+1
] we need some Taylor expansion estimate :

Lemma 5.2. For V := S + U , (U, η) ∈ (H n+2 ) 2 such that V L ∞ + η L ∞ ≤ inf ρ S /2 (non vacuum condition), then J∂ x δH[V + η] -J∂ x δH[V ] -J∂ x δ 2 H[V ]η H n ≤ C( η H n+2 + U H n+2 ) η 2 H n+2 , with C continuous. Proof. We set V = ρ v , U = r u , η = γ ω .
Elementary computations lead to

J∂ x δH[V + η] -J∂ x δH[V ] -J∂ x δ 2 H[V ]η = -∂ x       γω ω 2 2 + g(ρ + γ) -g(ρ) -γg ′ (ρ) -K(ρ + γ) -K(ρ) ∂ 2 x γ + K(ρ) + γK ′ (ρ) -K(ρ + γ) ∂ 2 x ρ -K ′ (ρ + γ) -K ′ (ρ) ∂ x ρ∂ x γ -1 2 K ′ (ρ + γ)(∂ x γ) 2       .
We have using the rules in 2.2

∂ x (γω) H n+1 γ H n+2 ω H n+2 ≤ η 2 H n+2 .
For the second coordinate, we only treat a few terms. Thanks to the smallness assumption in L ∞ , inf ρ, inf ρ + γ ≥ inf ρ S /2 so that we can use the composition rules again

∂ x (K(ρ + γ) -K(ρ))∂ 2 x γ H n K(ρ + γ) -K(ρ) H n+1 γ H n+3 γ 2 H n+3 , ∂ x K(ρ) + γK ′ (ρ) -K(ρ + γ))∂ 2 x ρ H n K(ρ) + γK ′ (ρ) -K(ρ + γ) H n+1 r H n+3 γ 2 H n+2 .
After similar estimates for the other terms, we end up with

J∂ x δH[U + η] -J∂ x δH[U ] -J∂ x δ 2 H[U ]η H n ω 2 H n+2 + γ 2 H n+3 = η 2 H n+2 .
Proof of theorem 5.1. We fix some k ∈ N large, and iterate Newton's algorithm k times.

The proof is divided in the following steps : 1. Control of f 0 , 2. Control of η 1 , 3. Iteration argument.

Step 1: Control of f 0 . We use the partition of unity (4.14):

∂ t S -J∂ x δH[S] = j,k, j =k χ 2 j J∂ x δH[V k ] + j χ 2 j J∂ x δH[V j ] -δH S .
We set

V j = ρ j v j , S = ρ S v S
, and explicit the second term

δH[V j ] -δH[S] =     -(ρ S -ρ j )v j -ρ S (v S -v j ) (u S -u j )(u j + u S ) + K(ρ j ) -K(S) ∂ 2 x ρ j +K S ∂ 2 x ρ j -S + 1 2 K ′ (ρ j )(∂ x ρ j ) 2 -1 2 K ′ (S)(∂ x S) 2     .
Since all terms are smooth and spatially decorrelated, they are small and exponentially decaying in t for any Sobolev norms, for example

χ j ∂ x (K(ρ j ) -K(ρ S ))∂ 2 x ρ j H 2n+2k ∂ x ρ j H 2n+2(k+1) K(ρ j ) -K(S) H 2n+2k+1 (supp(χ j ))
e -αc 0 t A .

With similar computations, we find the existence of C 0 only depending on S, n, k such that

f 0 H 2n+2k ≤ C 0 e -αc 0 t A .
Step 2: Control of η 1 and η 2 . At this point some care with the constants is required. We denote R η the resolvent operator associated to J∂ x δ 2 H[S + η]. Following the notation and result of theorem 4.7, we set C e = max n≤p≤n+k C e,p . There exists a constant C R such that for 1 ≤ l ≤ k, under the conditions

C e (1/A + η X 2n+2l ) 1/4 ≤ αc 0 /4, 1/A + η X 2n+2l ≤ min 1≤j≤k ε n+j , (5.1)
and using te t ≤ e 2t then

R η (t, s) L(H 2n+2l ) ≤ C(1 + |t -s|)e αc 0 |t-s|/4 ≤ C R e αc 0 |t-s|/2 . (5.2)
According to lemma 5.2, there exists C T ayl such that for

j ≤ n + k, η ∈ H 2j+2 , U ∈ H 2j+2 , V = S + U, η L ∞ + U L ∞ ≤ inf S/2, (5.3)
⇒ J∂ x δH[V + η] -J∂ x δH[V ] -J∂ x δ 2 H[V ]η H 2j ≤ C T ayl η 2 H 2j+2 . (5.4)
With these notations we bound η 1 using the Duhamel formula,

η 1 = ∞ t R 0 (t, s)f 0 (s)ds, ⇒ η 1 H 2n+2k ≤ ∞ t C 0 C R e αc 0 2 |t-s| e -αc 0 s A ds = 2C 0 C R Aαc 0 e -αc 0 t .
We define δ = 2C 0 C R /(Aαc 0 ), which can be as small as needed.

To bound

f 1 = J∂ x δH[S] + J∂ x δH[S]η 1 -J∂ x δH[S + η 1 ]
we can use estimate (5.4) (note that up to decreasing min 1≤j≤k ε n+j , condition (5.1) is stronger than (5.3))

f 1 H 2n+2(k-1) ≤ C T ayl δ 2 e -2αc 0 t .
Next to use the resolvent estimate (5.2), we need to bound η 1 X 2n+2(k-1) . This is done thanks to a general estimate : if

∂ t η = J∂ x δH[S + U ]η + f , then using (4.35) ∀ N ≥ 2, ∂ t η H N-2 ≤ C X,N ( U H N ) η H N + f H N-2 .
(5.5)

We define

C X = max 1≤j≤k C X,2n+2j (ε n+j ). Since ∂ t η 1 = J∂ x δ 2 H[S]η 1 -f 0 , (5.5) gives ∂ t η 1 H 2n+2(k-1) ≤ C X η 1 H 2n+2k + C 0 e -αc 0 t A ≤ C X δ + C 0 A .
In particular, η 1 X 2n+2(k-1) ≤ (C X + 1)δ + C 0 /A. Therefore (up to increasing A) condition (5.1) is satisfied (with l = k), Duhamel's formula gives again

η 2 H 2n+2(k-1) ≤ ∞ t C R C T ayl δ 2 e αc 0 |t-s|/2 e -2αc 0 s ds ≤ C R C T ayl δ αc 0 δe -2αc 0 t := qδe -2αc 0 t .
We note for later use that for A large enough, we have q ≤ 1/2 so that j≥0 q j ≤ 1.

Step 3. Induction. Assume we have constructed (η i ) 1≤i≤j for some j < k, with

η 1 H 2n+2k ≤ δe -αc 0 t , ∀ i ≥ 2, η i H 2n+2(k-i+1) ≤ q 2 i-1 δe -2 i-1 αc 0 t .
In particular, j 1 η i H 2n+2(k-i+1) ≤ 2δ and from (5.4)

∀ 1 ≤ i ≤ j, f i (t) H 2n+2(k-i) ≤ C T ayl q 2 i δ 2 e -2 i αc 0 t .
Estimating ∂ t η i as for ∂ t η 1 we have

∂ t ( j i=1 η i ) H 2n+2(k-j) ≤ j i=1 C X η i H 2n+2(k-i+1) + f i-1 H 2n+2(k-i) ≤ 2C X δ + 2C T ayl δ 2 + C 0 A , therefore j 1 η i X 2n+2(k-j) ≤ 2(C X + 1)δ + 2C T ayl δ 2 + C 0 /A,
and the smallness conditions (5.1) are satisfied (with l = k -j) for A large enough independent of j. We can use the uniform resolvent estimate (5.2) and Taylor estimate (5.4) as for the construction of η 2

η j+1 H 2n+2(k-j) ≤ ∞ t C R C T ayl q 2 j
δ 2 e αc 0 |t-s|/2 e -2 j αc 0 s ds ≤ q 2 j δe -2 j αc 0 t .

By induction, we obtain (η j ) 1≤j≤k , the function U a k 1 η j is sufficient to end the proof since by construction and estimate (5.4)

∂ t (S + k 1 η j ) = J∂ x δH[S + k 1 η j ] + f k , with f k H 2n ≤ C T ayl q 2 k δe -2 k t ,
so that the remainder f k is as small and rapidly decreasing as required for k large enough.

Proof of the main result

This section is a compactness argument. Let V a = ρ a v a = S + U a be an approximate solution given by theorem 5.1 with U a ∈ H 2n+2 and n, ε, C e to choose later. Define V k the solution of (1.1) with Cauchy data V k (k) = V a (k). According to the energy estimate of proposition 3.1, we have for ∆U

k := V k -V a d dt ∆U k H 2n ≤ C ∆U k H 2n + U a H 2n+2 + 1/ρ k + 1/ρ a L ∞ ∆U k H 2n + εe -Cet . Let m = inf (x,t)∈R×R + ρ a , we pick ε such that ∆U k H 2n < 2ε ⇒ inf ρ k ≥ m/2, and ∆U k H 2n ≤ C 1 ∆U k H 2n . Set C = C(2C 1 ε + U a H 2n+2 + 3/m)
, and fix C e ≥ 2C. We can assume C ≥ 1. Since ∆U k (k) = 0, the energy estimate backwards in time gives (as long as

U (t) H 2n ≤ 2ε) d dt e Ct ∆U k H 2n ≥ -εe -(Ce-C)t ⇒ ∆U k (t) 2 H 2n ≤ ε C e -C e -Cet ≤ εe -Cet .
From a (backwards) continuation argument, the solution is well defined on [0, k] for ε small enough, and independently of k

∀ 0 ≤ t ≤ k, ∆U k (t) H 2n ≤ C 1 εe -Cet .
For n large (actually n = 2 is enough), we have from the equation

∂ t V k ∈ C b (R + , H 2n-2
), indeed we recall that ∂ x S is smooth and rapidly decaying:

∂ t ρ k = -∂ x (ρ k u k ), with ρ k ∈ (ρ S + H 2n+1 ), u k ∈ (u S + H 2n ), thus ∂ t ρ k ∈ H 2n-1 , ∂ t u k = -∂ x g(ρ k ) + (u k ) 2 2 (u S ) 2 +H 2n -K∂ 2 x ρ k H 2n-1 - 1 2 K ′ (∂ x ρ k ) 2 H 2n ∈ H 2n-2 . Similarly, ∂ t U a ∈ C b (R + , H 2n ). We deduce that V k -S = U a +∆U k is bounded in C b ([0, k], H 2n ) and C 1 b ([0, k], H 2n-2
. By weak* compactness, up to an extraction V k -S converges weakly to some U ∈ L ∞ (R + , H 2n ). Moreover for any bounded interval J, we have the compact embedding H 2n-2 (J) ⊂ H 2n-3 (J), so using the Ascoli-Arzela theorem, up to another extraction V k -S converges to U in C loc (R + , H 2n-2 loc ). For 2n -3 ≥ 2 it is not hard to check that S + U is a solution of the Euler-Korteweg system (1.1). Now due to the uniform estimate

V k (t) -V a (t) H 2n ≤ εe -Cet (t ≤ k), passing to the (weak) limit (S + U )(t) -V a (t) H 2n ≤ εe -Cet (a.e.).
From theorem 5.1, we also know V a -S H 2n e -αc 0 t

A , therefore we can conclude lim t→∞ U (t) -S(t) H 2n = 0.

Remark 7. A priori, the pointwise H 2n convergence holds only almost everywhere in t, however using the well-posedness theorem 1.1 in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF], one can prove that U coincides with the C(R + , H 2n ) solution, and by continuity the convergence holds for all t.

A Complements on traveling waves

Existence of kinks A traveling wave satisfy    -c∂ x ρ + ∂ x (ρv) = 0,

-c∂ x v + ∂ x (v 2 /2) + ∂ x g(ρ) = ∂ x K∂ 2 x ρ + 1 2 K ′ (∂ x ρ) 2 .
A first integration gives    ρ(v -c) = j, (v -c) 2 2 + g(ρ) -K∂ 2 x ρ -

1 2 K ′ (∂ x ρ) 2 = q.
(we will see several interpretations of this condition). Conversely, assume (A.7),(A.9),(A.11) are satisfied, and that f only changes sign once on (ρ -, ρ + ). Due to (A.7), (A.11), f ′ (ρ ± > 0 thus f must be positive then negative on (ρ -, ρ + ), and from (A.9), F remains positive on (ρ -, ρ + ), but vanishes at second order at ρ ± . The existence of a kink then just follows from the integration of ± √ K∂ x ρ/ 2F (ρ) = 1 (with a choice of sign adapted to the one of ρ --ρ + ). To summarize, provided this equation is satisfied c is a free parameter, and ρ + or ρ -is used to fully parametrize the traveling waves. Kinks should thus form locally two dimensional manifolds.

Remark 8. As the construction of the profile ρ depends on (v + -c) 2 , we can assume c-v + > 0.

The speed of kinks, some geometry The momentum equation is

j 2 2ρ 2 - j 2 2ρ 2 + + g(ρ) = K∂ 2 x ρ + 1 2 K ′ (∂ x ρ) 2 , ⇒ - j 2 (ρ -ρ + ) 2 2ρρ 2 + + G(ρ) = 1 2 K(∂ x ρ) 2 ≥ 0. (A.12)
Letting x → ±∞, from the sign condition we find again (A.11)

j 2 ρ 3 ± = (v ± -c) 2 ρ ± ≤ g ′ (ρ ± ). (A.13)
This inequality gives a geometric interpretation of (A.7), that we rewrite g(ρ -) = -j 2 2ρ 2 -+ q, g(ρ + ) = -j 2 2ρ 2 + + q, meaning that ρ ± are intersection points of the curves g, -j 2 /2ρ 2 + q, and conditions (A.11) mean that the curves intersect transversally at ρ ± . Condition (A.9) means that the total signed area between the two curves from ρ -to ρ + must be zero. See figure A. When g follows a Van Der Waals law, such conditions can be met we refer to [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] for some relevant examples.

The dimension of families of kinks. There exists a kink provided equations (A.7), (A.9) are met ((A.11) is open and therefore plays no role for the dimension), namely

f (ρ + ) = f (ρ -) = ρ + ρ - f (
ρ)dρ = 0, where f depends on ρ ± , c, j, q.

Consider the application ϕ : (ρ ± , j, q, c) → Figure 1: Full line g, dashed line -j 2 /(2ρ 2 ) + q, the two shaded areas should be equal.

  f (ρ + ) f (ρ -) ρ + ρ -f (ρ)dρ   , we have Dϕ =   -j 2 /ρ 3 -+ g ′ (ρ -) 0 j/ρ 2 - -1 0 0 -j 2 /ρ 3 + + g ′ (ρ + ) j/ρ 2 + -1 0 -f (ρ -) f (ρ + )
According to the sign condition (A.11), in the generic case -j 2 /ρ 3 ± + g ′ (ρ ± ) > 0, so the rank of the matrix is three and the kinks form a manifold of dimension two.

The case of solitons Kinks can not provide a nontrivial soliton in the limit ρ -→ ρ + , indeed kinks are monotonous therefore the "soliton" limit of a kink is actually a constant solution. Nevertheless, the construction of solitons follows the same lines. We denote ρ + = lim ±∞ ρ. Since g is a primitive of g ′ , we can assume g(ρ + ) = 0. Equation (A.7) gives

f (ρ + ) = j 2 2ρ 2 + -q = 0 ⇒ f (ρ) = j 2 2ρ 2 - j 2 2ρ 2 + + g(ρ).
Then (A.9) is free so

1 2 K(ρ ′ ) 2 = -j 2 2ρρ 2 + (ρ + -ρ) 2 + G(ρ) = -(c -v + ) 2 2ρ (ρ + -ρ) 2 + G(ρ). (A.14)
For j 2 < ρ 3 + g ′ (ρ + ) ⇔ (v + -c) 2 < ρ + g ′ (ρ + ), we can define

ρ m = sup ρ < ρ + : -j 2 2ρρ 2 + (ρ + -ρ) 2 + G(ρ) = 0 .
From basic ODE arguments, there exists a homoclinic orbit to ρ + with minimal value ρ m ; a "bubble" decreasing from ρ + to ρ m then increasing back to ρ + . Remark 9. We recall that a kink of speed c k and right endstate (ρ + , v + ) satisfies (v + -c k ) 2 < ρ + g ′ (ρ + ), and since its construction depends on (v + -c k ) 2 rather than v + -c k , we may assume c k -v + ≥ 0. In particular since there exists solitons of speed c s with (v + -c s ) 2 arbitrarily close to ρ + g ′ (ρ + ), there always exists solitons faster than the kink and sharing the same endstate.

j 2 2 2 /ρ 2 - 2 2 2

 22222 (1/ρ --1/ρ + ) ρ --ρ + 0 /ρ 3 -+ g ′ (ρ -) 0 j/ρ 3 + + g ′ (ρ + ) j/ρ 2 + (1/ρ --1/ρ + ) ρ --ρ + 0

  small enough so that M ε ≤ Cε 1/4 /2). (4.32) has the form (e -δ|t-s| ϕ(t)) ′ ≤ M e µ|t-s| , so by integration on [s, t]

	1/4 |t-s|/2 +	Cε 1/4 2	n k=1	|β k (t)|.	(4.32)
	(for ε n
	k=1

Assuming lim ±∞ ρ = ρ ± , lim ±∞ v = v ± , we have

This implies q = j 2 2ρ 2 + g(ρ) -K∂ 2 x ρ -

Set f (ρ) = j 2 2ρ 2 -q + g(ρ), we get two conditions

Multiplying (A.5) by ∂ x ρ and integrating from ρ -to ρ

with G the primitive of g such that G(ρ -) = 0. From this integrated momentum equation we get one condition :

This condition can be written only in term of ρ -, ρ + :

Lastly according to (A.2) ρ satisfies the following system of ODE

up to a change of variable it is hamiltonian (with energy F (ρ)) therefore steady states can only be centers or saddles, and a traveling wave connects two saddle points. So (ρ ± , 0) should be a saddle point, which leads to a last condition: the characteristic equation at (ρ ± , 0) is

and the roots in λ are real with opposite sign under the condition

Existence of kink-stable solitons configuration According to remark 9, given a kink with right endstate (ρ + , v + ), there exists solitons with same endstate and larger speed satisfying c -v + > 0. We are left to check wether such solitons are stable.

We assume here that the asymptotic state (ρ + , v + ) is fixed, so that solitons only depend on the speed c, and we also assume g ′′ (ρ + ) ≥ 0 (this is true for the Van Der Waals case).

For consistency, we first prove that the stability condition dP/dc < 0 is indeed equivalent to the stability condition of Benzoni et al [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF]. To do so, we recall the definition of momentum of instability from [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF]. The equations satisfied by a soliton are

dx, and recalling P = (ρ -ρ + )(v -v + ), they can be expressed in an abstract way

where

The momentum of instability is then

and the stability condition of [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] is m ′′ (c) > 0.

Lemma A.1. The condition m ′′ (c) > 0 is equivalent to

Proof. Denote ′ the derivative with respect to c, using (A.15) we have

We differentiate again and use the identity

The condition m ′′ > 0 gives the expected result.

The so-called transonic limit corresponds to j 2 /ρ 2 + = (v + -c) 2 → ρ + g ′ (ρ + ), so we set

From numerical computations it was conjectured in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] that solitons are stable in the transonic limit, and this is rigorously proved with the following result. As it gives the existence of stable solitons with speed arbitrarily close to ρ + g ′ (ρ + ), it also provides the existence of kink-stable soliton configurations.

Lemma A.2. For ε small enough and g ′′ (ρ + ) > 0, "bubble" solitons of speed ρ + g ′ (ρ + )(1 -ε) are stable.

Proof. The condition dP/dc < 0 is equivalent to dP/dε > 0 and equation (A.14) reads

Note that α > 0, in the limit ε → 0+, solitons have an amplitude ρ + -ρ m ∼ ε/α → 0, where ρ m (ε) is the minimum of ρ, and in this regime ρ ′ m (ε) < 0. Up to translation, we can assume that the minimum of ρ is reached at x = 0, and ρ is strictly decreasing on (-∞, 0). Using on x ∈ (0, ∞) the change of variable ρ(x) = ρ, dx = K 2F dρ we find

As is expectable, the situation is somewhat degenerate at ε = 0, as one can check that P (c) =

). This is handled by a factorization of F (see (A.8)):

Here H(ρ + ) = 0 and by construction H(ρ m (ε)) + ε = 0. The condition α > 0 implies H ′ (ρ + ) > 0, so ϕ(ρ, ε) := (H + ε)/(ρ -ρ m ) is well defined near (ρ, ε) = (ρ + , 0), smooth and does not cancel. To summarize, F = ρ + g ′ (ρ + )(ρ-ρ + ) 2 (ρ-ρm) From ρ ′ m (ε) < 0, ε → δ(ε) is locally invertible and the stability condition is equivalent to dP/dδ > 0, but it is clear from the formula that dP/dδ = 3δ 1/2 2 1 0 r(c -v + ) ρ ρK ρ + g ′ (ρ + )(1 -r)ϕ(r, ε) dr + O(δ 3/2 ), which is positive for δ small enough.