Orbital stability of a chain of dark solitons for general nonintegrable Schrödinger equations with non-zero condition at infinity.

Jordan Berthoumieu¹¹1CY Cergy Paris Université, Laboratoire Analyse, Géométrie, Modélisation, F-95302 Cergy-Pontoise, France. Website: https://berthoumieujordan.wordpress.com E-mail: jordan.berthoumieu@cyu.fr

Abstract

In this article, we focus on the stability of dark solitons for a general one-dimensional nonlinear Schrödinger equation. More precisely, we prove the orbital stability of a chain of travelling waves whose speeds are well ordered, taken close to the speed of sound $c_{s}$ and such that the solitons are initially localized far away from each other. The proof relies on the arguments developed by F. Béthuel, P. Gravejat and D. Smets in [4] and first introduced in [14] by Y. Martel, F. Merle and T.-P. Tsai.

1 Introduction

We are interested in the defocusing nonlinear Schrödinger equation

i\partial_{t}\Psi+\partial_{x}^{2}\Psi+\Psi f(|\Psi|^{2})=0\quad\text{on }\mathbb{R}\times\mathbb{R}.

(

NLS

)

This equation appears as a relevant model in condensed matter physics. In particular, it arises in the context of the Bose-Einstein condensation or of superfluidity (see [1, 7]), but also in nonlinear optics (see [10, 11]), when the natural condition at infinity is

|\Psi(t,x)|\underset{|x|\rightarrow+\infty}{\longrightarrow}1.

(1)

In the latter context, this condition expresses the presence of a nonzero background. It differs from the case of null condition at infinity, in the sense that the resulting dispersion relation is different.
In ( $NLS$ ), the nonlinearity $f$ can be taken equal to $f(\rho)=1-\rho$ and we then obtain the so-called Gross-Pitaevskii equation, but we can also take many other functions $f$ satisfying the condition $f(1)=0$ . This provides possible alternative behaviours as enumerated by D. Chiron in [5]. In the sequel, we restrict our attention to the defocusing case, meaning that we assume that

f^{\prime}(1)<0.

(2)

If $\Psi$ does not vanish, we can lift it as $\Psi=\rho e^{i\varphi}$ where $\rho$ and $\varphi$ are as smooth as $f$ is. Setting the new variables $\eta=1-\rho^{2}$ and $v=-\partial_{x}\varphi$ , we obtain the hydrodynamical form of the equation

\left\{\begin{array}[]{l}\partial_{t}\eta=-2\partial_{x}\big{(}v(1-\eta)\big{)},\\ \partial_{t}v=-\partial_{x}\bigg{(}f(1-\eta)-v^{2}-\dfrac{\partial_{x}^{2}\eta}{2(1-\eta)}-\dfrac{(\partial_{x}\eta)^{2}}{4(1-\eta)^{2}}\bigg{)}.\\ \end{array}\right.

(

NLS_{hy}

)

The linearized system around the trivial solution $(\rho,v)=(1,0)$ reduces, in the long wave approximation, to the free wave equation with the sound speed

c_{s}=\sqrt{-2f^{\prime}(1)}.

(3)

The plane wave solutions of this linearized system satisfy indeed the dispersion relation

\omega(\xi)=\pm\sqrt{\xi^{4}+c_{s}^{2}\xi^{2}}.

For $k\geq 0$ , we shall write $\mathcal{X}_{hy}^{k}(\mathbb{R}):=H^{k+1}(\mathbb{R})\times H^{k}(\mathbb{R})$ and endow this space with the associated euclidean norm

\|(\eta,v)\|_{\mathcal{X}^{k}}^{2}=\|\eta\|_{H^{k+1}}^{2}+\|v\|_{H^{k}}^{2}.

Let us also introduce the non-vanishing associated subset

\mathcal{NX}_{hy}^{k}(\mathbb{R}):=\big{\{}(\eta,v)\in\mathcal{X}_{hy}^{k}(\mathbb{R})\big{|}\max_{\mathbb{R}}\eta<1\big{\}}.

We will label $\mathcal{X}_{hy}(\mathbb{R}):=\mathcal{X}_{hy}^{k}(\mathbb{R})$ and $\mathcal{NX}_{hy}(\mathbb{R}):=\mathcal{NX}_{hy}^{k}(\mathbb{R})$ when $k=0$ . This functional setting is related to several quantities, which are at least formally, conserved along the flow. These are the energy

E(\eta,v):=\int_{\mathbb{R}}e(\eta,v):=\dfrac{1}{8}\int_{\mathbb{R}}\dfrac{(\partial_{x}\eta)^{2}}{1-\eta}+\dfrac{1}{2}\int_{\mathbb{R}}(1-\eta)v^{2}+\dfrac{1}{2}\int_{\mathbb{R}}F(1-\eta),

(4)

with

F(r):=\int_{r}^{1}f(\rho)d\rho,

and (the renormalized) momentum

p(\eta,v):=\int_{\mathbb{R}}\pi(\eta,v):=\dfrac{1}{2}\int_{\mathbb{R}}\eta v.

(5)

As it is mentionned in [3], the assumptions that we will make on the nonlinearity $f$ give to $\mathcal{NX}_{hy}(\mathbb{R})$ the status of the energy set of ( $NLS_{hy}$ ). Note also that, in view of the previous expression, $p$ is smooth on $\mathcal{X}_{hy}(\mathbb{R})$ . Moreover, if $f\in\mathcal{C}^{k}(\mathbb{R}_{+})$ , $E$ is $\mathcal{C}^{k+1}$ on $\mathcal{NX}_{hy}(\mathbb{R})$ . Concerning the existence of solutions to ( $NLS_{hy}$ ), we recall the following local wellposedness theorem.

Theorem 1.1 (Gallo [8]).

We assume that $f\in\mathcal{C}^{3}(\mathbb{R}_{+})$ is such that for all $\rho\in\mathbb{R}$ ,

\dfrac{c_{s}^{2}}{4}(1-\rho)^{2}\leq F(\rho).

(H1)

Let $k\in\{0,1,2\}$ and let $(\eta_{0},v_{0})\in\mathcal{NX}_{hy}^{k}(\mathbb{R})$ . There exist $T_{\max}>0$ and a unique solution $(\eta,v)\in\mathcal{C}^{0}\big{(}[0,T_{\max}),\mathcal{NX}_{hy}^{k}(\mathbb{R})\big{)}$ to ( $NLS_{hy}$ ) with initial datum $(\eta_{0},v_{0})$ . The maximal time $T_{\max}$ is continuous with respect to the initial datum and is characterized by

\lim_{t\rightarrow T_{\max}^{-}}\sup_{x\in\mathbb{R}}\eta(t,x)=1.

Moreover, the flow map is continuous on $\mathcal{NX}_{hy}^{k}(\mathbb{R})$ and the energy and the momentum are conserved along the flow.

1.1 Travelling wave solutions and minimizing property

In this article, we aim at understanding the dynamics of a chain of dark solitons when their speeds are ordered and taken close to the sound of speed $c_{s}$ . "Dark soliton" is a general term designating special solutions of ( $NLS$ ). In our framework, they are travelling waves of nonzero speed¹¹1They are also labelled grey solitons by opposition with the black soliton with speed $c=0$ . of the form $\Psi(t,x)=u(x-ct)$ . We plug this ansatz in ( $NLS$ ) and we obtain the ordinary differential equation satisfied by the profile $u$ , which is

-icu^{\prime}+u^{\prime\prime}+uf(|u|^{2})=0.

(

TW_{c}

)

Under suitable conditions on the nonlinearity $f\in\mathcal{C}^{3}(\mathbb{R}_{+})$ there exist travelling waves of given momentum in a certain range $(0,\mathfrak{q}_{*})$ (see [3]) and they are orbitally stable. We assume that these conditions hold true in the sequel. Namely

•

For all $\rho\in\mathbb{R}$ ,

$\dfrac{c_{s}^{2}}{4}(1-\rho)^{2}\leq F(\rho).$ (H1)
•

There exist $M\geq 0$ and $q\in[2,+\infty)$ such that for all $\rho\geq 2$ ,

$F(\rho)\leq M|1-\rho|^{q}.$ (H2)
•

$f^{\prime\prime}(1)+3f^{\prime}(1)\neq 0.$ (H3)

This result relies on a variational argument which consists on solving the minimization problem

E_{\min}(\mathfrak{p}):=\inf\big{\{}E(v)\big{|}v\in\mathcal{N}\mathcal{X}(\mathbb{R}),p(v)=\mathfrak{p}\big{\}},

(6)

where $E$ and $p$ denote the energy and the momentum associated with the formulas (4) and (5). The Euler-Lagrange equation

\nabla E(v)-c\nabla p(v)=0

(7)

indeed reduces to ( $TW_{c}$ ) where the speed $c$ is the Lagrange multiplier of the problem. The question of uniqueness of the minimizer goes beyond the scope of this article but we shall touch upon a partial result in this way in Section 1.2. For $\mathfrak{p}\in(0,\mathfrak{q}_{*})$ , we shall label $\mathfrak{v}_{c(\mathfrak{p})}$ a dark soliton minimizing (6) and mention that, due to condition (H3) (see Theorem 5.1 and the remark just after in [13]), $c(\mathfrak{p})$ necessarily lies in $(-c_{s},c_{s})\setminus\{0\}$ .

Since $c(\mathfrak{p})\neq 0$ , the solutions $\mathfrak{v}_{c(\mathfrak{p})}$ do not vanish on $\mathbb{R}$ (see Remark 4.6 in [3]). Plugging the Madelung transform in ( $TW_{c}$ ), we obtain the hydrodynamical version of ( $TW_{c}$ ) satisfied by the variables $(\eta_{c},v_{c}):=(1-|\mathfrak{v}_{c}|^{2},-\varphi_{c}^{\prime})$ ,

\left\{\begin{array}[]{l}\dfrac{c}{2}\eta_{c}=v_{c}(1-\eta_{c}),\\ cv_{c}=f(1-\eta_{c})-v_{c}^{2}-\dfrac{\eta_{c}^{\prime\prime}}{2(1-\eta_{c})}-\dfrac{(\eta_{c}^{\prime})^{2}}{4(1-\eta_{c})^{2}}.\\ \end{array}\right.

(

TW_{c,hy}

)

For the sake of completeness, we will check in Subsection 2.2 the bound

\max_{\mathbb{R}}\eta_{c}<1,

(8)

for solitons with speed $c$ close to $c_{s}$ . In the sequel, we will also take into account the invariance by translation of the equation by setting

Q_{c}:=(\eta_{c},v_{c})\quad\text{and}\quad Q_{c,a}:=(\eta_{c,a},v_{c,a}):=\big{(}\eta_{c}(.-a),v_{c}(.-a)\big{)},

for $a\in\mathbb{R}$ .

1.2 Chain of solitons in the transonic regime

In [5], D. Chiron proved the existence of a continuous branch of travelling waves in the transonic limit, i.e. with speed close to $c_{s}$ . We slightly improve this result in the next theorem.

Theorem 1.2.

There exists a critical speed $c_{0}>0$ such that for $c\in(c_{0},c_{s})$ , there exists a non constant smooth solution $Q_{c}$ of ( $TW_{c,hy}$ ), that is unique up to translations. Furthermore, the mapping $c\in(c_{0},c_{s})\mapsto Q_{c}$ belongs to $\mathcal{C}^{2}\big{(}(c_{0},c_{s}),\mathcal{NX}^{2}(\mathbb{R})\big{)}$ and for $c\in(c_{0},c_{s})$ , we have

\dfrac{d}{dc}\big{(}p(Q_{c})\big{)}<0.

(9)

Moreover there exists $a_{d},K_{d}>0$ independent of $c\in(c_{0},c_{s})$ and $x\in\mathbb{R}$ such that,

\sum_{\begin{subarray}{c}0\leq k_{1}\leq 3\\ 0\leq k_{2}\leq 2\\ 0\leq j\leq 2\end{subarray}}(c_{s}^{2}-c^{2})^{j-1}\Big{(}|\partial_{c}^{j}\partial_{x}^{k_{1}}\eta_{c}(x)|+c^{1+2j+2k_{2}}|\partial_{c}^{j}\partial_{x}^{k_{2}}v_{c}(x)|\Big{)}\leq K_{d}e^{-a_{d}\sqrt{c_{s}^{2}-c^{2}}|x|}.

(10)

Remark 1.3.

We can show that the branch is actually $\mathcal{C}^{3}\left((c_{0},c_{s}),\big{(}H^{2}(\mathbb{R})\times L^{2}(\mathbb{R})\big{)}\cap\big{(}\mathcal{C}_{loc}^{5}(\mathbb{R})\big{)}^{2}\right)$ . Nevertheless, the choice of the specific exponents in Theorem 1.2 is sharp for the study in this article.

Remark 1.4.

The restriction to positive speeds is not an arbitrary choice. Indeed note that if $Q_{c}$ is a hydrodynamical travelling wave of speed $c$ , then $(\eta_{c},-v_{c})$ is a hydrodynamical travelling wave of speed $-c$ and our results extend naturally to these travelling waves.

In [9], M. Grillakis, J. Shatah and W.A. Strauss gave some sharp conditions for orbital stability (and instability) for a general class of Hamiltonian systems. D. Chiron showed in [6] that this general result could be specified to travelling waves for nonlinear Schrödinger type equations. The aforementioned condition for orbital stability is precisely (9).

Combining Theorem 1.2 with the existence of minimizing travelling waves stated in Section 1.1, we can exhibit a unique branch of solitons with transonic speed that are orbitally stable. This allows us to consider the orbital stability of a chain of solitons the speeds of which are taken in this regime. That²²2In the following definition and throughout this article, we will earmark $\mathfrak{c},\mathfrak{a}$ for $N$ -dimensional vectors and script letters for real numbers. is why we introduce the set of well ordered admissible speeds

\mathrm{Adm}_{N}:=\big{\{}\mathfrak{c}:=(c_{1},...,c_{N})\in(c_{0},c_{s})^{N}\big{|}\ c_{1}<...<c_{N}\big{\}}.

Each speed is here chosen such that the corresponding travelling wave is unique and orbitally stable. Another crucial condition for the orbital stability to hold lies in the initial length between the solitons. This induces to define the set

\mathrm{Pos}_{N}(L):=\big{\{}\mathfrak{a}:=(a_{1},...,a_{N})\in\mathbb{R}^{N}\big{|}a_{k+1}-a_{k}>L,\ \forall k\in\{1,...,N-1\}\big{\}},

for $L>0$ . For $\mathfrak{c}=(c_{1},...,c_{N})\in\mathrm{Adm}_{N}$ , and $\mathfrak{a}=(a_{1},...,a_{N})\in\mathbb{R}^{N}$ , we finally define the sum of solitons

R_{\mathfrak{c},\mathfrak{a}}:=(\eta_{\mathfrak{c},\mathfrak{a}},v_{\mathfrak{c},\mathfrak{a}})=\sum_{i=1}^{N}Q_{c_{i},a_{i}}.

With these notations, we can state our main result.

Theorem 1.5.

Let $\mathfrak{c}^{*}\in\mathrm{Adm}_{N}$ . There exists $\alpha_{*},L_{*},A_{*},\tau_{*}>0$ , such that the following holds. If $Q_{0}=(\eta_{0},v_{0})\in\mathcal{NX}_{hy}(\mathbb{R})$ is such that for some $\mathfrak{a}^{0}:=(a_{1}^{0},...,a_{N}^{0})\in\mathrm{Pos}_{N}(L_{0})$ with $L_{0}\geq L_{*}$ ,

\alpha_{0}:=\left\|Q_{0}-R_{\mathfrak{c}^{*},\mathfrak{a}^{0}}\right\|_{\mathcal{X}}\leq\alpha_{*},

then, the unique solution $Q(t)=\big{(}\eta(t),v(t)\big{)}$ to ( $NLS_{hy}$ ) associated with the initial datum $(\eta_{0},v_{0})$ is globally defined and there exists $(\mathfrak{a},\mathfrak{c})\in\mathcal{C}^{1}(\mathbb{R}_{+},\mathbb{R}^{2N})$ such that for any $t\in\mathbb{R}_{+}$ ,

\left\|Q(t)-R_{\mathfrak{c}^{*},\mathfrak{a}(t)}\right\|_{\mathcal{X}}\leq A_{*}K(\alpha_{0},L_{0}),

where $K(\alpha_{0},L_{0})=\alpha_{0}+e^{-\tau_{*}L_{0}}$ . Regarding the modulation parameters, we have

\left|\mathfrak{a}^{\prime}(t)-\mathfrak{c}(t)\right|+\left|\mathfrak{c}^{\prime}(t)\right|\leq A_{*}K(\alpha_{0},L_{0}).

Remark 1.6.

Since every norm on a finite dimensional Banach space are topologically equivalent, we use, here as in the sequel, one unique notation to designate every norm on $\mathbb{R}^{N}$ or $M_{N}(\mathbb{R})\sim\mathbb{R}^{N^{2}}$ . We shall indeed write $\left|\mathfrak{x}\right|$ for any $\mathfrak{x}:=(x_{1},...,x_{N})\in\mathbb{R}^{N}$ .

1.3 Sketch of the proof

The proof can be summarized in the following several steps.

1.3.1 Coercivity around the soliton $Q_{c}$

As it is stated in Section 1.1, being a solution to ( $TW_{c}$ ) means that the Euler-Lagrange equation $\nabla(E-cp)(Q_{c})=0$ is satisfied. We introduce the bilinear form $\mathcal{H}_{c}:=\nabla^{2}(E-cp)(Q_{c})$ defined on $H^{2}(\mathbb{R})\times L^{2}(\mathbb{R})$ and $H_{c}(\varepsilon):=\left\langle\mathcal{H}_{c}(\varepsilon),\varepsilon\right\rangle_{L^{2}}$ the corresponding quadratic form, which is well-defined on $\mathcal{X}_{hy}(\mathbb{R})$ . By linearizing ( $TW_{c,hy}$ ), we obtain the following expression

\displaystyle\mathcal{H}_{c}=\begin{pmatrix}\mathcal{L}_{c}&-\dfrac{c}{2(1-\eta_{c})}\\ -\dfrac{c}{2(1-\eta_{c})}&1-\eta_{c}\end{pmatrix}

with

\mathcal{L}_{c}(\varepsilon_{\eta})=-\Big{(}\dfrac{\varepsilon_{\eta}^{\prime}}{4(1-\eta_{c})}\Big{)}^{\prime}-\dfrac{c^{2}+2F(1-\eta_{c})+2(1-\eta_{c})f(1-\eta_{c})+2(1-\eta_{c})^{2}f^{\prime}(1-\eta_{c})}{4(1-\eta_{c})^{2}}\varepsilon_{\eta}.\\

The quadratic form $H_{c}$ is coercive under the following orthogonality constraints (see [16] for similar arguments).

Proposition 1.7.

Let $c\in(0,c_{s})$ . There exists $l_{c}>0$ such that for all $\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})$ , satisfying the orthogonal conditions

0=\left\langle\varepsilon,Q_{c}^{\prime}\right\rangle_{L^{2}\times L^{2}}=\nabla p(Q_{c}).\varepsilon,

(11)

then

H_{c}(\varepsilon)\geq l_{c}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}.

(12)

The proof of Proposition 1.7 requires to deal with derivatives like $\partial_{c}Q_{c}$ . This explains why we first need to verify (see Section 2) that the branch of solitons of speed $c$ is smooth. Apart from the smoothness of the considered functions, the proof of Proposition 1.7 is reminiscent from [4]. We obtain a formula for the essential spectrum that generalize the one in [4], that is

\sigma_{ess}(\mathcal{H}_{c})\subset\left[\dfrac{c_{s}^{2}-c^{2}}{1+c_{s}^{2}+\sqrt{(1-c_{s}^{2})^{2}+4c^{2}}},+\infty\right).

Furthermore, we observe that the linearized operator $\mathcal{H}_{c}$ owns a unique negative direction and a unique vanishing direction. Eliminating these directions by adding orthogonality conditions (11) eventually leads to (12). We refer to the proof of Proposition 1 in [4] for more details. Spectral considerations allow us to precise the choice of $l_{c}$ , that can be taken as follows

l_{c}:=\inf_{\begin{subarray}{c}\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})\setminus\{0\}\\ \varepsilon\text{ satisfies\leavevmode\nobreak\ \eqref{proposition condition d'orthogonalité sur varepsilon}. }\end{subarray}}\dfrac{|H_{c}(\varepsilon)|}{\left\|\varepsilon\right\|_{\mathcal{X}}^{2}}>0.

(13)

1.3.2 Almost minimizing property of a sum of solitons

A first consequence of Proposition 1.7 is that, if $\varepsilon$ satisfies the orthogonal condition (11), then by the minimizing property of $Q_{c,a}$ , we get

(E-cp)(Q_{c,a}+\varepsilon)\geq(E-cp)(Q_{c,a})+l_{c}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}+\mathcal{O}\big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{3}\big{)}.

(14)

Now, we ought to find the same kind of estimate when we perturb $R_{\mathfrak{c},\mathfrak{a}}$ instead of $Q_{c,a}$ . Let $\mathfrak{c}:=(c_{1},...,c_{N})\in\mathrm{Adm}_{N}$ and $\mathfrak{a}\in\mathrm{Pos}_{N}(L)$ with some $L>0$ . Let us set

\displaystyle\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L):=\left\{R_{\mathfrak{c},\mathfrak{a}}+\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})\bigg{|}\begin{array}[]{l}\left\langle\varepsilon,Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}=\nabla p(Q_{c_{k},a_{k}}).\varepsilon=0,\forall k\in\{1,...,N\}\\ \mathfrak{a}\in\mathrm{Pos}_{N}(L)\end{array}\right\},

(17)

and consider a perturbation $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ . Instead of considering $E-cp$ for a single soliton of speed $c$ , let $\mathfrak{c}^{*}:=(c_{1}^{*},...,c_{N}^{*})$ and let us construct a function

G(Q):=E(Q)-\sum_{k=1}^{N}c_{k}^{*}p_{k}(Q).

(18)

that resembles $E-c_{k}^{*}p$ around each soliton $Q_{c_{k}^{*}}$ , and that will eventually satisfy some coercivity property around the chain of solitons $R_{\mathfrak{c},\mathfrak{a}}$ . We refer to (20) for the definition of $p_{k}$ .

With this goal in mind, we introduce the two following partitions of unity. Set $\tau>0$ . For $k\in\{1,...,N\}$ , we define the functions

\Phi_{k}(x):=\dfrac{1}{2}\bigg{(}\tanh\Big{(}\tau\big{(}x-a_{k}+\frac{L}{4}\big{)}\Big{)}-\tanh\Big{(}\tau\big{(}x-a_{k}-\frac{L}{4}\big{)}\Big{)}\bigg{)},

and

\Phi_{k,k+1}(x)=\left\{\begin{array}[]{l}\dfrac{1}{2}\bigg{(}1-\tanh\Big{(}\tau\big{(}x-a_{1}+\frac{L}{4}\big{)}\Big{)}\bigg{)}\quad\text{if }k=0,\\ \dfrac{1}{2}\bigg{(}\tanh\Big{(}\tau\big{(}x-a_{k}-\frac{L}{4}\big{)}\Big{)}-\tanh\Big{(}\tau\big{(}x-a_{k+1}+\frac{L}{4}\big{)}\Big{)}\bigg{)}\quad\text{if }k\in\{1,...,N-1\},\\ \dfrac{1}{2}\bigg{(}1+\tanh\Big{(}\tau\big{(}x-a_{N}-\frac{L}{4}\big{)}\Big{)}\bigg{)}\quad\text{if }k=N.\\ \end{array}\right.

We have, by construction,

\sum_{k=1}^{N}\Phi_{k}+\sum_{k=0}^{N}\Phi_{k,k+1}=1,

(19)

and we define

\varepsilon_{k}:=\varepsilon(.+a_{k})\sqrt{\Phi_{k}(.+a_{k})}\quad\text{and}\quad\varepsilon_{k,k+1}:=\varepsilon\sqrt{\Phi_{k,k+1}}.

In addition, choose $\tau_{0}>0$ and for $k\in\{1,...,N+1\}$ , set

\chi_{k}(x)=\left\{\begin{array}[]{l}1\quad\text{if }k=1,\\ \dfrac{1}{2}\bigg{(}1+\tanh\left(\tau_{0}\Big{(}x-\frac{a_{k}+a_{k-1}}{2}\Big{)}\right)\bigg{)}\quad\text{if }k\in\{2,...,N\},\\ 0\quad\text{if }k=N+1.\\ \end{array}\right.

Note that we have

\sum_{k=1}^{N}(\chi_{k}-\chi_{k+1})=1.

The choice of $\tau_{0}$ and $\tau$ shall be fixed later. Namely $\tau$ shall be fixed in the proof of Corollary 1.13 and $\tau_{0}$ in the proof of Proposition 1.17. This choice shall only depend on the nonlinearity $f$ and $\mathfrak{c}^{*}$ , that is why we allow us to shrink their value throughout the article. In order to focus on each solitons, we define

p_{k}(Q):=\int_{\mathbb{R}}\pi(Q)(\chi_{k}-\chi_{k+1}).

(20)

The quantity defined in (20) can be interpreted as a momentum localized around the soliton $Q_{c_{k},a_{k}}$ . Indeed, provided that the $a_{k}$ are sufficiently far away from each other, we can formally write that $\chi_{k}(x)-\chi_{k+1}(x)\sim 1$ for $x$ close to $a_{k}$ and $\chi_{k}(x)-\chi_{k+1}(x)\sim 0$ for $x$ close to $a_{j}$ for $j\neq k$ . Plugging this into the expression of $p_{k}$ , this could be regarded as $p_{k}(Q)=p(Q)$ if $Q\sim Q_{c_{k},a_{k}}$ . We also define

\nu_{\mathfrak{c}}:=\min\Big{\{}\nu_{c_{k}}:=\sqrt{c_{s}^{2}-c_{k}^{2}}\Big{|}k\in\{1,...,N\}\Big{\}},

(21)

and

l_{\mathfrak{c}}:=\min\big{\{}l_{c_{k}}\big{|}k\in\{1,...,N\}\big{\}},

(22)

according to Proposition 1.7. We now write the precise approximation of the energy and the (localized) momentum of a perturbed chain of solitons $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ , when $\mathfrak{a}\in\mathrm{Pos}_{N}(L)$ . Henceforth, we introduce the notation $g=\mathcal{O}\left(h\right)$ that means that there exists a constant $C>0$ that only depends on the parameter $\mathfrak{c}^{*}$ introduced above such that $|g|\leq C|h|$ .

Proposition 1.8.

For $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ , and $\tau>0$ such that $2\tau<\nu_{\mathfrak{c}}$ , we have

	$\displaystyle E$	$\displaystyle(Q)=\sum_{k=1}^{N}E(Q_{c_{k}})+\dfrac{1}{2}\bigg{(}\sum_{k=1}^{N}\nabla^{2}E(Q_{c_{k}})(\varepsilon_{k},\varepsilon_{k})+\sum_{k=0}^{N}\nabla^{2}E(0)(\varepsilon_{k,k+1},\varepsilon_{k,k+1})\bigg{)}$
		$\displaystyle+\mathcal{O}\Big{(}\Lambda(L,\mathfrak{c})e^{-a_{d}\nu_{\mathfrak{c}}L}\Big{)}+\mathcal{O}\big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}e^{-a_{d}\nu_{\mathfrak{c}}L}\big{)}+\mathcal{O}(\tau\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2})+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\tau L}{2}}\Big{)}+\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon),$

where

\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)=\int_{0}^{1}\dfrac{(1-t)^{2}}{2}\nabla^{3}E(R_{\mathfrak{c},\mathfrak{a}}+t\varepsilon)(\varepsilon,\varepsilon,\varepsilon)dt,

(23)

and

\Lambda(L,\mathfrak{c})=\dfrac{1}{\nu_{\mathfrak{c}}}+L.

Proposition 1.9.

For $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ , and $\tau,\tau_{0}>0$ such that $\tau_{0}<2\tau<\nu_{\mathfrak{c}}$ , we have

	$\displaystyle p_{k}$	$\displaystyle(Q)=\ p(Q_{c_{k}})+\dfrac{1}{2}\Big{(}\nabla^{2}p(Q_{c_{k}}).(\varepsilon_{k},\varepsilon_{k})+\nabla^{2}p_{k}(0).(\varepsilon_{k,k+1},\varepsilon_{k,k+1})+\nabla^{2}p_{k}(0).(\varepsilon_{k-1,k},\varepsilon_{k-1,k})\Big{)}$
		$\displaystyle+\mathcal{O}\Big{(}\Lambda(L,\mathfrak{c})(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\Big{)}+\mathcal{O}\big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\big{)}+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\tau L}{2}}\Big{)}.$

1.3.3 Orthogonal decomposition

Now, we fix $\mathfrak{c}^{*}$ and set

\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L):=\left\{Q\in\mathcal{X}_{hy}(\mathbb{R})\big{|}\inf_{\mathfrak{a}\in\mathrm{Pos}_{N}(L)}\left\|Q-R_{\mathfrak{c}^{*},\mathfrak{a}}\right\|_{\mathcal{X}}<\alpha\right\}.

(24)

We decompose any function $Q\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L)$ as a chain of solitons, the speeds and localisations of which depend smoothly on $Q$ , plus an extra perturbation. Here, the speeds and localisations are constructed implicitly such that the orthogonal conditions (11) hold true. We shall also see that these implicit functions, called modulation parameters, are controlled in accordance of the parameters $(\alpha,L)$ of the set $\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L)$ . We first introduce a notation that will be used throughout the article. For real numbers $\alpha,\alpha^{\prime},L$ and $L^{\prime}$ , we define

(\alpha,L)\prec(\alpha^{\prime},L^{\prime})\ \Longleftrightarrow\ \alpha\leq\alpha^{\prime}\text{ and }L\geq L^{\prime}.

Proposition 1.10.

Set $\mathfrak{c}^{*}=(c_{1}^{*},...,c_{N}^{*})\in\mathrm{Adm}_{N}$ . There exist constants $\alpha_{1},L_{1},K_{1}>0$ , only depending on $\mathfrak{c}^{*}$ , and two functions

(\mathfrak{C},\mathfrak{A})=\big{(}(c_{1},...,c_{N}),(a_{1},...,a_{N})\big{)}\in\mathcal{C}^{1}\big{(}\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{1},L_{1}),\mathbb{R}^{2N}\big{)}

such that for any $Q=(\eta,v)\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{1},L_{1})$ , the perturbation

\varepsilon:=\varepsilon(Q):=Q-R_{\mathfrak{C}(Q),\mathfrak{A}(Q)}

(25)

satisfies the orthogonality conditions

\left\langle\varepsilon,\partial_{x}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}=\nabla p(Q_{c_{k},a_{k}}).\varepsilon=0,

(26)

for any $k\in\{1,...,N\}$ . Moreover, if for some $(\alpha,L)\prec(\alpha_{1},L_{1})$ , and some $\mathfrak{a}^{*}\in\mathrm{Pos}_{N}(L)$ , we have

\left\|Q-R_{\mathfrak{c}^{*},\mathfrak{a}^{*}}\right\|_{\mathcal{X}}\leq\alpha,

(27)

then

\left\|\varepsilon(Q)\right\|_{\mathcal{X}}+\left|\mathfrak{C}(Q)-\mathfrak{c}^{*}\right|+\left|\mathfrak{A}(Q)-\mathfrak{a}^{*}\right|\leq K_{1}\alpha.

(28)

Also, we obtain some uniform control (with respect to $Q=(\eta,v)$ ) on several elementary positive quantities, defined for $\mathfrak{c}\in\mathrm{Adm}_{N}$ , e.g. (21) and (22), but also

\mu_{\mathfrak{c}}:=\min\big{\{}c_{k}-c_{0}\big{|}k\in\{1,...,N\}\big{\}},

and

\kappa_{\mathfrak{c}}:=\min\left\{-\dfrac{d}{dc}\Big{(}p(Q_{c})\Big{)}_{|c=c_{k}}\Big{|}k\in\{1,...,N\}\right\}.

In this direction, we need to introduce first the following lemma, that provides a uniform bound from below between a chain of solitons and the constant function $1$ . The proof of this lemma is in Appendix A.

Lemma 1.11.

There exist $(\alpha_{2},L_{2})\prec(\alpha_{1},L_{1})$ and $\beta^{*}\in(0,1)$ such that for any $(\alpha,L)\prec(\alpha_{2},L_{2})$ and $\mathfrak{a}\in\mathrm{Pos}_{N}(L)$ , we have

\alpha+\beta^{*}<1\quad\text{and}\quad\left\|\eta_{\mathfrak{c}^{*},\mathfrak{a}}\right\|_{L^{\infty}}\leq\beta^{*}.

(29)

Now, we state the uniform bounds on $\mu_{\mathfrak{c}(Q)},\nu_{\mathfrak{c}(Q)},\kappa_{\mathfrak{c}(Q)}$ and $l_{\mathfrak{c}(Q)}$ .

Corollary 1.12.

Under the orthogonality conditions of Proposition 1.10, and taking possibly a larger $L_{2}$ and a smaller $\alpha_{2}$ , we have

\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{2},L_{2})\subset\mathcal{NX}_{hy}(\mathbb{R}).

(30)

Moreover, there exists $l_{*}>0$ such that for all $Q=(\eta,v)\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{2},L)$ with $L\geq L_{2}$ , we have

1-\eta\geq\dfrac{1-\beta^{*}}{2},

(31)

\mathfrak{A}(Q)\in\mathrm{Pos}_{N}(L-1),

(32)

and

\nu_{\mathfrak{C}(Q)}\geq\dfrac{\nu_{\mathfrak{c}^{*}}}{2},\ \mu_{\mathfrak{C}(Q)}\geq\dfrac{\mu_{\mathfrak{c}^{*}}}{2},\ \kappa_{\mathfrak{C}(Q)}\geq\dfrac{\kappa_{\mathfrak{c}^{*}}}{2},\ l_{\mathfrak{C}(Q)}\geq l_{*}.

(33)

The previous decomposition can be partially summarized as the fact that for any $L\geq L_{2}\geq L_{1}+1$ ,

\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{1},L)\subset\mathcal{U}^{\perp}_{\mathfrak{C}(Q),\mathfrak{A}(Q)}(L-1)\subset\mathcal{U}^{\perp}_{\mathfrak{C}(Q),\mathfrak{A}(Q)}(L_{1}).

(34)

From now on, we impose $\tau_{0}<2\tau<\frac{\nu_{\mathfrak{c}^{*}}}{2}$ . As a consequence of this choice, we can highlight the almost minimizing property of a chain of solitons $R_{\mathfrak{c},\mathfrak{a}}$ for the functional $G$ defined in (18).

Corollary 1.13.

There exists $\widetilde{l}_{*}>0$ such that for any $Q\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L)$ decomposed as $Q=R_{\mathfrak{C}(Q),\mathfrak{A}(Q)}+\varepsilon$ according to Proposition 1.10, with $(\alpha,L)\prec(\alpha_{2},L_{2})$ , we have

\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{*}})-c_{k}^{*}p(Q_{c_{k}^{*}})\big{)}+\dfrac{\widetilde{l}_{*}}{4}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}+\mathcal{O}(\left\|\varepsilon\right\|_{\mathcal{X}}^{3})+\mathcal{O}\big{(}\left|\mathfrak{C}(Q)-\mathfrak{c}^{*}\right|^{2}\big{)}+\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}L}\Big{)}\leq G(Q),

and

G(Q)\leq\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{*}})-c_{k}^{*}p(Q_{c_{k}^{*}})\big{)}+\mathcal{O}(\left\|\varepsilon\right\|_{\mathcal{X}}^{2})+\mathcal{O}\big{(}\left|\mathfrak{C}(Q)-\mathfrak{c}^{*}\right|^{2}\big{)}+\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}L}\Big{)}.

1.3.4 Evolution in time

In the two following subsections, we deal with the time evolution of the perturbation which will eventually conclude the proof of the main result. The first step consists in implementing this evolution in the orthogonal decomposition of Section 1.3.3. Whenever we take $(\alpha,L)\prec(\alpha_{1},L_{1})$ , there exists, by local wellposedness, a solution $Q$ to ( $NLS_{hy}$ ) in $\mathcal{C}^{0}\left([0,T],\mathcal{NX}_{hy}(\mathbb{R})\right)$ with $0<T=T(\alpha,L)<T_{\max}$ such that for any $t\in[0,T]$ ,

Q(t)\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L).

(35)

Then Proposition 1.10 enables us to define the functions

\mathfrak{c}(t):=\big{(}c_{1}(t),...,c_{N}(t)\big{)}:=\mathfrak{C}\big{(}\eta(t),v(t)\big{)}\text{ and }\mathfrak{a}(t):=\big{(}a_{1}(t),...,a_{N}(t)\big{)}:=\mathfrak{A}\big{(}\eta(t),v(t)\big{)},

and also

\varepsilon(t):=\big{(}\eta(t),v(t)\big{)}-R_{\mathfrak{c}(t),\mathfrak{a}(t)},

which depend continuously on $t$ and satisfy the orthogonality conditions (26). Bear also in mind that Corollary 1.12 holds, so that we have uniform bounds for $\nu_{\mathfrak{c}(t)},\mu_{\mathfrak{c}(t)}$ , etc. We write $E(t):=E\big{(}Q(t)\big{)}$ , $p_{k}(t):=p_{k}\big{(}Q(t)\big{)}$ and $G(t):=G\big{(}Q(t)\big{)}$ for $t\in[0,T]$ . We shall see that up to taking further $(\alpha_{2},L_{2})$ we can improve the smoothness of $\mathfrak{c},\mathfrak{a}$ and give a uniform control in time of these functions. This shall make a crucial use of the fact that the speed $\mathfrak{c}^{*}:=(c_{1}^{*},...,c_{N}^{*})\in\mathrm{Adm}_{N}$ is ordered. In this sense, we define

\sigma^{*}:=\min\left\{c_{k+1}^{*}-c_{k}^{*}\ \big{|}\ k\in\{1,...,N-1\}\right\}>0.

Proposition 1.14.

There exists $(\alpha_{3},L_{3})\prec(\alpha_{2},L_{2})$ such that if $(\alpha,L)\prec(\alpha_{3},L_{3})$ , then $(\mathfrak{c},\mathfrak{a})\in\mathcal{C}^{1}([0,T],\mathrm{Adm}_{N}\times\mathbb{R}^{N})$ and we have for any $t\in[0,T]$

\left|\mathfrak{a}^{\prime}(t)-\mathfrak{c}(t)\right|+\left|\mathfrak{c}^{\prime}(t)\right|=\mathcal{O}(\left\|\varepsilon(t)\right\|_{\mathcal{X}})+\mathcal{O}\big{(}Le^{-\frac{a_{d}\nu_{\mathfrak{c}^{*}}L}{2}}\big{)}.

(36)

Moreover, we also have for any $t\in[0,T]$ and $k\in\{1,...,N-1\}$ ,

a_{k+1}(t)-a_{k}(t)\geq a_{k+1}(0)-a_{k}(0)+\sigma^{*}t>L-1+\sigma^{*}t,

(37)

and for any $k\in\{1,...,N\}$ ,

\sqrt{c_{s}^{2}-\big{(}a_{k}^{\prime}(t)\big{)}^{2}}\geq\dfrac{\nu_{\mathfrak{c}^{*}}}{2}.

(38)

Another crucial ingredient in the proof is a monotonicity formula for the momentum. More precisely, we consider now for $k\in\{1,...,N\}$ , the quantity

\widetilde{p}_{k}(Q):=\int_{\mathbb{R}}\pi(Q)\chi_{k},

and $\widetilde{p}_{k}(t):=\widetilde{p}_{k}\big{(}Q(t)\big{)}$ .

Remark 1.15.

Whereas $p_{k}(Q)$ could be read as the amount of momentum provided by the single soliton $Q_{c_{k}}$ , $\widetilde{p}_{k}(Q)$ approximates the amount of momentum provided by the sum of solitons $Q_{c_{k},a_{k}},...,Q_{c_{N},a_{N}}$ . This claim is all the more consistent from the property that

\widetilde{p}_{k}(Q)=\sum_{j=k}^{N}p_{j}(Q).

In particular, $\widetilde{p}_{1}(Q)=p(Q)$ .

Remark 1.16.

Since the modulation parameter $\mathfrak{a}$ now depends on time, so do implicitly the functions $\chi_{k}$ for any $k\in\{1,...,N\}$ .

The conservation of the momentum implies that $\widetilde{p}_{1}$ is conserved. Regarding $\widetilde{p}_{k}$ for $k\geq 2$ , and provided again that $c_{1}^{*}<...<c^{*}_{N}$ , we can prove the following monotonicity result.

Proposition 1.17.

There exists $(\alpha_{4},L_{4})\prec(\alpha_{3},L_{3})$ such that, if $(\alpha,L)\prec(\alpha_{4},L_{4})$ , $\widetilde{p}_{k}$ is differentiable and we have for any $t\in[0,T]$ and $k\in\{1,...,N\}$ ,

-\dfrac{d}{dt}\big{(}\widetilde{p}_{k}(t)\big{)}\leq\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}(L-1+\sigma^{*}t)}\Big{)},

(39)

and

\dfrac{d}{dt}\big{(}\mathcal{G}(t)\big{)}\leq\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}(L-1+\sigma^{*}t)}\Big{)}.

(40)

1.3.5 Proof of the orbital stability

Figure 1: Illustration of the continuation argument of Theorem 1.5. Here we represent the sets of the form

\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L)

as cylinders with smaller size when

\alpha

and

\frac{1}{L}

are smaller. The black curve represents a solution

Q(t)

with initial condition in

\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{5},L_{0}-2)

, this solution must leave the neighborhood at some maximal time

T^{*}

. However, taking

(\alpha_{*},L_{*})

such that the effects of the perturbative method are well controlled (see Proposition 1.18), we manage to extend the solution

Q(t)

for a larger time

T^{*}+\theta

and such that the latter still lies in the initial neighborhood.

To conclude as for the orbital stability, we are going to use the control of the perturbation $\varepsilon(t)$ in terms of the distance between each solitons at time $t\in[0,T]$ , the initial perturbation $\varepsilon(0)$ and $\left|\mathfrak{c}(0)-\mathfrak{c}^{*}\right|$ . That is the aim of the following proposition, the proof of which is at the end of Section 5.

Proposition 1.18.

There exists $(\alpha_{5},L_{5})\prec(\alpha_{4},L_{4})$ such that if $(\alpha,L)\prec(\alpha_{5},L_{5})$ , we have for any $t\in[0,T]$ ,

\mathfrak{a}(t)\in\mathrm{Pos}_{N}(L-1),

(41)

\left|\mathfrak{c}(t)-\mathfrak{c}^{*}\right|=\mathcal{O}\left(\left\|\varepsilon(0)\right\|_{\mathcal{X}}^{2}\right)+\left|\mathfrak{c}(0)-\mathfrak{c}^{*}\right|+\mathcal{O}\left(\left\|\varepsilon(t)\right\|_{\mathcal{X}}^{2}\right)+\mathcal{O}\left(Le^{-a_{d}\tau_{0}L}\right),

(42)

\left\|\varepsilon(t)\right\|_{\mathcal{X}}^{2}\leq\mathcal{O}(\left\|\varepsilon(0)\right\|_{\mathcal{X}}^{2})+\left|\mathfrak{c}(0)-\mathfrak{c}^{*}\right|+\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}L}\Big{)}.

(43)

We finally prove the main result, by a continuation argument. An illustration is displayed in Figure 1 to describe what the argument consists on.

Proof of Theorem 1.5.

Consider an initial datum $Q_{0}$ such that $\left\|Q_{0}-R_{\mathfrak{c}^{*},\mathfrak{a}^{0}}\right\|_{\mathcal{X}}:=\alpha_{0}$ for some $\mathfrak{a}^{0}\in\mathrm{Pos}_{N}(L_{0})$ with $(\alpha_{0},L_{0})\prec(\alpha_{*},L_{*})$ and $(\alpha_{*},L_{*})\prec(\frac{\alpha_{5}}{2},L_{5}+2)$ to be precised later. Let $Q(t)$ be the corresponding solution locally existing on some interval $[0,T]$ such that for any $t\in[0,T]$ ,

Q(t)\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{5},L_{0}-2).

Since $(\alpha_{5},L_{0}-2)\prec(\alpha_{i},L_{i})$ for any $i\in\{1,...,5\}$ , we can exhibit $\mathfrak{a}(t),\mathfrak{c}(t),\varepsilon(t)$ such that all the estimates in the results in Section 1.3.4 hold for any $t\in[0,T]$ . Then we set

I:=\left\{T\geq 0\big{|}\ \forall t\in[0,T],Q(t)\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{5},L_{0}-2)\right\}\quad\text{and}\quad T^{*}:=\sup I.

Suppose by contradiction that $T^{*}<+\infty$ . By (30), we have $0\leq T^{*}\leq T_{\max}$ . In addition, $T^{*}>0$ , by continuity of the flow. Now we proceed in two steps, first we show that $T^{*}<T_{\max}$ and then that there exists $\theta>0$ such that $0<T^{*}+\theta<T_{\max}$ and $T^{*}+\theta\in I$ .
Step 1. By contradiction, if $T^{*}=T_{\max}$ in case $T_{\max}<+\infty$ , $\left\|\eta(t)\right\|_{L^{\infty}}$ would tend to 1 as $t$ tends to $T^{*}=T_{\max}$ by local well-posedness. However, for any $t\in[0,T^{*})$ , there exists $\mathfrak{a}^{t}\in\mathrm{Pos}_{N}(L_{0}-2)$ such that $\left\|Q(t)-R_{\mathfrak{c}^{*},\mathfrak{a}^{t}}\right\|_{\mathcal{X}}<\alpha_{5}$ . Thus, taking possibly a smaller $\alpha_{5}$ to deal with the best multiplying constant in the one-dimensional Sobolev embedding, we can suppose that $\left\|\eta(t)-\eta_{\mathfrak{c}^{*},\mathfrak{a}^{t}}\right\|_{L^{\infty}}<\alpha_{5}$ . Since $L_{0}-2\geq L_{5}$ , Lemma 1.11 yields

\displaystyle\left\|\eta(t)\right\|_{L^{\infty}}\leq\alpha_{5}+\beta^{*},

and we obtain a contradiction by passing to the limit $t\rightarrow T^{*}$ .
Step 2. We use (42) and (43) in Proposition 1.18 to obtain

$\displaystyle\left\\|Q(t)-R_{\mathfrak{c}^{*},\mathfrak{a}(t)}\right\\|_{\mathcal{X}}$	$\displaystyle\leq\left\\|\varepsilon(t)\right\\|_{\mathcal{X}}+\left\\|R_{\mathfrak{c}(t),\mathfrak{a}(t)}-R_{\mathfrak{c}^{*},\mathfrak{a}(t)}\right\\|_{\mathcal{X}}$
	$\displaystyle\leq\mathcal{O}\left(\left\\|\varepsilon(0)\right\\|_{\mathcal{X}}+\left\|\mathfrak{c}(0)-\mathfrak{c}^{}\right\|+L_{0}e^{-a_{d}\tau_{0}L_{0}}+\big{\|}\mathfrak{c}(t)-\mathfrak{c}^{}\big{\|}\right)$
	$\displaystyle\leq K_{6}\left(\alpha_{0}+e^{-a_{d}\frac{\tau_{0}}{2}L_{0}}\right).$	(44)

By Proposition 1.14 and the choice of $\tau_{0}$ before Corollary 1.13, we deduce

\left|\mathfrak{a}^{\prime}(t)-\mathfrak{c}(t)\right|+\left|\mathfrak{c}^{\prime}(t)\right|\leq K_{7}\left(\alpha_{0}+e^{-a_{d}\frac{\tau_{0}}{2}L_{0}}\right),

(45)

for any $t\in[0,T^{*})$ . Using successively (41) and then (44), we infer that there exists $t^{n}\rightarrow T^{*}$ with $t^{n}<T^{*}$ such that for any $n$ ,

	$\displaystyle\inf_{\mathfrak{a}\in\mathrm{Pos}_{N}(L_{0}-2)}\left\\|Q(T^{})-R_{\mathfrak{c}^{},\mathfrak{a}}\right\\|_{\mathcal{X}}$	$\displaystyle\leq\left\\|Q(T^{})-R_{\mathfrak{c}^{},\mathfrak{a}(t^{n})}\right\\|_{\mathcal{X}}$
		$\displaystyle\leq\left\\|Q(T^{})-Q(t^{n})\right\\|_{\mathcal{X}}+\left\\|Q(t^{n})-R_{\mathfrak{c}^{},\mathfrak{a}(t^{n})}\right\\|_{\mathcal{X}}$
		$\displaystyle\leq\left\\|Q(T^{*})-Q(t^{n})\right\\|_{\mathcal{X}}+K_{6}\left(\alpha_{0}+e^{-a_{d}\frac{\tau_{0}}{2}L_{0}}\right).$

Now passing to the limit $n\rightarrow+\infty$ , and imposing additionally

K_{6}\left(\alpha_{*}+e^{-a_{d}\frac{\tau_{0}}{2}L_{*}}\right)<\dfrac{\alpha_{5}}{2},

we obtain that $Q(T^{*})\in\mathcal{U}_{\mathfrak{c}^{*}}\left(\frac{\alpha_{5}}{2},L_{0}-2\right)$ . By local wellposedness, we show the same way that there exists $\theta>0$ such that $T^{*}+\theta<T_{\max}$ and $Q(T^{*}+\theta)\in\mathcal{U}_{\mathfrak{c}^{*}}\left(\alpha_{5},L_{0}-2\right)$ which contradicts the maximality of $T^{*}$ . We have shown that $T^{*}=+\infty$ , thus the solution $Q(t)$ is globally defined and the estimates (44), (45) hold for any $t\in\mathbb{R}_{+}$ . We conclude by taking $\tau_{*}=\frac{a_{d}\tau_{0}}{2}$ and $A^{*}$ larger than both $K_{6}$ and $K_{7}$ . ∎

2 Existence of a branch of smooth travelling waves

To prove Theorem 1.2, we proceed in three steps. First we cite D. Chiron’s result that provides a unique local branch of solutions of ( $TW_{c}$ ) in the transonic limit.

Theorem 2.1 (Theorem 4 in [5]).

Let $f\in\mathcal{C}^{3}(\mathbb{R}_{+})$ and assume that (H3) holds. There exist $\delta>0$ and $0<c_{0}<c_{s}$ such that for any $c\in(c_{0},c_{s})$ , there exists a solution $\mathfrak{v}_{c}$ to ( $TW_{c}$ ) such that $\left\|1-|\mathfrak{v}_{c}|\right\|_{L^{\infty}}\leq\delta$ . Morevover, this solution is unique up to a translation and a constant phase shift of modulus one. Furthermore, we have the asymptotic estimate

p(\mathfrak{v}_{c})\underset{c\rightarrow c_{s}}{\sim}\dfrac{6c_{s}\nu_{c}^{3}}{k^{2}},

(2.1)

where $k:=2f^{\prime\prime}(1)+6f^{\prime}(1)$ .

In Subsection 2.1, we provide a slight improvement of the previous theorem, showing that the branch $c\mapsto Q_{c}$ is actually smooth. Once we know that there exists locally a unique smooth hydrodynamical travelling wave for $c$ close to $c_{s}$ , we show that it decays exponentially. Using this decay at infinity, we eventually improve the smoothness of the map $c\mapsto Q_{c}$ to the space $\mathcal{C}^{2}\big{(}(c_{0},c_{s}),\mathcal{NX}^{2}(\mathbb{R})\big{)}$ .

2.1 A priori existence of a branch of travelling waves

In our framework, $f\in\mathcal{C}^{3}(\mathbb{R}_{+})$ , then by Theorem 2.1, we exhibit $c_{0}$ such that for $c\in(c_{0},c_{s})$ there exists a unique solution $\mathfrak{v}_{c}$ to ( $TW_{c}$ ), up to a translation and a constant phase shift. This travelling wave lies in the energy set and does not vanish. As a consequence, we can construct the corresponding hydrodynamical travelling wave, unique up to translations, which belongs to $\mathcal{NX}_{hy}(\mathbb{R})$ . As explained in [5], we can freeze the invariance by translation such that the resulting hydrodynamical travelling wave $(\eta_{c},v_{c})$ is even. For $c\in(c_{0},c_{s})$ , we set $X(c,Q):=\nabla E(Q)-c\nabla p(Q)$ . We can check from (4) that $X$ is a $\mathcal{C}^{3}$ function (because $f\in\mathcal{C}^{3}(\mathbb{R}_{+})$ ) on $\mathbb{R}\times\left\{(\eta,v)\in H^{2}(\mathbb{R})\times L^{2}(\mathbb{R})\big{|}\eta,v\text{ are even and }\max_{\mathbb{R}}\eta<1\right\}$ the subset of the Banach space $\mathbb{R}\times H^{2}\times L^{2}$ . By definition, we deduce $\partial_{2}X(c,Q_{c}).\varepsilon=\mathcal{H}_{c}(\varepsilon)$ . Since $\ker(\mathcal{H}_{c})=\mathrm{Span}(\partial_{x}Q_{c})$ is spanned by an odd function, then $\ker\big{(}\partial_{2}X(c,Q_{c})\big{)}=\{0\}$ . Since $\mathcal{H}_{c}$ is self-adjoint, we infer from the Fredholm alternative that $\partial_{2}X(c,Q_{c})$ is in fact invertible. Thus, by the implicit function theorem, there exists a neighborhood of $c$ on which $X(c,Q_{c})=0$ owns a unique solution. By uniqueness of the solution (up to translation) of ( $TW_{c,hy}$ ) at fixed speed, this solution is $Q_{c}$ and the map $c\mapsto Q_{c}\in\mathcal{C}^{3}\big{(}(c_{0},c_{s}),H^{2}(\mathbb{R})\times L^{2}(\mathbb{R})\big{)}$ . Moreover $\eta_{c}$ and $v_{c}$ satisfy

-\partial_{x}^{2}\eta_{c}=\dfrac{1}{2}\mathcal{N}_{c}^{\prime}(\eta_{c}),

(2.2)

where

\mathcal{N}_{c}(x)=c^{2}x^{2}-4(1-x)F(1-x),

(2.3)

and

v_{c}=\dfrac{c\eta_{c}}{2(1-\eta_{c})}.

(2.4)

By standard arguments on smooth dependence (with respect to $c$ ) for ODEs depending on a parameter, the branch is $\mathcal{C}^{3}$ on $(c_{0},c_{s})$ with values in $\big{(}\mathcal{C}_{loc}^{5}(\mathbb{R})\big{)}^{2}$ (see for instance [15]).

2.2 Decay at infinity

By Subsection 2.1, there exists a unique branch of solution to ( $TW_{c,hy}$ ) lying in the set

\mathcal{I}:=\mathcal{C}^{3}\Big{(}(c_{0},c_{s}),\big{(}H^{2}(\mathbb{R})\times L^{2}(\mathbb{R})\big{)}\cap\big{(}\mathcal{C}_{loc}^{3}(\mathbb{R})\times\mathcal{C}_{loc}^{2}(\mathbb{R})\big{)}\Big{)}.

(2.5)

The exponents in the definition of $\mathcal{I}$ are set in accordance with establishing (10). We state additional equations that are also satisfied by $(\eta_{c},v_{c})\in\mathcal{NX}(\mathbb{R})$ in order to study the exponential decay. Integrating (2.2), we obtain

-(\partial_{x}\eta_{c})^{2}=\mathcal{N}_{c}(\eta_{c}).

(2.6)

Moreover, note that

\mathcal{N}_{c_{s}}(x)=c_{s}^{2}x^{2}-4(1-x)F(1-x)=\mathcal{O}(x^{3}).

(2.7)

We use a result established in [12] (see also [5]) that gives a sufficient and necessary criterion for the existence and uniqueness of a solution satisfying (2.2). It essentially adapts a general result concerning ordinary differential equations due to H. Berestycki and P.-L. Lions (see Theorem 5 in [2]).

Proposition 2.2 ([12],[5]).

Let $c\in(c_{0},c_{s})$ . There exists a unique (up to translations) non constant solution $\eta_{c}$ of (2.2) vanishing at infinity if and only if there exists $\xi_{c}\in(0,1)$ such that $\mathcal{N}_{c}(\xi_{c})=0$ , $\mathcal{N}_{c}(x)<0$ on $(0,\xi_{c})\text{ and }\mathcal{N}^{\prime}_{c}(\xi_{c})>0$ . In that case, $\eta_{c}$ is even, reaches its maximum in $0$ and $\eta_{c}(0)=\xi_{c}$ .

Remark 2.3.

The zero $\xi_{c}$ can be supposed minimal for this property, so we can see later that $\xi_{c}$ tends to 0 as $c$ tends to $c_{s}$ .

We first give a bound from below (equal to $c/c_{s}$ in the Gross-Pitaevskii case) on $|\mathfrak{v}_{c}|$ .

Proposition 2.4.

There exists $\delta\in(0,1)$ , independent of $c\in(c_{0},c_{s})$ , such that

\min_{\mathbb{R}}|\mathfrak{v}_{c}|=|\mathfrak{v}_{c}(0)|=\sqrt{1-\eta_{c}(0)}\geq\delta\dfrac{c}{c_{s}}.

In terms of the hydrodynamic variables, this reads

\max_{\mathbb{R}}\eta_{c}=\eta_{c}(0)=\xi_{c}\leq 1-\delta^{2}\dfrac{c^{2}}{c_{s}^{2}}.

(2.8)

Proof.

The function $\eta_{c}$ achieves its maximum in $0$ (up to a translation). Furthermore, by (2.6), we have $0=\mathcal{N}_{c}\big{(}\eta_{c}(0)\big{)}$ , i.e. $0=c^{2}\eta_{c}(0)^{2}-4\big{(}1-\eta_{c}(0)\big{)}F\big{(}1-\eta_{c}(0)\big{)}$ , and using a second order Taylor’s formula with remainder, we obtain

\dfrac{c^{2}}{c_{s}^{2}}=(1-\eta_{c}(0))\Big{(}1+\dfrac{\eta_{c}(0)}{2c_{s}^{2}}\int_{0}^{1}(1-t)f^{\prime\prime}\big{(}1-t\eta_{c}(0)\big{)}dt\Big{)}.

(2.9)

In particular, we get

\dfrac{c^{2}}{c_{s}^{2}}\dfrac{1}{1+\|f^{\prime\prime}\|_{L^{\infty}([0,1])}/4c_{s}^{2}}\leq|\mathfrak{v}_{c}(0)|^{2},

which provides the estimates in Proposition 2.4. ∎

Remark 2.5.

In order to simplify the notations, we occasionally write $f_{1}(x)\lesssim f_{2}(x)$ that stands for $f_{1}(x)=\mathcal{O}\big{(}f_{2}(x)\big{)}$ .

Lemma 2.6.

We have $\eta_{c}\in\mathcal{C}^{3}(\mathbb{R})\cap H^{3}(\mathbb{R})$ and $v_{c}\in\mathcal{C}^{2}(\mathbb{R})\cap H^{2}(\mathbb{R})$ , and

\sum_{\begin{subarray}{c}0\leq k_{1}\leq 3\\ 0\leq k_{2}\leq 2\end{subarray}}|\partial_{x}^{k_{1}}\eta_{c}(x)|+c^{1+2k_{2}}|\partial_{x}^{k_{2}}v_{c}(x)|\lesssim e^{-\nu_{c}|x|}.

(2.10)

Proof.

Step 1: Exponential decay of $\eta_{c}$ . The exponential decay of $\eta_{c}$ is proved in [6] and is stated as follows,

|\eta_{c}(x)|\lesssim M_{c}e^{-\nu_{c}|x|},

where

M_{c}=\xi_{c}e^{\int^{\xi_{c}}_{0}\frac{\mathcal{N}_{c_{s}}(\xi)}{\sqrt{-\xi^{2}\mathcal{N}_{c}(\xi)}\big{(}\sqrt{-\mathcal{N}_{c}(\xi)}+\sqrt{\nu_{c}^{2}\xi^{2}}\big{)}}d\xi}.

(2.11)

We need to improve the previous estimate. In order to do that, we shall handle the dependence in $c$ in the constant $M_{c}$ . We first prove the following intermediate result. We define $\widetilde{k}:=-\frac{3}{k}$ with $k$ defined in Theorem 2.1.

Claim 2.7.

We have

\xi_{c}\underset{c\rightarrow c_{s}}{=}\widetilde{k}\nu_{c}^{2}+\mathcal{O}(\nu_{c}^{4}).

(2.12)

Moreover,

\partial_{c}\xi_{c}=-\dfrac{2c\xi_{c}^{2}}{\mathcal{N}_{c}^{\prime}(\xi_{c})}\underset{c\rightarrow c_{s}}{=}-2c_{s}\widetilde{k}+\mathcal{O}(\nu_{c}^{2}),

(2.13)

and

\partial_{c}^{2}\xi_{c}\underset{c\rightarrow c_{s}}{\sim}-2\widetilde{k}.

Proof.

We show that the first positive zero $\xi_{c}\in(0,1)$ satisfies $\xi_{c}=-3\nu_{c}^{2}/k+\mathcal{O}(\nu_{c}^{3})$ . We first write a Taylor expansion of $\mathcal{N}_{c}$ at $0$ :

\mathcal{N}_{c}(\xi)\underset{\xi\rightarrow 0}{=}-\xi^{2}\big{(}\nu_{c}^{2}+\dfrac{k}{3}\xi+\mathcal{O}(\xi^{2})\big{)}.

(2.14)

Now we show that $\xi_{c}$ strictly decreases to $0$ as $c\rightarrow c_{s}$ , so we can plug $\xi_{c}\neq 0$ into (2.14) and get $0\underset{c\rightarrow c_{s}}{=}\nu_{c}^{2}+\dfrac{k}{3}\xi_{c}+\mathcal{O}(\xi_{c}^{2})$ . We would then deduce simultaneously that $\xi_{c}=\mathcal{O}(\nu_{c}^{2})$ and (2.12). To do this, we consider $c\mapsto\xi_{c}=\eta_{c}(0)$ which is smooth (see Subsection 2.1), and differentiating $\mathcal{N}_{c}(\xi_{c})=0$ leads to the first identity in (2.13). Since $\mathcal{N}_{c}^{\prime}(\xi_{c})>0$ , this means that $\xi_{c}$ strictly decreases with respect to $c$ . Moreover it lies in $(0,1)$ for any $c$ , so that there exists $\xi_{*}<1$ such that

\xi_{c}\underset{c\rightarrow c_{s}}{\longrightarrow}\xi_{*}.

We assume by contradiction that $\xi_{*}\neq 0$ . Passing to the limit $c\rightarrow c_{s}$ into $\mathcal{N}_{c}(\xi_{c})=0$ and using (H1), we obtain $c_{s}^{2}\xi_{*}^{2}\geq c_{s}^{2}(1-\xi_{*})\xi_{*}^{2}$ , then $\xi_{*}\geq 1$ . Thus we must have $\xi_{*}=0$ . To conclude the proof, we differentiate $\partial_{c}\xi_{c}$ with respect to $c$ and use (2.12) and (2.13). ∎

We can control the integral in (2.11), by using successively (2.7), Proposition 2.2, the substitution $\xi=\nu_{c}^{2}\eta$ and (2.14),

	$\displaystyle\bigg{\|}\int^{\xi_{c}}_{0}\dfrac{\mathcal{N}_{c_{s}}(\xi)}{\sqrt{-\xi^{2}\mathcal{N}_{c}(\xi)}(\sqrt{\mathcal{-N}_{c}(\xi)}+\sqrt{\nu_{c}^{2}\xi^{2}})}d\xi\bigg{\|}$	$\displaystyle\leq\int_{0}^{\xi_{c}}\dfrac{\|\mathcal{N}_{c_{s}}(\xi)\|}{\sqrt{-\xi^{4}\nu_{c}^{2}\mathcal{N}_{c}(\xi)}}d\xi\lesssim\int_{0}^{\xi_{c}}\dfrac{\xi}{\nu_{c}\sqrt{-\mathcal{N}_{c}(\xi)}}d\xi$
		$\displaystyle=\nu_{c}^{3}\int_{0}^{\frac{\xi_{c}}{\nu_{c}^{2}}}\dfrac{\eta d\eta}{\sqrt{-\mathcal{N}_{c}(\nu_{c}^{2}\eta)}}\underset{c\rightarrow c_{s}}{\sim}\int_{0}^{\widetilde{k}}\dfrac{d\eta}{\sqrt{1-\frac{\eta}{\widetilde{k}}}}=2\widetilde{k},$

where $\widetilde{k}=-\frac{3}{k}>0$ . Therefore we conclude from the previous claim that,

|\eta_{c}(x)|\lesssim\nu_{c}^{2}e^{-\nu_{c}|x|}.

(2.15)

Step 2: Exponential decay of the remaining terms. In view of (2.3), $\mathcal{N}^{\prime}_{c}\in\mathcal{C}^{3}((-\infty,1])$ so it is continuous. Since $\eta_{c}\in H^{1}(\mathbb{R})$ , by using the one-dimensional Sobolev embedding and (2.2), we obtain that $\eta_{c}\in\mathcal{C}^{2}(\mathbb{R})$ . In particular, by a standard bootstrap argument $\partial_{x}^{2}\eta_{c}\in\mathcal{C}^{3}(\mathbb{R})$ i.e. $\eta_{c}\in\mathcal{C}^{5}(\mathbb{R})$ . Now we deal with the exponential decay. Using (2.6) and (2.7), we get $|\partial_{x}\eta_{c}|\leq\nu_{c}|\eta_{c}|+\mathcal{O}\left(|\eta_{c}|^{\frac{3}{2}}\right)\lesssim|\eta_{c}|$ . Regarding the other derivatives, we write a Taylor expansion of $\mathcal{N}_{c}^{\prime}$ , so that we obtain

-\partial_{x}^{2}\eta_{c}=\dfrac{\eta_{c}}{2}\int_{0}^{1}\mathcal{N}_{c}^{\prime\prime}(t\eta_{c})dt.

(2.16)

By (2.15), $\eta_{c}$ is uniformly bounded with respect to $c\in(0,c_{s})$ and $x\in\mathbb{R}$ . In view of

\mathcal{N}_{c}^{\prime\prime}(t\eta_{c})=2c^{2}+8f(1-t\eta_{c})+4(1-t\eta_{c})f^{\prime}(1-t\eta_{c}),

and since $f^{\prime}$ is continuous, the integral in (2.16) is uniformly bounded with respect to $x\in\mathbb{R}$ and $c\in(0,c_{s})$ . Therefore, we have shown $|\partial_{x}^{2}\eta_{c}|\lesssim|\eta_{c}|$ and thus the exponential decay of $\partial_{x}^{2}\eta_{c}$ . We can differentiate (2.2) with respect to $x$ and with a bootstrap argument, we control $\partial^{3}_{x}\eta_{c}$ and $\partial_{x}^{4}\eta_{c}$ in terms of $|\eta_{c}|$ . Regarding the derivatives of $v_{c}$ , by (2.4) and Proposition 2.4, we have $c|v_{c}|\lesssim|\eta_{c}|$ and

\partial_{x}v_{c}=\dfrac{c}{2}\dfrac{\partial_{x}\eta_{c}}{(1-\eta_{c})^{2}},

so that $c^{3}|\partial_{x}v_{c}|\lesssim|\partial_{x}\eta_{c}|$ . By induction, we show that $c^{1+2k}|\partial_{x}^{k}v_{c}|\lesssim|\eta_{c}|$ , for $k\in\{0,...,2\}$ . ∎

Now we deal with the derivatives of $\eta_{c}$ and $v_{c}$ with respect to $c$ . We obtain estimates similar to $(2.9)$ in [4].

Lemma 2.8.

There exists $K_{d}>0$ and $a_{d}>0$ independent of $c$ and $x$ such that

\sum_{\begin{subarray}{c}0\leq k_{1}\leq 3\\ 0\leq k_{2}\leq 2\\ 1\leq j\leq 3\end{subarray}}\nu_{c}^{2(j-1)}\Big{(}|\partial_{c}^{j}\partial_{x}^{k_{1}}\eta_{c}(x)|+c^{1+2j+2k_{2}}|\partial_{c}^{j}\partial_{x}^{k_{2}}v_{c}(x)|\Big{)}\leq K_{d}e^{-a_{d}\nu_{c}|x|}.

(2.17)

Proof.

First, we prove

|\partial_{c}\eta_{c}|+c^{3}|\partial_{c}v_{c}|\lesssim e^{-\frac{\nu_{c}x}{2}}.

We compute

\partial_{c}v_{c}=\dfrac{v_{c}}{c}+\dfrac{c}{2}\dfrac{\partial_{c}\eta_{c}}{(1-\eta_{c})^{2}},

then by Proposition 2.4, we infer that $c^{3}\partial_{c}v_{c}$ enjoys the same decay than $c^{2}|v_{c}|+|\partial_{c}\eta_{c}|$ . We already handled the decay of $v_{c}$ in Lemma 2.6, so it remains to deal with $\partial_{c}\eta_{c}$ . We consider $K_{c}(x):=(\partial_{c}\eta_{c}(x))^{2}$ . By (2.2), we obtain

K_{c}^{\prime\prime}=2(\partial_{c}\partial_{x}\eta_{c})^{2}+2\partial_{c}\eta_{c}\partial_{c}\partial_{x}^{2}\eta_{c}\geq-K_{c}\mathcal{N}^{\prime\prime}_{c}(\eta_{c})-4c\eta_{c}\partial_{c}\eta_{c}.

Since $\eta_{c}(x)$ tends to $0$ as $x$ tends to $\pm\infty$ , we have

-\mathcal{N}^{\prime\prime}_{c}(\eta_{c})\underset{|x|\rightarrow+\infty}{\longrightarrow}2\nu_{c}^{2}.

Then, there exists $R_{*}>0$ such that for any $|x|\geq R_{*}$ , $\nu_{c}^{2}\leq-\mathcal{N}^{\prime\prime}_{c}(\eta_{c})$ hence $K_{c}^{\prime\prime}\geq\nu_{c}^{2}K_{c}-4c\eta_{c}\partial_{c}\eta_{c}$ . Thus

K_{c}^{\prime\prime}-\nu_{c}^{2}K_{c}\geq-4c\sqrt{2}\dfrac{\eta_{c}}{\nu_{c}}\sqrt{\dfrac{1}{2}}\nu_{c}\partial_{c}\eta_{c}.

Using the exponential decay of $\eta_{c}$ in (2.15), we get

K_{c}^{\prime\prime}(x)-\dfrac{\nu_{c}^{2}}{2}K_{c}(x)\geq-\dfrac{8c^{2}\eta_{c}^{2}(x)}{\nu_{c}^{2}}\gtrsim-c^{2}\nu_{c}^{2}e^{-2\nu_{c}|x|},

hence there exists a constant $C>0$ such that

-K_{c}^{\prime\prime}(x)+\dfrac{\nu_{c}^{2}}{2}K_{c}(x)\leq C\nu_{c}^{2}e^{-\frac{\nu_{c}}{\sqrt{2}}|x|}\leq C\nu_{c}^{2}e^{-\frac{\nu_{c}}{2\sqrt{2}}|x|}.

We introduce a general lemma for the exponential decay of solutions to elliptic differential inequalities, the proof of which is at the end of this subsection.

Lemma 2.9.

Let $g$ be a non negative function in $\mathcal{C}^{2}(\mathbb{R})$ . Set $\omega>0$ and $0<a<1$ and assume that there exists $C>0$ such that for any $x\in\mathbb{R}$ , we have $-g^{\prime\prime}(x)+\omega^{2}g(x)\leq C\omega^{2}e^{-a\omega|x|}$ and $e^{-2\omega|x|}\big{(}g(x)e^{\omega|x|}\big{)}^{\prime}$ tends to $0$ as $|x|$ tends to $+\infty$ . Then there exists $\widetilde{C}>0$ independent of $\omega$ such that for all $x\in\mathbb{R}$ ,

g(x)\leq\widetilde{C}e^{-a\omega|x|},

with $\widetilde{C}=g(0)+\frac{C}{1-a^{2}}$

We apply this lemma with $a=1/2$ , $\omega(c):=\frac{\nu_{c}}{\sqrt{2}}$ and $g=K_{c}$ . It remains to check that

e^{-2\omega|x|}\big{(}K_{c}(x)e^{\omega|x|}\big{)}^{\prime}\underset{|x|\rightarrow+\infty}{\longrightarrow}0.

Indeed, at fixed speed $c\in(c_{0},c_{s})$ , $\partial_{c}\eta_{c},\partial_{c}\partial_{x}\eta_{c}\in H^{1}(\mathbb{R})$ (the branch of solitons lies in the set defined in (2.5)) so that $K_{c}$ and $K_{c}^{\prime}$ tend to $0$ as $|x|$ tend to $+\infty$ . Finally using (2.13), we infer that $\big{(}\partial_{c}\eta_{c}(0)\big{)}^{2}$ is bounded by a constant that does not depend on $c$ . Therefore, there exists $\widetilde{C}>0$ independent of $c$ and $x$ such that

K_{c}(x)\leq\widetilde{C}e^{-\frac{\nu_{c}}{2\sqrt{2}}|x|},

which provides the exponential decay of $\partial_{c}\eta_{c}$ .

Now we deal with the coupled derivative $\partial_{c}\partial_{x}\eta_{c}$ . We deduce the exponential decay of $\partial_{c}\partial_{x}^{2}\eta_{c}$ from differentiating (2.2) with respect to $c$ and using the exponential decay of $\partial_{c}\eta_{c}$ and $\eta_{c}$ . The exponential decay of $\partial_{c}\partial_{x}\eta_{c}$ can be then derived from integrating $\partial_{c}\partial_{x}^{2}\eta_{c}$ between $x>0$ and $+\infty$ (and between $-\infty$ and $x<0$ ), since $\partial_{c}\partial_{x}\eta_{c}\in H^{1}(\mathbb{R})$ .

Next we set $L_{c}:=(\partial_{c}^{2}\eta_{c})^{2}$ . Similarly to $K_{c}$ , we differentiate (2.2) twice with respect to $c$ , and using the exponential decay of $\eta_{c}$ and $\partial_{c}\eta_{c}$ , we deduce that there exists $a_{d}\in(0,1)$ , $R_{*}>0$ and $C>0$ such that for $|x|\geq R_{*}$ ,

-L_{c}^{\prime\prime}+\nu_{c}^{2}L_{c}\leq Ce^{-a_{d}\nu_{c}|x|}.

We verify that the assumptions of Lemma 2.9 are satisfied, and thus we have the bound, up to taking a larger constant $C>0$ ,

|\partial_{c}^{2}\eta_{c}(x)|\leq\dfrac{C}{\nu_{c}^{2}}e^{-a_{d}\nu_{c}|x|},

where $C$ does not depend on $c$ because of the asymptotics for $\partial_{c}^{2}\xi_{c}$ in Claim 2.7. Since $f\in\mathcal{C}^{3}(\mathbb{R}_{+})$ , we can repeat the same type of arguments, and deduce the exponential decay of the remaining derivatives $\partial_{c}^{j}\partial_{x}^{3}\eta$ when $j\in\{0,...,2\}$ and $\partial_{c}^{3}\partial_{x}^{k}\eta_{c}$ when $k\in\{0,...,3\}$ . We conclude the proof by deriving the decay of the derivatives $\partial^{j}_{c}\partial_{x}^{k}v_{c}$ from the formulae (2.4). ∎

We finish this section by the proof of Lemma 2.9.

Proof.

By hypothesis, we have for $x\geq 0$ ,

-\Big{(}\big{(}g(x)e^{\omega x})^{\prime}e^{-2\omega x}\Big{)}^{\prime}e^{\omega x}=-g^{\prime\prime}(x)+\omega^{2}g(x)\leq C\omega^{2}e^{-a\omega x}.

Integrating this equation between $y$ positive and $+\infty$ provides, by using the limit in the hypotheses,

\big{(}g(y)e^{\omega y}\big{)}^{\prime}\leq\dfrac{C\omega}{a+1}e^{-(a-1)\omega y}.

We integrate between $0$ and $z$ positive and we get

g(z)\leq\Big{(}g(0)+\dfrac{C}{1-a^{2}}\big{(}e^{-(a-1)\omega y}-1\big{)}\Big{)}e^{-\omega z}\leq\widetilde{C}e^{-a\omega z}.

The proof is similar for $z$ negative.

∎

2.3 Proof of Theorem 1.2

Using the decay at infinity and the a priori existence of a unique branch in $\mathcal{I}$ , we now conclude and show that the branch is $\mathcal{C}^{2}\big{(}(c_{0},c_{s}),\mathcal{NX}^{2}(\mathbb{R})\big{)}$ . We proceed locally. Set $c_{1}\in(c_{0},c_{s})$ and $\delta>0$ such that $(c_{1}-\delta,c_{1}+\delta)\subset(c_{0},c_{s})$ . We show that $c\mapsto\eta_{c}\in\mathcal{C}^{0}\big{(}(c_{1}-\delta,c_{1}+\delta),H^{3}(\mathbb{R})\big{)}$ . Let $\varepsilon>0$ and $c_{2}\in(c_{1}-\delta,c_{1}+\delta)$ , we first have for all $R>0$ ,

\displaystyle\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{3}}^{2}\leq\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{3}(-R,R)}^{2}+\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{3}(\mathbb{R}\setminus[-R,R])}^{2}.

On the one hand, we have by Lemma 2.8,

	$\displaystyle\\|\eta_{c_{1}}-\eta_{c_{2}}\\|_{H^{3}(\mathbb{R}\setminus[-R,R])}^{2}$	$\displaystyle\leq\sum_{k=0}^{3}2\big{(}\\|\partial_{x}^{k}\eta_{c_{1}}\\|_{L^{2}(\mathbb{R}\setminus[-R,R])}^{2}+\\|\partial_{x}^{k}\eta_{c_{2}}\\|_{L^{2}(\mathbb{R}\setminus[-R,R])}^{2}\big{)}$
		$\displaystyle\leq 24K_{d}^{2}\\|e^{-2a_{d}\nu_{c_{1}+\delta}\|x\|}\\|_{L^{2}(\mathbb{R}\setminus[-R,R])}^{2}.$

We can find $R>0$ large enough such that $\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{5}(\mathbb{R}\setminus[-R,R])}^{2}\leq\varepsilon^{2}/2$ . On the other hand, by Lemma 2.6 with $j=1$ and $k\in\{0,...,3\}$ and doing a first order Taylor expansion with respect to $c$ , we have in particular that

\big{|}\partial_{x}^{k}\eta_{c_{1}}(x)-\partial_{x}^{k}\eta_{c_{2}}(x)\big{|}=\left|\int_{c_{1}}^{c_{2}}\partial_{c}\partial_{x}^{k}\eta_{c}(x)dc\right|\leq K_{d}|c_{1}-c_{2}|.

Since

\displaystyle\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{3}(-R,R)}^{2}\leq\sum_{k=0}^{3}2R\|\partial_{x}^{k}\eta_{c_{1}}-\partial_{x}^{k}\eta_{c_{2}}\|_{L^{\infty}(-R,R)}^{2},

for $c_{2}$ close enough to $c_{1}$ , we have $\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{3}(-R,R)}^{2}\leq\varepsilon^{2}/2$ . In conclusion, we can find $\delta>0$ such that, if $|c_{1}-c_{2}|\leq\delta$ , then $\|\eta_{c_{1}}-\eta_{c_{2}}\|_{H^{3}}\leq\varepsilon$ . Using that we can control the third derivative with respect to $c$ in Lemma 2.8, we deal the same way for the remaining derivatives $\partial_{c}^{j}\partial_{x}^{k}\eta_{c_{1}}$ for $j\leq 2$ and $k\leq 3$ . The fact that $c\mapsto v_{c}\in\mathcal{C}^{2}\left((c_{0},c_{s}),H^{2}(\mathbb{R})\right)$ then follows from (2.4).

Finally, we prove that

\dfrac{d}{dc}\big{(}p(Q_{c})\big{)}\underset{c\rightarrow c_{s}}{\sim}\dfrac{-18c_{s}^{2}\nu_{c}}{k^{2}}.

(2.18)

As a consequence of (2.6) and Proposition 2.2, and doing the substitution $\xi=\eta_{c}(0)$ in the definition of the momentum (5), we get the following formula, and the following asymptotics already known (we refer to [5]):

p(Q_{c})=\dfrac{c}{2}\int_{0}^{\xi_{c}}\dfrac{\xi^{2}}{(1-\xi)\sqrt{-\mathcal{N}_{c}(\xi)}}d\xi=\mathcal{O}(\nu_{c}^{3}).

(2.19)

Now we write

\displaystyle p(Q_{c})=\underbrace{\int_{0}^{\xi_{c}}(g-\widetilde{g})(\xi,c)d\xi}_{I_{c}}+\underbrace{\int_{0}^{\xi_{c}}\widetilde{g}(\xi,c)d\xi}_{II_{c}},

(2.20)

where

g(\xi,c)=\dfrac{c}{2}\dfrac{\xi^{2}}{(1-\xi)\sqrt{-\mathcal{N}_{c}(\xi)}}\quad\text{and}\quad\widetilde{g}(\xi,c)=\dfrac{c}{2}\dfrac{\xi_{c}^{2}}{(1-\xi_{c})\sqrt{(\xi_{c}-\xi)\mathcal{N}_{c}^{\prime}(\xi_{c})}}.

Using (2.12), we state a few asymptotics that will help us later. Recall that $\widetilde{k}:=-\frac{3}{k}$ , we have

\xi_{c}\underset{c\rightarrow c_{s}}{=}\widetilde{k}\nu_{c}^{2}+\mathcal{O}(\nu_{c}^{3}),

(2.21)

\mathcal{N}_{c}^{\prime}(\xi_{c})\underset{c\rightarrow c_{s}}{=}\widetilde{k}\nu_{c}^{4}+\mathcal{O}(\nu_{c}^{6}),

(2.22)

\mathcal{N}_{c}^{\prime\prime}(\xi_{c})\underset{c\rightarrow c_{s}}{=}4\nu_{c}^{2}+\mathcal{O}(\nu_{c}^{4}).

(2.23)

On the one hand, we compute

II_{c}=\int_{0}^{\xi_{c}}\widetilde{g}(\xi,c)d\xi=\dfrac{c\xi_{c}^{\frac{5}{2}}}{(1-\xi_{c})\sqrt{\mathcal{N}^{\prime}_{c}(\xi_{c})}}\underset{c\rightarrow c_{s}}{\sim}c_{s}\widetilde{k}^{2}\nu_{c}^{3},

(2.24)

so that

	$\displaystyle\dfrac{d}{dc}\big{(}II_{c}\big{)}=\dfrac{1}{c}II_{c}+$	$\displaystyle\dfrac{\frac{5}{2}c\xi_{c}^{\frac{3}{2}}\partial_{c}\xi_{c}}{(1-\xi_{c})\sqrt{\mathcal{N}^{\prime}_{c}(\xi_{c})}}+\dfrac{c\xi_{c}^{\frac{5}{2}}\partial_{c}\xi_{c}}{(1-\xi_{c})^{2}\sqrt{\mathcal{N}^{\prime}_{c}(\xi_{c})}}$
		$\displaystyle-\dfrac{2c^{2}\xi_{c}^{\frac{7}{2}}}{(1-\xi_{c})\big{(}\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}-\dfrac{c\xi_{c}^{\frac{5}{2}}\partial_{c}\xi_{c}\mathcal{N}_{c}^{\prime\prime}(\xi_{c})}{2(1-\xi_{c})\big{(}\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}.$

We recall that, by Claim 2.7,

\partial_{c}\xi_{c}=-\dfrac{2c\xi_{c}^{2}}{\mathcal{N}_{c}^{\prime}(\xi_{c})},

therefore we obtain the asymptotics

\dfrac{d}{dc}\big{(}II_{c}\big{)}\underset{\xi\rightarrow\xi_{c}}{=}-3c_{s}^{2}\widetilde{k}^{2}\nu_{c}+\mathcal{O}\left(\nu_{c}^{3}\right).

(2.25)

On the other hand, we notice

g(\xi,c)-\widetilde{g}(\xi,c)\underset{\xi\rightarrow\xi_{c}}{\sim}\dfrac{c\xi_{c}\sqrt{\xi_{c}-\xi}(\xi_{c}-2)}{2\sqrt{\mathcal{N}^{\prime}_{c}(\xi_{c})}},

(2.26)

so that the function $(g-\widetilde{g})(.,c)$ extends to $\xi_{c}$ by continuity. We infer, that

	$\displaystyle\dfrac{d}{dc}\big{(}I_{c}\big{)}$	$\displaystyle=\partial_{c}\xi_{c}\lim_{\xi\rightarrow\xi_{c}}(g-\widetilde{g})(\xi,c)+\int_{0}^{\xi_{c}}\partial_{c}(g-\widetilde{g})(\xi,c)d\xi$
		$\displaystyle=0+\underbrace{\int_{0}^{\xi_{c}}l(\xi,c)d\xi}_{p_{1}(c)}+\underbrace{\int_{0}^{\xi_{c}}c\partial_{c}l(\xi,c)d\xi}_{p_{2}(c)},$

with

l(\xi,c)=\dfrac{1}{c}\big{(}g(\xi,c)-g(\xi_{c},c)\big{)}=\dfrac{1}{2}\Bigg{(}\dfrac{\xi^{2}}{(1-\xi)\sqrt{-\mathcal{N}_{c}(\xi)}}-\dfrac{\xi_{c}^{2}}{(1-\xi_{c})\sqrt{(\xi_{c}-\xi)\mathcal{N}^{\prime}_{c}(\xi_{c})}}\Bigg{)}.

We have, by (2.19) and (2.24),

p_{1}(c)=\dfrac{p(Q_{c})}{c}-\dfrac{II_{c}}{c}=\mathcal{O}(\nu_{c}^{3}),

and we derive, using the expression of $\partial_{c}\xi_{c}$ , that

	$\displaystyle p_{2}(c)=\underbrace{\dfrac{c^{2}}{2}\int_{0}^{\xi_{c}}\Bigg{(}\dfrac{\xi^{4}}{(1-\xi)\big{(}-\mathcal{N}_{c}(\xi)\big{)}^{\frac{3}{2}}}-\dfrac{\xi_{c}^{4}}{(1-\xi_{c})\big{(}(\xi_{c}-\xi)\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}\Bigg{)}d\xi}_{p_{2}^{1}(c)}$
	$\displaystyle+\underbrace{c^{2}\bigg{(}\dfrac{\xi_{c}^{3}}{(1-\xi_{c})\big{(}\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}-\dfrac{\xi_{c}^{4}\mathcal{N}^{\prime\prime}_{c}(\xi_{c})}{2(1-\xi_{c})\big{(}\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{5}{2}}}+\dfrac{\xi_{c}^{3}}{\big{(}\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}\Big{(}\dfrac{2}{1-\xi_{c}}+\frac{\xi_{c}}{(1-\xi_{c})^{2}}\Big{)}\bigg{)}\int_{0}^{\xi_{c}}\dfrac{d\xi}{\sqrt{\xi_{c}-\xi}}}_{p_{2}^{2}(c)}.$

The second term reduces to

p_{2}^{2}(c)=2c_{s}^{2}\widetilde{k}^{2}\nu_{c}+\mathcal{O}(\nu_{c}^{3}).

(2.27)

On the other hand, we decompose the first one as

$\displaystyle p_{2}^{1}(c)$	$\displaystyle=\dfrac{c^{2}}{2}\int_{0}^{\xi_{c}}\Bigg{(}\dfrac{\xi^{4}}{(1-\xi)\big{(}-\mathcal{N}_{c}(\xi)\big{)}^{\frac{3}{2}}}-\dfrac{\xi^{4}}{(1-\xi_{c})\big{(}-\mathcal{N}_{c}(\xi)\big{)}^{\frac{3}{2}}}\Bigg{)}d\xi$
	$\displaystyle+\dfrac{c^{2}}{2}\int_{0}^{\xi_{c}}\Bigg{(}\dfrac{\xi^{4}}{(1-\xi_{c})\big{(}-\mathcal{N}_{c}(\xi)\big{)}^{\frac{3}{2}}}-\dfrac{\xi^{4}}{(1-\xi_{c})\big{(}(\xi_{c}-\xi)\big{(}\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}\Bigg{)}d\xi$
	$\displaystyle+\dfrac{c^{2}}{2}\int_{0}^{\xi_{c}}\Bigg{(}\dfrac{\xi^{4}}{(1-\xi_{c})\big{(}(\xi_{c}-\xi)\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}-\dfrac{\xi_{c}^{4}}{(1-\xi_{c})\big{(}(\xi_{c}-\xi)\mathcal{N}^{\prime}_{c}(\xi_{c})\big{)}^{\frac{3}{2}}}\Bigg{)}d\xi$
	$\displaystyle=-c_{s}^{2}\widetilde{k}^{2}\nu_{c}+\mathcal{O}(\nu_{c}^{3}).$	(2.28)

Now adding (2.25), (2.27) and (2.28) , we obtain, according to decomposition (2.20),

\dfrac{d}{dc}\big{(}p(\mathfrak{v}_{c})\big{)}\underset{c\rightarrow c_{s}}{\sim}-2c_{s}^{2}\nu_{c}\widetilde{k}^{2}.

(2.29)

To conclude there exists a threshold $c_{0}$ such that for $c\in(c_{0},c_{s})$ , condition (9) is fulfilled.

3 Perturbation of a sum of solitons and minimizing properties

Before we get to the proof of Proposition 1.8, we need to settle some tools that will simplify the computations. For $M\geq 2$ , we define the following linear subspace of the set of multivariate real polynomials:

\displaystyle\mathcal{P}_{M}=\mathcal{P}[X_{1},...,X_{M}]:=\Big{\{}\sum_{|\alpha|\leq m}p_{\alpha}X^{\alpha}\Big{|}m\in\mathbb{N},p_{ke_{i}}=0\ ,\forall(k,i)\in\{0,...,m\}\times\{1,...,M\}\Big{\}},

where $(e_{1},...,e_{M})$ is the canonical base of $\mathbb{R}^{M}$ . This vector space is constructed such that the monomials are not contained in it and is also stable under the usual multiplication. One shall list a few of these remarkable polynomials. For instance, the elementary symmetric polynomials are contained in $\mathcal{P}_{M}$ , and will be labelled

S^{k,M}:=\sum_{1\leq j_{1}<...<j_{k}\leq M}X_{j_{1}}...X_{j_{k}},

(3.1)

for $k\in\{1,...,M\}$ . To simplify, we will denote $P(\eta_{\mathfrak{c},\mathfrak{a}}):=P(\eta_{c_{1},a_{1}},...,\eta_{c_{N},a_{N}})$ and $P(v_{\mathfrak{c},\mathfrak{a}}):=P(v_{c_{1},a_{1}},...,v_{c_{N},a_{N}})$ , for $P\in\mathcal{P}[X_{1},...,X_{N}]$ and $P(Q_{\mathfrak{c},\mathfrak{a}}):=P(\eta_{c_{1},a_{1}},...,\eta_{c_{N},a_{N}},v_{c_{1},a_{1}},...,v_{c_{N},a_{N}})$ for $P\in\mathcal{P}[X_{1},...,X_{N},Y_{1},...,Y_{N}]$ . Furthermore, for polynomials $P\in\mathcal{P}[X_{\sigma(1)},...,X_{\sigma(M)}]$ with $M<N$ and $\sigma\in\mathfrak{S}_{M}$ , we will still use the notation $P(\eta_{\mathfrak{c},\mathfrak{a}})$ to designate

P(\eta_{\mathfrak{c},\mathfrak{a}})=P(\eta_{c_{\sigma(1)},a_{\sigma(1)}},...,\eta_{c_{\sigma(M)},a_{\sigma(M)}}).

For $M\geq 2$ , we can also define

B^{M}:=\sum_{k=1}^{M}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{M}X^{2}_{k}X_{j}\in\mathcal{P}_{M}\quad\text{and}\quad\widetilde{B}^{2M}:=\sum_{k=1}^{M}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{M}Y^{2}_{k}X_{j}\in\mathcal{P}_{2M}

and

C^{M}:=\sum_{k=1}^{M}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{M}X^{3}_{k}X_{j}\in\mathcal{P}_{M},

and even

D^{M}:=\prod_{k=1}^{M}(1-X_{k})-(1-\sum_{k=1}^{M}X_{k})=\sum_{k=2}^{M}(-1)^{k}S^{k,M}\in\mathcal{P}_{M},

(3.2)

D_{k}^{M-1}:=\prod_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{M}(1-X_{j})-(1-\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{M}X_{j})\in\mathcal{P}_{M-1}.

(3.3)

Now we introduce a lemma (proven in Section B) which provides a control of the $L^{p}$ -norms of such polynomials, when they are evaluated at functions that decay exponentially.

Lemma 3.1.

For any $M\geq 2$ , $\mathfrak{a}\in\mathrm{Pos}_{M}(L),\mathfrak{b}\in(\mathbb{R}_{+}^{*})^{M}$ , $P\in\mathcal{P}[X_{1},...,X_{M}]$ , and functions $(f_{k})_{k\in\{1,...,M\}}$ such that

f_{k}(x)=\mathcal{O}\left(e^{-b_{k}|x|}\right),

(3.4)

we have for any $p\in[1,+\infty]$ ,

\left\|P\big{(}\tau_{a_{1}}f_{1},...,\tau_{a_{M}}f_{M}\big{)}\right\|_{L^{p}}=\mathcal{O}\bigg{(}\Big{(}\frac{2}{p\min_{k}(b_{k})}+L\Big{)}^{\frac{1}{p}}e^{-\min_{k}(b_{k})L}\bigg{)},

(3.5)

where $\tau_{a_{k}}f_{k}:=f_{k}(.-a_{k})$ .

Remark 3.2.

An important illustration of this lemma is to replace the functions $f_{k}$ by any derivatives or powers of $\eta_{c_{k},a_{k}}$ or $v_{c_{k},a_{k}}$ which satisfy (3.4) because of their decay.

Proof of Proposition 1.8.

We write

E(Q)=E(R_{\mathfrak{c},\mathfrak{a}}+\varepsilon)=E(R_{\mathfrak{c},\mathfrak{a}})+\nabla E(R_{\mathfrak{c},\mathfrak{a}}).\varepsilon+\dfrac{\nabla^{2}E(R_{\mathfrak{c},\mathfrak{a}})(\varepsilon,\varepsilon)}{2}+\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon),

where the remainder $\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)$ can be explicited as follows,

\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)=\int_{0}^{1}\dfrac{(1-t)^{2}}{2}\nabla^{3}E(R_{\mathfrak{c},\mathfrak{a}}+t\varepsilon)(\varepsilon,\varepsilon,\varepsilon)dt.

We also compute the higher order derivatives

\nabla E(R_{\mathfrak{c},\mathfrak{a}}).\varepsilon=\int_{\mathbb{R}}\bigg{(}\dfrac{\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}}\partial_{x}\varepsilon_{\eta}}{4(1-\eta_{\mathfrak{c},\mathfrak{a}})}+\dfrac{(\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})^{2}\varepsilon_{\eta}}{8(1-\eta_{\mathfrak{c},\mathfrak{a}})}+(1-\eta_{\mathfrak{c},\mathfrak{a}})v_{\mathfrak{c},\mathfrak{a}}\varepsilon_{v}+\dfrac{1}{2}\big{(}f(1-\eta_{\mathfrak{c},\mathfrak{a}})-v_{\mathfrak{c},\mathfrak{a}}^{2}\big{)}\varepsilon_{\eta}\bigg{)},

(3.6)

and

	$\displaystyle\nabla^{2}E(R_{\mathfrak{c},\mathfrak{a}})(\varepsilon,\varepsilon)=$	$\displaystyle\int_{\mathbb{R}}\bigg{(}\dfrac{(\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})^{2}\varepsilon_{\eta}^{2}}{4(1-\eta_{\mathfrak{c},\mathfrak{a}})^{3}}+\dfrac{\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}}\varepsilon_{\eta}\partial_{x}\varepsilon_{\eta}}{2(1-\eta_{\mathfrak{c},\mathfrak{a}})^{2}}+\dfrac{(\partial_{x}\varepsilon_{\eta})^{2}}{4(1-\eta_{\mathfrak{c},\mathfrak{a}})}$
		$\displaystyle+(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{v}^{2}-2v_{\mathfrak{c},\mathfrak{a}}\varepsilon_{\eta}\varepsilon_{v}-\dfrac{f^{\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}})}{2}\varepsilon_{\eta}^{2}\bigg{)}.$		(3.7)

Claim 3.3.

E(R_{\mathfrak{c},\mathfrak{a}})=\sum_{k=1}^{N}E(Q_{c_{k}})+\mathcal{O}\Big{(}\Lambda(L,\mathfrak{c})e^{-a_{d}\nu_{\mathfrak{c}}L}\Big{)}.

Proof.

Recall that

E(R_{\mathfrak{c},\mathfrak{a}})=\dfrac{1}{8}\int_{\mathbb{R}}\dfrac{(\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})^{2}}{1-\eta_{\mathfrak{c},\mathfrak{a}}}+\dfrac{1}{2}\int_{\mathbb{R}}(1-\eta_{\mathfrak{c},\mathfrak{a}})v_{\mathfrak{c},\mathfrak{a}}^{2}+\dfrac{1}{2}\int_{\mathbb{R}}F(1-\eta_{\mathfrak{c},\mathfrak{a}}).

(3.8)

First, we study the kinetic energy, namely, the first and second terms in (3.8), by writing

(1-\eta_{\mathfrak{c},\mathfrak{a}})v_{\mathfrak{c},\mathfrak{a}}^{2}=\sum_{k=1}^{N}(1-\eta_{c_{k},a_{k}})v_{c_{k},a_{k}}^{2}-B^{2N}(Q_{\mathfrak{c},\mathfrak{a}})+2S^{2,N}(v_{\mathfrak{c},\mathfrak{a}})-2S^{2,N}(v_{\mathfrak{c},\mathfrak{a}})S^{1,N}(\eta_{\mathfrak{c},\mathfrak{a}}).

(3.9)

Using in addition Proposition 2.4, we deal with the first term of $E(R_{\mathfrak{c},\mathfrak{a}})$ by writing

	$\displaystyle\dfrac{(\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})^{2}}{1-\eta_{\mathfrak{c},\mathfrak{a}}}$	$\displaystyle=\sum_{k=1}^{N}\dfrac{(\partial_{x}\eta_{c_{k},a_{k}})^{2}}{(1-\eta_{c_{k},a_{k}})\Big{(}1-\sum_{\begin{subarray}{c}l=1\\ l\neq k\end{subarray}}^{N}\frac{\eta_{c_{l},a_{l}}}{1-\eta_{c_{k},a_{k}}}\Big{)}}+\dfrac{2}{1-\eta_{\mathfrak{c},\mathfrak{a}}}S^{2,N}(\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})$
		$\displaystyle=\sum_{k=1}^{N}\dfrac{(\partial_{x}\eta_{c_{k},a_{k}})^{2}}{1-\eta_{c_{k},a_{k}}}+\mathcal{O}\big{(}\widetilde{B}^{2N}(\eta_{\mathfrak{c},\mathfrak{a}},\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}S^{2,N}(\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}})\big{)}.$		(3.10)

As a consequence of Lemma B.2 in the appendix, we obtain

F(1-\eta_{\mathfrak{c},\mathfrak{a}})=\sum_{k=1}^{N}F(1-\eta_{c_{k},a_{k}})+\mathcal{O}\big{(}S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}S^{3,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}B^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}C^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}.

(3.11)

We integrate (3.9), (3.10) and (3.11) on $\mathbb{R}$ . Therefore, by exponential decay (10) and using Lemma 3.1 with $p=1$ , we conclude the proof of the claim. ∎

Now we deal with $\nabla E(R_{\mathfrak{c},\mathfrak{a}}).\varepsilon$ . As a matter of example, we only handle the term associated with the nonlinearity $f$ and we refer to [4] concerning the other terms. By Lemma B.2, we have

f(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}=\sum_{k=1}^{N}f(1-\eta_{c_{k},a_{k}})\varepsilon_{\eta}+\mathcal{O}\Big{(}\varepsilon_{\eta}B^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\Big{)}+\mathcal{O}\Big{(}\varepsilon_{\eta}S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})\Big{)}.

Integrating the previous equation on $\mathbb{R}$ , using the Cauchy-Schwarz inequality and eventually Lemma 3.1 with $p=2$ , leads to

	$\displaystyle\int_{\mathbb{R}}f(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}$	$\displaystyle=\sum_{k=1}^{N}\int_{\mathbb{R}}f(1-\eta_{c_{k},a_{k}})\varepsilon_{\eta}+\mathcal{O}\Big{(}\left\\|\varepsilon_{\eta}\right\\|_{L^{2}}\left\\|S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})\right\\|_{L^{2}}\Big{)}+\mathcal{O}\Big{(}\left\\|\varepsilon_{\eta}\right\\|_{L^{2}}\left\\|B^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\right\\|_{L^{2}}\Big{)}$
		$\displaystyle=\sum_{k=1}^{N}\int_{\mathbb{R}}f(1-\eta_{c_{k},a_{k}})\varepsilon_{\eta}+\mathcal{O}\Big{(}\\|\varepsilon\\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}e^{-a_{d}\nu_{\mathfrak{c}}L}\Big{)}.$

The other terms can be dealt with similarly, so that we infer

\nabla E(R_{\mathfrak{c},\mathfrak{a}}).\varepsilon=\sum_{k=1}^{N}\nabla E(Q_{c_{k},a_{k}}).\varepsilon+\mathcal{O}\big{(}\|\varepsilon\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}e^{-a_{d}\nu_{\mathfrak{c}}L}\big{)}.

Since $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ and $Q_{c_{k},a_{k}}$ solves (7) with $c=c_{k}$ , we have for any $k\in\{1,...,N\}$ ,

\nabla E(Q_{c_{k},a_{k}}).\varepsilon=\big{(}\nabla E(Q_{c_{k},a_{k}})-c_{k}\nabla p(Q_{c_{k},a_{k}})\big{)}.\varepsilon=0,

so that

\nabla E(R_{\mathfrak{c},\mathfrak{a}}).\varepsilon=\mathcal{O}\big{(}\|\varepsilon\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}e^{-a_{d}\nu_{\mathfrak{c}}L}\big{)}.

(3.12)

Now we handle the quadratic term, we first define

\nabla^{2}E(R_{\mathfrak{c},\mathfrak{a}}).(\varepsilon,\varepsilon)=:\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon),

and we decompose according to the partition of the unity (19),

\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon)=\sum_{k=1}^{N}\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon)\Phi_{k}+\sum_{k=0}^{N}\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon)\Phi_{k,k+1}.

We write

Claim 3.4.

For $\tau$ such that $2\tau<\nu_{\mathfrak{c}}$ , we have

\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon)\Phi_{k}=\int_{\mathbb{R}}J(Q_{c_{k},a_{k}},\varepsilon)\Phi_{k}+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\tau L}{2}}\Big{)}

and

\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon)\Phi_{k,k+1}=\int_{\mathbb{R}}J(0,\varepsilon)\Phi_{k,k+1}+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\tau L}{2}}\Big{)}.

Proof.

Like we did for $\nabla E(R_{\mathfrak{c},\mathfrak{a}}).\varepsilon$ , we only deal with the term in which the nonlinearity $f$ intervenes. In view of Lemma B.2, we write

f^{\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}^{2}\Phi_{k}=f^{\prime}(1-\eta_{c_{k},a_{k}})\varepsilon_{\eta}^{2}\Phi_{k}+\mathcal{O}\Big{(}A_{k}^{N}\big{(}\eta_{c_{1},a_{1}},...,\eta_{c_{k-1},a_{k-1}},\Phi_{k},\eta_{c_{k+1},a_{k+1}},...,\eta_{c_{N},a_{N}}\big{)}\varepsilon_{\eta}^{2}\Big{)},

where $G^{N}_{k}=X_{k}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}X_{j}\in\mathcal{P}[X_{1},...,X_{N}]$ . Since $\Phi_{k}(x)=\mathcal{O}\big{(}e^{-2\tau|x-a_{k}-\frac{L}{4}|}\big{)}$ and $\eta_{c_{j},a_{j}}(x)=\mathcal{O}\big{(}e^{-\nu_{\mathfrak{c}}|x-a_{j}|}\big{)}$ for any $j\neq k$ , and since $(a_{1},...,a_{k-1},a_{k}+L/4,a_{k+1},...,a_{N})\in\mathrm{Pos}_{N}\big{(}\frac{3L}{4}\big{)}$ , we can apply Lemma 3.1 with $P=A_{k}^{N}$ and $p=+\infty$ , and we deduce

\int_{\mathbb{R}}f^{\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}^{2}\Phi_{k}=\int_{\mathbb{R}}f^{\prime}(1-\eta_{c_{k},a_{k}})\varepsilon_{\eta}^{2}\Phi_{k}+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{3a_{d}\min(2\tau,\nu_{\mathfrak{c}})L}{4}}\Big{)}.

(3.13)

In a similar manner, we write the following Taylor expansion of the first order,

	$\displaystyle f^{\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}^{2}\Phi_{k,k+1}$	$\displaystyle=f^{\prime}(1)\varepsilon_{\eta}^{2}\Phi_{k,k+1}$
		$\displaystyle-\int_{0}^{1}\varepsilon_{\eta}^{2}\Big{(}\eta_{c_{k},a_{k}}\Phi_{k,k+1}+A_{k}^{N}(\eta_{c_{1},a_{1}},...,\Phi_{k,k+1},...,\eta_{c_{N},a_{N}})\Big{)}f^{\prime\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})dt.$

We have for $\widetilde{k}\leq k,\eta_{c_{\widetilde{k}},a_{\widetilde{k}}}(x)=\mathcal{O}\big{(}e^{-\nu_{\mathfrak{c}}(x-a_{\widetilde{k}})^{+}}\big{)}$ and $\Phi_{k,k+1}(x)=\mathcal{O}\big{(}e^{-2\tau(x-a_{k}-\frac{L}{4})^{-}}\big{)}$ , and for $\widetilde{k}\geq k+1,\eta_{c_{\widetilde{k}},a_{\widetilde{k}}}(x)=\mathcal{O}\big{(}e^{-\nu_{\mathfrak{c}}(x-a_{\widetilde{k}})^{-}}\big{)}$ and $\Phi_{k,k+1}(x)=\mathcal{O}\big{(}e^{-2\tau(x-a_{k+1}+\frac{L}{4})^{+}}\big{)}$ . Now using Lemma 3.1 with $P=S^{2,2}\in\mathcal{P}[X_{1},X_{2}]$ and $p=+\infty$ , we get

\int_{\mathbb{R}}\varepsilon_{\eta}^{2}\eta_{c_{k},a_{k}}\Phi_{k,k+1}=\mathcal{O}\big{(}\left\|\varepsilon_{\eta}\right\|_{L^{2}}^{2}\left\|\eta_{c_{k},a_{k}}\Phi_{k,k+1}\right\|_{L^{\infty}}\big{)}=\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\min(2\tau,\nu_{\mathfrak{c}})L}{4}}\Big{)}.

(3.14)

On the other hand, we also use Lemma 3.1 with $P=A_{k}^{N}$ and $p=+\infty$ , so that

\int_{\mathbb{R}}\varepsilon_{\eta}^{2}A_{k}^{N}(\eta_{c_{1},a_{1}},...,\Phi_{k,k+1},...,\eta_{c_{N},a_{N}})=\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\min(2\tau,\nu_{\mathfrak{c}})L}{4}}\Big{)}.

(3.15)

As a consequence of both (3.14) and (3.15), we derive

\int_{\mathbb{R}}f^{\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}^{2}\Phi_{k,k+1}=\int_{\mathbb{R}}f^{\prime}(1)\varepsilon_{\eta}^{2}\Phi_{k,k+1}+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\min(2\tau,\nu_{\mathfrak{c}})L}{4}}\Big{)}.

(3.16)

We use the same type of consideration for the other terms, then by hypothesis on $\tau$ , and combining (3.13) and (3.16), it eventually proves Claim 3.4:

\int_{\mathbb{R}}J(R_{\mathfrak{c},\mathfrak{a}},\varepsilon)\Phi_{k,k+1}=\int_{\mathbb{R}}J(0,\varepsilon)\Phi_{k,k+1}+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\tau L}{2}}\Big{)}.

∎

The rest of the argument does not depend on the nonlinearity $f$ , but only on the construction of the partition of the unity (19), so we just take $\tau$ like in Claim 3.4 and deduce that

\nabla^{2}E(R_{\mathfrak{c},\mathfrak{a}}).(\varepsilon,\varepsilon)=\nabla^{2}E(Q_{c_{k}}).(\varepsilon_{k},\varepsilon_{k})+\nabla^{2}E(0).(\varepsilon_{k,k+1},\varepsilon_{k,k+1})+\mathcal{O}(\tau\left\|\varepsilon\right\|_{\mathcal{X}}^{2})+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}e^{-\frac{a_{d}\tau L}{2}}\Big{)}.

(3.17)

Now summing the estimates on each order of the derivative of the energy, we conclude the proof of Proposition 1.8. ∎

Contrary to the previous proofs in this section, the proofs of Proposition 1.9 does not depend on the shape of the nonlinearity $f$ and can be written with the methods from [4].

4 Orthogonal decomposition and control on the functional $G$

The proof of Proposition 1.10 follows the lines of the proof of the analogous Proposition 2 in [4], provided the nonlinearity $f$ satisfies the suitable properties. The only crucial property to check is the exponential decay of the travelling waves, and of its first spatial derivatives, which is provided by Theorem 1.2. We refer to [4] for more details. Throughout this section and to suit the framework of Subsection 1.3.2, we shall use the notation $(\mathfrak{c},\mathfrak{a}):=\big{(}\mathfrak{C}(Q),\mathfrak{A}(Q)\big{)}$ with $\mathfrak{C},\mathfrak{A}$ and $\varepsilon$ the functions given by Proposition 1.10.

Proof of Corollary 1.12.

As for Proposition 1.10, there are similarities with [4], in particular concerning the proof of the first and second inequalities in (33) for which we refer to [4]. Take $L\geq L_{2}$ . Since $Q\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{2},L)$ , there exists $\mathfrak{a}^{*}\in\mathrm{Pos}_{N}(L)$ such that $\left\|Q-R_{\mathfrak{c}^{*},\mathfrak{a}^{*}}\right\|_{\mathcal{X}}<\alpha_{2}$ . Taking $\alpha_{2}$ small enough, (28) and Lemma 1.11 yield to

1-\eta(x)\geq 1-\left\|\eta_{\mathfrak{c}^{*},\mathfrak{a}^{*}}\right\|_{L^{\infty}}-\alpha_{2}\geq\dfrac{1-\beta^{*}}{2}.

This implies (30) and (31). We deduce also (32) from taking $\alpha_{2}$ small enough in (28).

Regarding $\kappa_{\mathfrak{c}}$ , we first use that $c\mapsto Q_{c}$ is $\mathcal{C}^{2}$ by writing that, for any $k\in\{1,...,N\}$ ,

\left|\dfrac{d}{dc}\Big{(}p(Q_{c})\Big{)}_{|c=c_{k}^{*}}-\dfrac{d}{dc}\Big{(}p(Q_{c})\Big{)}_{|c=c_{k}}\right|\leq\left\|\dfrac{d^{2}}{dc^{2}}\Big{(}p(Q_{c})\Big{)}\right\|_{L^{\infty}([c_{k}^{*},c_{k}])}|c_{k}^{*}-c_{k}|,

thus

-\dfrac{d}{dc}\Big{(}p(Q_{c})\Big{)}_{|c=c_{k}}\geq\kappa_{\mathfrak{c}^{*}}-\left\|\dfrac{d^{2}}{dc^{2}}\Big{(}p(Q_{c})\Big{)}\right\|_{L^{\infty}([c_{k}^{*},c_{k}])}|c_{k}^{*}-c_{k}|,

so it remains to bound $\|\frac{d^{2}}{dc^{2}}\big{(}p(Q_{c})\big{)}\|_{L^{\infty}([c_{k}^{*},c_{k}])}$ by a constant that only depends on $\mathfrak{c}^{*}$ . The exponential decay in Theorem 1.2 and Lemma A.3 provide a positive constant $\widetilde{K}$ such that for any $c\in[c_{k}^{*},c_{k}]$ ,

\displaystyle\left|\dfrac{d^{2}}{dc^{2}}\Big{(}p(Q_{c})\Big{)}\right|

\displaystyle\leq\left|\nabla p(Q_{c}).\partial_{c}^{2}Q_{c}\right|+\left|\nabla^{2}p(Q_{c}).(\partial_{c}Q_{c},\partial_{c}Q_{c})\right|\leq\widetilde{K},

(4.1)

where $\widetilde{K}$ can be explicited in terms of $\mathfrak{c}^{*},\nu_{\mathfrak{c}^{*}}$ and $\mu_{\mathfrak{c}^{*}}$ , provided that $|\mathfrak{c}-\mathfrak{c}^{*}|\leq\frac{\mu_{\mathfrak{c}^{*}}}{2}$ . We then deduce the third inequality in (33) by taking $\alpha_{2}$ sufficiently small so that the previous condition is satisfied and such that $\kappa_{\mathfrak{c}^{*}}-\widetilde{K}K_{1}\alpha_{2}\geq\frac{\kappa_{\mathfrak{c}^{*}}}{2}$ .
Now we prove the last inequality in (33). According to definition (13), take $\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})$ satisfying (11) with $c=c_{k}$ , and write its orthogonal decomposition

\varepsilon=\lambda\partial_{x}Q_{c_{k}^{*}}+\mu\nabla p(Q_{c_{k}^{*}})+\varepsilon^{*},

(4.2)

where $\lambda,\mu\in\mathbb{R}$ , $\varepsilon^{*}$ satisfies the orthogonal conditions (11) with $c=c_{k}^{*}$ , and where $\nabla p(Q_{c_{k}^{*}})$ is identified to its representing vector in $L^{2}(\mathbb{R})\times L^{2}(\mathbb{R})$ . Since $\varepsilon$ satisfies (11), we compute

\left\{\begin{array}[]{l}\left\langle\varepsilon,\partial_{x}Q_{c_{k}^{*}}-\partial_{x}Q_{c_{k}}\right\rangle_{L^{2}\times L^{2}}=\left\langle\varepsilon,\partial_{x}Q_{c_{k}^{*}}\right\rangle_{L^{2}\times L^{2}}=\lambda\left\|\partial_{x}Q_{c_{k}^{*}}\right\|_{L^{2}}^{2},\\ \left\langle\varepsilon,\nabla p(Q_{c_{k}^{*}})-\nabla p(Q_{c_{k}})\right\rangle_{L^{2}\times L^{2}}=\left\langle\varepsilon,\nabla p(Q_{c_{k}^{*}})\right\rangle_{L^{2}\times L^{2}}=\mu\left\|\nabla p(Q_{c_{k}^{*}})\right\|_{L^{2}}^{2}.\\ \end{array}\right.

Since the quantities $\partial_{x}Q_{c_{k}^{*}}$ and $\nabla p(Q_{c_{k}^{*}})$ are different from zero, using Lemmas A.1 and A.3 for the momentum, there exists a constant $C_{*}>0$ , only depending on $\mathfrak{c}^{*}$ such that

|\lambda|+|\mu|\leq C_{*}\left\|\varepsilon\right\|_{\mathcal{X}}\left|\mathfrak{c}^{*}-\mathfrak{c}\right|.

(4.3)

On the other hand, noticing in addition that $\left\langle\nabla p(Q_{c_{k}^{*}}),\partial_{x}Q_{c_{k}^{*}}\right\rangle_{L^{2}\times L^{2}}=0$ , we have

$\displaystyle H_{c_{k}}(\varepsilon)$	$\displaystyle=\lambda^{2}H_{c_{k}}(\partial_{x}Q_{c_{k}^{}})+\mu^{2}H_{c_{k}}\big{(}\nabla p(Q_{c_{k}^{}})\big{)}+H_{c_{k}}(\varepsilon^{*})$
	$\displaystyle+2\lambda\left\langle\mathcal{H}_{c_{k}}(\partial_{x}Q_{c_{k}^{}}),\varepsilon^{}\right\rangle_{L^{2}\times L^{2}}+2\mu\left\langle\mathcal{H}_{c_{k}}\big{(}\nabla p(Q_{c_{k}^{}})\big{)},\varepsilon^{}\right\rangle_{L^{2}\times L^{2}}$	(4.4)
	$\displaystyle+2\lambda\mu\left\langle\mathcal{H}_{c_{k}}(\partial_{x}Q_{c_{k}^{}}),\nabla p(Q_{c_{k}^{}})\right\rangle_{L^{2}\times L^{2}}.$

We notice, by definition of $\mathcal{H}_{c}$ , that

\mathcal{H}_{c_{k}}=(c_{k}^{*}-c_{k})\nabla^{2}p(Q_{c_{k}})+\mathcal{H}_{c_{k}^{*}}.

(4.5)

Taking benefit of (4.5), there exists a positive constant $\widetilde{C}_{*}$ such that

H_{c}(\varepsilon)\geq H_{c_{k}^{*}}(\varepsilon^{*})-\left|\mathfrak{c}-\mathfrak{c}^{*}\right|\left\|\varepsilon^{*}\right\|_{\mathcal{X}}^{2}-\widetilde{C}_{*}\left((\lambda^{2}+\mu^{2})(\left|\mathfrak{c}-\mathfrak{c}^{*}\right|+1)+(|\lambda|+|\mu|)\left\|\varepsilon^{*}\right\|_{\mathcal{X}}\right).

(4.6)

Using the control (4.3), the orthogonality conditions on $\varepsilon^{*}$ , the coercivity of $H_{c_{k}^{*}}$ , and up to taking a larger constant $\widetilde{C}_{*}$ , we obtain

H_{c}(\varepsilon)\geq\big{(}l_{c_{k}^{*}}-\left|\mathfrak{c}-\mathfrak{c}^{*}\right|\big{)}\left\|\varepsilon^{*}\right\|_{\mathcal{X}}^{2}-\widetilde{C}_{*}\left|\mathfrak{c}-\mathfrak{c}^{*}\right|\left\|\varepsilon\right\|_{\mathcal{X}}\big{(}\left|\mathfrak{c}-\mathfrak{c}^{*}\right|\left\|\varepsilon\right\|_{\mathcal{X}}+\left\|\varepsilon^{*}\right\|_{\mathcal{X}}\big{)}.

Moreover, we verify that $\left\|\varepsilon\right\|_{\mathcal{X}}^{2}=\mathcal{O}\left(\lambda^{2}+\mu^{2}\right)+\mathcal{O}\left((|\lambda|+|\mu|)\left\|\varepsilon^{*}\right\|_{L^{2}}\right)+\left\|\varepsilon^{*}\right\|_{\mathcal{X}}^{2}$ . Up to taking a smaller $\alpha_{2}$ so that $|\mathfrak{c}^{*}-\mathfrak{c}|$ is small enough, we infer from (4.3) that $\left\|\varepsilon\right\|_{\mathcal{X}}=\mathcal{O}\left(\left\|\varepsilon^{*}\right\|_{\mathcal{X}}\right)$ . Plugging this into (4.6), we deduce that there exists $l_{*}>0$ , only depending on $\mathfrak{c}^{*}$ such that for any $\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})$ satisfying (11) and any $k\in\{1,...,N\}$ ,

H_{c_{k}}(\varepsilon)\geq l_{*}\left\|\varepsilon\right\|_{\mathcal{X}}^{2},

which leads to the last inequality in (33). ∎

We have imposed that $\tau_{0}<2\tau<\frac{\nu_{\mathfrak{c}^{*}}}{2}$ . Before we pass to the proof of Corollary 1.13, we combine Proposition 1.8 and 1.9 and deduce the following corollary concerning the functional $G$ defined in (18).

Corollary 4.1.

Let $L\geq L_{2}$ . For $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ , we have

	$\displaystyle G(Q)=$	$\displaystyle\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{}})-c_{k}^{}p(Q_{c_{k}^{*}})\big{)}+\dfrac{1}{2}\bigg{(}\sum_{k=1}^{N}H_{c_{k}}(\varepsilon_{k})+\sum_{k=0}^{N}H_{0}^{k}(\varepsilon_{k,k+1},\varepsilon_{k,k+1})\bigg{)}$
		$\displaystyle+\mathcal{O}\big{(}\left\|\mathfrak{c}-\mathfrak{c}^{*}\right\|^{2}\big{)}+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}\big{(}\tau+e^{-\frac{a_{d}\tau L}{2}}\big{)}\Big{)}+\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)+\mathcal{O}\left(\left\\|\varepsilon\right\\|_{\mathcal{X}}^{4}\right)$
		$\displaystyle+\mathcal{O}\Big{(}\Lambda(L,\mathfrak{c})(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\Big{)}+\mathcal{O}\big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\big{)},$

where

H_{0}^{k}(\varepsilon_{k,k+1})=\left\{\begin{array}[]{l}\nabla^{2}(E-c_{1}p_{1})(0).(\varepsilon_{0,1},\varepsilon_{0,1})\quad\text{if }k=0,\\ \nabla^{2}(E-c_{k}p_{k}-c_{k+1}p_{k+1})(0).(\varepsilon_{k,k+1},\varepsilon_{k,k+1})\quad\text{if }k\in\{1,...,N-1\},\\ \nabla^{2}(E-c_{N}p_{N})(0).(\varepsilon_{N,N+1},\varepsilon_{N,N+1})\quad\text{if }k=N.\\ \end{array}\right.

Proof.

We just combine Proposition 1.8 and 1.9. We first deal with the combination of the second order terms appearing in the expression of both previous propositions. We recognize the desired terms $H_{c_{k}}(\varepsilon_{k})$ and $H_{0}^{k}(\varepsilon_{k,k+1})$ except that the speeds are $c_{k}^{*}$ instead of $c_{k}$ . This can be overcome by using the same kind of decomposition than in (4.5). Indeed, we write that $\nabla^{2}(E-c_{k}^{*}p)=\nabla^{2}(E-c_{k}p)+(c_{k}-c_{k}^{*})\nabla^{2}p$ , therefore we deal with the terms involving $\nabla^{2}p$ by using Lemma A.3, by using that $\left\|\varepsilon_{k}\right\|_{\mathcal{X}}\lesssim\left\|\varepsilon\right\|_{\mathcal{X}}$ and by using the standard Young’s inequality. This can be summarized in the following inequalities: $\big{|}(c_{k}-c_{k}^{*})\nabla^{2}p(Q_{c_{k}}).(\varepsilon_{k},\varepsilon_{k})\big{|}\lesssim|c_{k}-c_{k}^{*}|\left\|\varepsilon\right\|_{\mathcal{X}}^{2}\lesssim\left|\mathfrak{c}-\mathfrak{c}^{*}\right|^{2}+\left\|\varepsilon\right\|_{\mathcal{X}}^{4}$ . As for the terms involving $\nabla^{2}p_{k}$ , the same decomposition than previously and a straightforward computation of the second derivative of $p_{k}$ leads to the same control. It remains to study the term $\sum_{k=1}^{N}\big{(}E(Q_{c_{k}})-c_{k}^{*}p(Q_{c_{k}})\big{)}$ . We use a second order Taylor expansion between $Q_{c_{k}}$ and $Q_{c_{k}^{*}}$ . The first order term cancels because of (7), and the second order term can be dealt by using Lemma A.1 and A.3, hence the resulting $\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{*}})-c_{k}^{*}p(Q_{c_{k}^{*}})\big{)}+\mathcal{O}\left(|\mathfrak{c}-\mathfrak{c}^{*}|^{2}\right)$ . ∎

Recall that $H_{c}$ is coercive whenever $\varepsilon$ satisfies some orthogonality conditions. We make now use of this property to state some almost coercivity property along the $\varepsilon_{k}$ .

Corollary 4.2.

Let $L\geq L_{2}$ . For $Q\in\mathcal{U}^{\perp}_{\mathfrak{c},\mathfrak{a}}(L)$ , there exists $\widetilde{l}_{\mathfrak{c}}>0$ proportional³³3As usual in this framework, the proportionality constant only depends on $\mathfrak{c}^{*}$ . to $\min(l_{\mathfrak{c}},\nu_{\mathfrak{c}})$ such that for any $k\in\{0,...,N\}$ ,

H_{0}^{k}(\varepsilon_{k,k+1})\geq\widetilde{l}_{\mathfrak{c}}\left\|\varepsilon_{k,k+1}\right\|_{\mathcal{X}}^{2},

(4.7)

and for any $k\in\{1,...,N\}$ ,

H_{c_{k}}(\varepsilon_{k})\geq\widetilde{l}_{\mathfrak{c}}\left\|\varepsilon_{k}\right\|_{\mathcal{X}}^{2}+\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}e^{-\frac{a_{d}\tau L}{2}}\Big{)}.

(4.8)

Moreover,

\sum_{k=1}^{N}\left\|\varepsilon_{k}\right\|_{\mathcal{X}}^{2}+\sum_{k=0}^{N}\left\|\varepsilon_{k,k+1}\right\|_{\mathcal{X}}^{2}\geq\left\|\varepsilon\right\|_{\mathcal{X}}^{2}.

(4.9)

Regarding the proof of Corollary 4.2, we just add that it is reminiscent of the fact that the quadratic form $\nabla^{2}(E-c_{s}p)(0)$ is nonnegative. We refer to the proof of Lemma 1 in [4] for more details. Finally, we combine the almost coercivity property of Corollary 4.2 with Proposition 1.10 and we derive the proof of Corollary 1.13.

Proof of Corollary 1.13.

We first prove the lower bound. We show that $\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)=\mathcal{O}\left(\left\|\varepsilon\right\|_{\mathcal{X}}^{3}\right)$ . For that, we need to investigate the dependence in $\mathfrak{c},\mathfrak{a}$ of each terms controlling $\nabla^{3}E(R_{\mathfrak{c},\mathfrak{a}}+t\varepsilon)$ in Lemma A.3 (applied with $l=3$ ). Let us begin with the term $\left\|R_{\mathfrak{c},\mathfrak{a}}+t\varepsilon\right\|_{\mathcal{X}}$ . Since $Q\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha,L)$ , there exists $\mathfrak{a}^{*}\in\mathrm{Pos}_{N}(L)$ , such that $\left\|Q-R_{\mathfrak{c}^{*},\mathfrak{a}^{*}}\right\|_{\mathcal{X}}\leq\alpha$ . Invoking (28) and taking possibly a smaller $\alpha_{2}$ such that $\left|\mathfrak{c}-\mathfrak{c}^{*}\right|\leq K_{1}\alpha_{2}<\delta^{*}$ with $\delta^{*}$ from Lemma A.1 leads to

	$\displaystyle\left\\|R_{\mathfrak{c},\mathfrak{a}}+t\varepsilon\right\\|_{\mathcal{X}}$	$\displaystyle\leq K_{lip}\big{(}\|\mathfrak{c}-\mathfrak{c}^{}\|+\|\mathfrak{a}-\mathfrak{a}^{}\|\big{)}+\left\\|R_{\mathfrak{c}^{},\mathfrak{a}^{}}\right\\|_{\mathcal{X}}+\left\\|\varepsilon\right\\|_{\mathcal{X}}$
		$\displaystyle\leq M^{}+\sum_{k=1}^{N}\left\\|Q_{c_{k}^{}}\right\\|_{\mathcal{X}},$

where $M^{*}:=\max(1,K_{lip})K_{1}\alpha_{2}$ only depends on $\mathfrak{c}^{*}$ . Secondly, using (31), we write for any $x\in\mathbb{R}$ ,

\displaystyle\big{|}1-\big{(}R_{\mathfrak{c},\mathfrak{a}}(x)+t\varepsilon(x)\big{)}\big{|}\geq 1-\eta(x)\geq\dfrac{1-\beta^{*}}{2}.

Finally, using that $f^{\prime\prime\prime}$ is continuous, we can write the Taylor expansion of $f^{\prime\prime}$ to the first order, for any $x\in\mathbb{R}$ ,

\displaystyle\big{|}f^{\prime\prime}\big{(}1-\eta_{\mathfrak{c},\mathfrak{a}}(x)-t\varepsilon(x)\big{)}\big{|}\leq\big{|}f^{\prime\prime}(1-\eta_{\mathfrak{c}^{*},\mathfrak{a}^{*}})\big{|}+M^{*}\|f^{\prime\prime\prime}\|_{L^{\infty}([-M^{*},M^{*}])}.

Passing to the $L^{\infty}$ -norm, the right-hand side of the previous inequality no longer depends on $\mathfrak{a}^{*}$ , hence $\left\|f^{\prime\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}}-t\varepsilon)\right\|_{L^{\infty}}$ is bounded by a constant that only depends on $\mathfrak{c}^{*}$ .

Secondly, by (34), and up to taking a larger $L_{2}$ , Corollary 4.2 applies and using additionally the property (33), there exists $\widetilde{l}_{*}>0$ that only depends on $\mathfrak{c}^{*}$ such that

	$\displaystyle\sum_{k=1}^{N}H_{c_{k}}(\varepsilon_{k})+\sum_{k=0}^{N}H_{0}^{k}(\varepsilon_{k,k+1})$	$\displaystyle\geq\widetilde{l}_{\mathfrak{c}}\left(\sum_{k=1}^{N}\left\\|\varepsilon_{k}\right\\|_{\mathcal{X}}^{2}+\sum_{k=0}^{N}\left\\|\varepsilon_{k,k+1}\right\\|_{\mathcal{X}}^{2}\right)+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}\Lambda(L_{1},\mathfrak{c})^{\frac{1}{2}}e^{-\frac{a_{d}\tau L_{1}}{2}}\Big{)}$
		$\displaystyle\geq\widetilde{l}_{*}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}\Lambda(L_{1},\mathfrak{c})^{\frac{1}{2}}e^{-\frac{a_{d}\tau L_{1}}{2}}\Big{)}.$		(4.10)

Now, plugging (4) and (33) in Corollary 4.1 yields to

	$\displaystyle G(Q)$	$\displaystyle\geq\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{}})-c_{k}^{}p(Q_{c_{k}^{}})\big{)}+\dfrac{\widetilde{l}_{}}{2}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}+\mathcal{O}\big{(}\left\|\mathfrak{c}-\mathfrak{c}^{*}\right\|^{2}\big{)}$
		$\displaystyle+\mathcal{O}\Big{(}L(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\Big{)}+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}\big{(}\tau+e^{-\frac{a_{d}\tau L_{1}}{2}}+L_{1}^{\frac{1}{2}}e^{-\frac{a_{d}\tau L_{1}}{2}}\big{)}\Big{)}+\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon),$

where we have also used that $\left\|\varepsilon\right\|_{\mathcal{X}}$ is bounded, and that $L^{\frac{1}{2}}$ and $\Lambda(L,\mathfrak{c})$ are controlled by $\mathcal{O}(L)$ . As a consequence of the both previous steps, we fix the values of $\tau$ and $\tau_{0}$ such that $\tau_{0}<2\tau<\frac{\nu_{\mathfrak{c}^{*}}}{2}\leq\nu_{\mathfrak{c}}$ , and $L_{2}$ large enough such that

\mathcal{O}\Big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}\big{(}\tau+e^{-\frac{a_{d}\tau L_{2}}{2}}+L_{2}e^{-\frac{a_{d}\tau L_{2}}{2}}\big{)}\Big{)}\leq\dfrac{\widetilde{l}_{*}}{4}\left\|\varepsilon\right\|_{\mathcal{X}}^{2},

and we can replace $\mathcal{O}\Big{(}L(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\Big{)}$ by $\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}L}\Big{)}$ . Thus, we get

G(Q)\geq\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{*}})-c_{k}^{*}p(Q_{c_{k}^{*}})\big{)}+\dfrac{\widetilde{l}_{*}}{4}\left\|\varepsilon\right\|_{\mathcal{X}}^{2}+\mathcal{O}\left(\left|\mathfrak{c}-\mathfrak{c}^{*}\right|^{2}\right)+\mathcal{O}\big{(}\left\|\varepsilon\right\|_{\mathcal{X}}^{3}\big{)}+\mathcal{O}\Big{(}Le^{-a_{d}\tau_{0}L})\Big{)}.

Let us now tackle the upper bound. It remains to bound $H_{0}$ and $H_{c_{k}}$ from above. By construction of $\Phi_{k}$ , we have that $\left\|\varepsilon_{k}\right\|_{\mathcal{X}}\lesssim\left\|\varepsilon\right\|_{\mathcal{X}}$ , and by the Cauchy-Schwarz inequality, $\big{|}H_{c_{k}}(\varepsilon_{k})\big{|}\lesssim\left\|\mathcal{H}_{c_{k}}(\varepsilon_{k})\right\|_{L^{2}}\left\|\varepsilon\right\|_{L^{2}}$ . Regarding $\nabla^{2}E(Q_{c_{k}^{*}})$ , we have

\displaystyle\nabla^{2}E(Q_{c_{k}})(\varepsilon_{k},\varepsilon_{k})

\displaystyle=\nabla^{2}E(Q_{c_{k}^{*}})(\varepsilon_{k},\varepsilon_{k})+\int_{0}^{1}\nabla^{3}E\big{(}(1-t)Q_{c_{k}^{*}}+tQ_{c_{k}}\big{)}.(Q_{c_{k}}-Q_{c_{k}^{*}},\varepsilon_{k},\varepsilon_{k})dt.

Similarly to $\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)$ , we use Lemma A.3 with $l=2,3$ to bound uniformly in $\mathfrak{c}$ the operator norm of $\nabla^{2}E(Q_{c_{k}^{*}})$ and $\nabla^{3}E\big{(}(1-t)Q_{c_{k}^{*}}+tQ_{c_{k}}\big{)}$ . We deduce

\displaystyle\nabla^{2}E(Q_{c_{k}})(\varepsilon_{k},\varepsilon_{k})

\displaystyle=\mathcal{O}\left(\left\|\varepsilon\right\|_{\mathcal{X}}^{2}\right).

The same way, we derive $\nabla^{2}p(Q_{c_{k}})(\varepsilon_{k},\varepsilon_{k})=\mathcal{O}\left(\left\|\varepsilon\right\|_{\mathcal{X}}^{2}\right)$ . As a consequence, we obtain that $H_{c_{k}}(\varepsilon_{k}),H^{0}_{c_{k}}(\varepsilon_{k})=\mathcal{O}\left(\left\|\varepsilon\right\|_{\mathcal{X}}^{2}\right)$ . We conclude by combining the previous bounds with Corollary 4.1. ∎

5 Dynamics of the modulation parameters

In this section, we prove Proposition 1.14. We refer to the beginning of Subsection 1.3.4 for the definition of the time-dependent functions $\varepsilon,\mathfrak{c},\mathfrak{a}$ .

Proof of Proposition 1.14.

First of all, we consider that the initial condition $(\eta_{0},v_{0})\in\mathcal{NX}^{2}(\mathbb{R})$ so that, in view of ( $NLS_{hy}$ ), the solution exists locally in time and belong to $\mathcal{C}^{1}([0,T],\mathcal{NX}(\mathbb{R}))$ . By composition with the functions $\mathfrak{C},\mathfrak{A}$ exhibited in Proposition 1.10, we infer that $(\mathfrak{c},\mathfrak{a})\in\mathcal{C}^{1}([0,T],\mathbb{R}^{2N})$ . Therefore, plugging $\varepsilon=(\eta,v)-R_{\mathfrak{c},\mathfrak{a}}$ in ( $NLS_{hy}$ ) and using that each soliton $(\eta_{c_{k}},v_{c_{k}})$ satisfies ( $TW_{c,hy}$ ) with $c=c_{k}$ , we obtain the equations

	$\displaystyle\partial_{t}\varepsilon_{\eta}=$	$\displaystyle\ 2\partial_{x}(\varepsilon_{\eta}v_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{v}(1-\eta_{\mathfrak{c},\mathfrak{a}})+\varepsilon_{\eta}\varepsilon_{v})+2\sum_{k=1}^{N}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}\partial_{x}(\eta_{c_{k},a_{k}}v_{c_{j},a_{j}})$
		$\displaystyle-\sum_{k=1}^{N}c_{k}^{\prime}\partial_{c}\eta_{c_{k},a_{k}}+\sum_{k=1}^{N}(a_{k}^{\prime}-c_{k})\partial_{x}\eta_{c_{k},a_{k}},$

and

	$\displaystyle\partial_{t}\varepsilon_{v}=$	$\displaystyle\ \partial_{x}\Big{(}2v_{\mathfrak{c},\mathfrak{a}}\varepsilon_{v}+\varepsilon_{v}^{2}+\varepsilon_{\eta}\int_{0}^{1}f^{\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})dt\Big{)}-\partial_{x}\Big{(}f(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{k=1}^{N}f(1-\eta_{c_{k},a_{k}})\Big{)}$
		$\displaystyle+\partial_{x}\bigg{(}\dfrac{\partial_{x}^{2}(\eta_{\mathfrak{c},\mathfrak{a}}+\varepsilon_{\eta})}{2(1-\eta_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{\eta})}-\sum_{k=1}^{N}\dfrac{\partial_{x}^{2}\eta_{c_{k},a_{k}}}{2(1-\eta_{c_{k},a_{k}})}\bigg{)}+\partial_{x}\bigg{(}\dfrac{\big{(}\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}}+\partial_{x}\varepsilon_{\eta}\big{)}^{2}}{4(1-\eta_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{\eta})^{2}}-\sum_{k=1}^{N}\dfrac{(\partial_{x}\eta_{c_{k},a_{k}})^{2}}{4(1-\eta_{c_{k},a_{k}})^{2}}\bigg{)}$
		$\displaystyle-\sum_{k=1}^{N}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}\partial_{x}(v_{c_{k},a_{k}}v_{c_{j},a_{j}})+\sum_{k=1}^{N}(a_{k}^{\prime}-c_{k})\partial_{x}v_{c_{k},a_{k}}-\sum_{k=1}^{N}c_{k}^{\prime}\partial_{c}v_{c_{k},a_{k}}.$

Now by differentiating the orthogonality conditions (26) on $\varepsilon(t)$ with respect to $t$ , and plugging both previous equations in it, we get for any $k\in\{1,...,N\}$ ,

\displaystyle M(t)\begin{pmatrix}\mathfrak{a}^{\prime}(t)-\mathfrak{c}(t)\\ \mathfrak{c}^{\prime}(t)\end{pmatrix}=\Phi(t),

(5.1)

with

	$\displaystyle M_{k,j}(t)=\left\langle\partial_{x}Q_{c_{j},a_{j}},\partial_{x}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}-\delta_{j,k}\left\langle\partial^{2}_{x}Q_{c_{k},a_{k}},\varepsilon\right\rangle_{L^{2}\times L^{2}}$
	$\displaystyle M_{k,j+N}(t)=-\left\langle\partial_{c}Q_{c_{j},a_{j}},\partial_{x}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}+\delta_{j,k}\left\langle\partial_{x}\partial_{c}Q_{c_{k},a_{k}},\varepsilon\right\rangle_{L^{2}\times L^{2}}$
	$\displaystyle M_{k+N,j}(t)=\nabla p(Q_{c_{k},a_{k}}).\partial_{x}Q_{c_{j},a_{j}}-\delta_{j,k}\nabla p(\partial_{x}Q_{c_{k},a_{k}}).\varepsilon,$
	$\displaystyle M_{k+N,j+N}(t)=-\nabla p(Q_{c_{k},a_{k}}).\partial_{c}Q_{c_{j},a_{j}}+\delta_{j,k}\nabla p(\partial_{c}Q_{c_{k},a_{k}}).\varepsilon$

	$\displaystyle\Phi_{k}(t)=\left\langle\begin{pmatrix}2v_{\mathfrak{c},\mathfrak{a}}+2\varepsilon_{v}+c_{k}&-2(1-\eta_{\mathfrak{c},\mathfrak{a}})\\ \int_{0}^{1}f^{\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})dt&2v_{\mathfrak{c},\mathfrak{a}}+2\varepsilon_{v}+c_{k}\end{pmatrix}\varepsilon,\partial_{x}^{2}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}$
	$\displaystyle+\sum_{j=1}^{N}\sum_{\begin{subarray}{c}l=1\\ l\neq j\end{subarray}}^{N}\left\langle\begin{pmatrix}2\eta_{c_{j},a_{j}}v_{c_{l},a_{l}}\\ -v_{c_{j},a_{j}}v_{c_{l},a_{l}}\end{pmatrix},\partial_{x}^{2}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}+\left\langle f(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{j=1}^{N}f(1-\eta_{c_{j},a_{j}}),\partial_{x}^{2}v_{c_{k},a_{k}}\right\rangle_{L^{2}}$
	$\displaystyle+\left\langle\dfrac{\partial_{x}^{2}(\eta_{\mathfrak{c},\mathfrak{a}}+\varepsilon_{\eta})}{2(1-\eta_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{\eta})}-\sum_{k=1}^{N}\dfrac{\partial_{x}^{2}\eta_{c_{k},a_{k}}}{2(1-\eta_{c_{k},a_{k}})}+\dfrac{\big{(}\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}}+\partial_{x}\varepsilon_{\eta}\big{)}^{2}}{4(1-\eta_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{\eta})^{2}}-\sum_{k=1}^{N}\dfrac{(\partial_{x}\eta_{c_{k},a_{k}})^{2}}{4(1-\eta_{c_{k},a_{k}})^{2}},\partial_{x}^{2}v_{c_{k},a_{k}}\right\rangle_{L^{2}},$

and

	$\displaystyle\Phi_{k+N}(t)=2\nabla p(\partial_{x}Q_{c_{k},a_{k}}).\begin{pmatrix}(2v_{\mathfrak{c},\mathfrak{a}}+2\varepsilon_{v}+c_{k})\varepsilon_{\eta}-2(1-\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{v}\\ \int_{0}^{1}f^{\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})\varepsilon_{\eta}dt+(2v_{\mathfrak{c},\mathfrak{a}}+2\varepsilon_{v}+c_{k})\varepsilon_{v}\end{pmatrix}$
	$\displaystyle+\sum_{j=1}^{N}\sum_{\begin{subarray}{c}l=1\\ l\neq j\end{subarray}}^{N}2\nabla p(\partial_{x}Q_{c_{k},a_{k}}).\begin{pmatrix}2\eta_{c_{j},a_{j}}v_{c_{l},a_{l}}\\ -v_{c_{j},a_{j}}v_{c_{l},a_{l}}\end{pmatrix}+\left\langle f(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{j=1}^{N}f(1-\eta_{c_{j},a_{j}}),\partial_{x}\eta_{c_{k},a_{k}}\right\rangle_{L^{2}}$
	$\displaystyle+\left\langle\dfrac{\partial_{x}^{2}(\eta_{\mathfrak{c},\mathfrak{a}}+\varepsilon_{\eta})}{2(1-\eta_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{\eta})}-\sum_{k=1}^{N}\dfrac{\partial_{x}^{2}\eta_{c_{k},a_{k}}}{2(1-\eta_{c_{k},a_{k}})}+\dfrac{\big{(}\partial_{x}\eta_{\mathfrak{c},\mathfrak{a}}+\partial_{x}\varepsilon_{\eta}\big{)}^{2}}{4(1-\eta_{\mathfrak{c},\mathfrak{a}}-\varepsilon_{\eta})^{2}}-\sum_{k=1}^{N}\dfrac{(\partial_{x}\eta_{c_{k},a_{k}})^{2}}{4(1-\eta_{c_{k},a_{k}})^{2}},\partial_{x}\eta_{c_{k},a_{k}}\right\rangle_{L^{2}}.$

We decompose $M(t)=D(t)+H(t)$ where all the entries of $D(t)$ are zero except the coefficients

	$\displaystyle D_{k,k}(t):=\left\\|\partial_{x}Q_{c_{k}}\right\\|_{L^{2}\times L^{2}}^{2},$
	$\displaystyle D_{k,k+N}(t):=-\left\langle\partial_{c}Q_{c_{k},a_{k}},\partial_{x}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}},$
	$\displaystyle D_{k+N,k}(t):=\nabla p(Q_{c_{k},a_{k}}).\partial_{x}Q_{c_{k},a_{k}}.$
	$\displaystyle D_{k+N,k+N}(t):=-\nabla p(Q_{c_{k}}).\partial_{c}Q_{c_{k}},$

Claim 5.1.

The matrix $D(t)$ is a diagonal matrix. Furthermore, for $\alpha$ small enough, $D(t)$ is invertible and $\left|D(t)^{-1}\right|$ is bounded by a constant that only depends on $\mathfrak{c}^{*}$ .

Proof.

First we show that the coefficients above and below the diagonal are zero. The coefficients below the diagonal are zero by doing an integration by part. The coefficients above the diagonal reads

-\left\langle\partial_{c}Q_{c_{k},a_{k}},\partial_{x}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}=-\int_{\mathbb{R}}\Big{(}\partial_{c}\eta_{c_{k}}\partial_{x}\eta_{c_{k}}+\partial_{c}v_{c_{k}}\partial_{x}v_{c_{k}}\Big{)}.

Since $\eta_{c}$ and $v_{c}$ are even, then the derivatives with respect to $x$ are odd and the derivatives with respect to $c$ are even. Thus the previous coefficient is zero by integrating on the real line an odd function. We obtain $D(t)=\text{diag}\big{(}D_{1,1}(t),...,D_{2N,2N}(t)\big{)}$ . For $\alpha\leq\alpha_{1}$ , (33) imply that for every $k\in\{1,...,N\}$ and $t\in[0,T]$ , $c_{k}(t)\in(c_{0},c_{s})$ . Moreover $D_{k+N,k+N}(t)=-\frac{d}{dc}\big{(}p(Q_{c})\big{)}_{|c=c_{k}}<0$ by Theorem 1.2. Then the diagonal coefficients of $D(t)$ do not vanish. Therefore, $D(t)$ is invertible and it remains to bound from below its operator norm. By Lemma A.1, we have for any $k\in\{1,...,N\}$ and $t\in[0,T]$ ,

\left\|\partial_{x}Q_{c_{k}^{*}}\right\|_{L^{2}}\leq K_{*}|\mathfrak{c}^{*}-\mathfrak{c}|+\sqrt{D_{k,k}(t)}.

(5.2)

and by (33),

\dfrac{\kappa_{\mathfrak{c}^{*}}}{2}\leq D_{k+N,k+N}(t).

(5.3)

There exists $\alpha_{3}\leq\alpha_{2}$ sufficiently small such that

K_{*}K_{1}\alpha_{3}\leq\min\left\{\dfrac{1}{2}\left\|\partial_{x}Q_{c_{k}^{*}}\right\|_{L^{2}},\dfrac{\kappa_{\mathfrak{c}^{*}}}{2}\right\}.

(5.4)

Since $Q(t)\in\mathcal{U}_{\mathfrak{c}^{*}}(\alpha_{1},L^{1})$ for any $t\in[0,T]$ , (28) applies and we obtain the desired uniform below bound (with respect to $t$ and $k$ ) for $D_{k,k}(t)$ and $D_{k+N,k+N}(t)$ . ∎

Now, since $Q_{\mathfrak{c},\mathfrak{a}}=\mathcal{O}(1)$ (by exponential decay and (33)),

H_{j,j}(t)=-\left\langle\partial^{2}_{x}Q_{c_{j},a_{j}},\varepsilon\right\rangle_{L^{2}\times L^{2}}=\mathcal{O}(\left\|\varepsilon\right\|_{\mathcal{X}}).

By Lemma 3.1, since $\mathfrak{a}(t)\in\mathrm{Pos}_{N}(L)$ with $L\geq L_{1}$ , we have, for $j\neq k\in\{1,...,N\}$ ,

H_{k,j}(t)=\left\langle\partial_{x}Q_{c_{j},a_{j}},\partial_{x}Q_{c_{k},a_{k}}\right\rangle_{L^{2}\times L^{2}}=\mathcal{O}\left(Le^{-\frac{a_{d}\nu_{\mathfrak{c}^{*}}L}{2}}\right),

where we used that $\Lambda(L,\mathfrak{c})$ is controlled by $\mathcal{O}\left(L\right)$ , by (33). We deal with the other terms the same way and we infer that $|H(t)|=\mathcal{O}(\left\|\varepsilon\right\|_{\mathcal{X}})+\mathcal{O}\left(Le^{-\frac{a_{d}\nu_{\mathfrak{c}^{*}}L}{2}}\right)$ . Moreover (28) is satisfied and taking possibly a smaller $\alpha_{3}$ and $L_{3}\geq L_{2}$ large enough, then for $\alpha\leq\alpha_{3}$ and $L\geq L_{3}$ , $\left|D(t)^{-1}H(t)\right|<1$ , so that $M(t)=D(t)\big{(}I+D(t)^{-1}H(t)\big{)}$ is invertible and $\left|M(t)^{-1}\right|$ is bounded by a constant that only depends on $\mathfrak{c}^{*}$ .

Now we show that $\left|\Phi(t)\right|=\mathcal{O}(\left\|\varepsilon\right\|_{\mathcal{X}})+\mathcal{O}\left(Le^{-\frac{a_{d}\nu_{\mathfrak{c}^{*}}L}{2}}\right)$ . This is the same argument than for $H(t)$ , since there only appear terms that involve $\varepsilon$ or multivariate polynomials in $\mathcal{P}_{N}$ evaluated in solitons. For sake of completeness, we deal with the terms involving the nonlinearity $f$ . By Lemma B.2, we get

	$\displaystyle\bigg{\|}\bigg{\langle}f(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{j=1}^{N}f(1-\eta_{c_{j},a_{j}}),\partial^{2}_{x}v_{c_{k},a_{k}}\bigg{\rangle}_{L^{2}}\bigg{\|}$	$\displaystyle=\mathcal{O}\left(\left\\|(S^{2,N}+B^{N})(\eta_{\mathfrak{c},\mathfrak{a}})\right\\|_{L^{1}}\left\\|\partial_{x}^{2}v_{c_{k}}\right\\|_{L^{\infty}}\right)$
		$\displaystyle=\mathcal{O}\Big{(}Le^{-\frac{a_{d}\nu_{\mathfrak{c}^{*}}L}{2}}\Big{)}.$

We use that $f^{\prime}$ is continuous and the fact that $\eta_{\mathfrak{c},\mathfrak{a}}=\mathcal{O}(1)$ in order to deduce

\left\langle\varepsilon_{\eta}\int_{0}^{1}f^{\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})dt,\partial_{x}^{2}v_{c_{k},a_{k}}\right\rangle_{L^{2}}=\mathcal{O}\left(\left\|\varepsilon\right\|_{\mathcal{X}}\right).

We have shown (36), provided that the initial condition $Q_{0}$ lies in $\mathcal{NX}^{2}(\mathbb{R})$ . Now taking $Q_{0}\in\mathcal{NX}(\mathbb{R})$ , we approach by density this initial value by a sequence in $\mathcal{NX}^{2}(\mathbb{R})$ and we argue as in [4]. The only point where this density approach differs from the Gross-Pitaevskii case is the nonlinearity $f$ , which appears in the equation of $\partial_{t}\varepsilon_{v}$ and in the expression of $\Phi(t)$ . By hypothesis, $f$ is at least $\mathcal{C}^{1}$ and in view of the expression of $M(t)$ and $\Phi(t)$ , this is sufficient to pass to the limit in (5.1) along the approaching sequence. Moreover the convergence is in $\mathcal{C}^{0}([0,T],\mathbb{R}^{2N})$ . This eventually shows that $\mathfrak{c},\mathfrak{a}\in\mathcal{C}^{1}\big{(}[0,T],\mathbb{R}^{2N}\big{)}$ and that for any $t\in[0,T]$ ,

\left|\mathfrak{a}^{\prime}(t)-\mathfrak{c}(t)\right|+\left|\mathfrak{c}^{\prime}(t)\right|=\mathcal{O}\Big{(}Le^{-\frac{a_{d}\nu_{\mathfrak{c}^{*}}L}{2}}\Big{)}+\mathcal{O}\left(\left\|\varepsilon(t)\right\|_{\mathcal{X}}\right).

(5.5)

Finally, we derive (37) and (38) from (5.5), by using Proposition 1.10, and taking possibly a further $(\alpha_{3},L_{3})$ . ∎

6 Monotonicity and final estimates

This section is dedicated to establishing the almost monotonicity in time of the quantity defined for $k\in\{1,...,N\}$ , by

\widetilde{p}_{k}(\eta,v)=\dfrac{1}{2}\int_{\mathbb{R}}\chi_{k}\eta v,

(6.1)

where

\chi_{k}(x)=\left\{\begin{array}[]{l}1\quad\text{if }k=1,\\ \dfrac{1}{2}\bigg{(}1+\tanh\left(\tau_{0}\Big{(}x-\frac{a_{k}+a_{k-1}}{2}\Big{)}\right)\bigg{)}\quad\text{if }k\in\{2,...,N\}.\\ \end{array}\right.

Before the proof, we state a conservation type formula for the momentum. Like in Section 5, this proof also relies on an approximation of the initial condition by a sequence in $\mathcal{NX}^{2}(\mathbb{R})$ and we refer to the proof of Corollary 3.1 in [4] for more details.

Lemma 6.1.

Let $(\eta,v)\in\mathcal{C}^{0}([0,T],\mathcal{NX}(\mathbb{R}))$ be a solution of ( $NLS_{hy}$ ) and $\widetilde{\chi}\in\mathcal{C}^{0}\big{(}[0,T],\mathcal{C}^{3}_{b}(\mathbb{R})\big{)}\cap\mathcal{C}^{1}\big{(}[0,T],\mathcal{C}^{0}_{b}(\mathbb{R})\big{)}$ , then $t\mapsto\left\langle\widetilde{\chi},\eta v\right\rangle_{L^{2}}$ is differentiable and its derivative is

	$\displaystyle\dfrac{d}{dt}\left(\int_{\mathbb{R}}\widetilde{\chi}\eta v\right)=$	$\displaystyle\int_{\mathbb{R}}·\partial_{t}\widetilde{\chi}\eta v+\int_{\mathbb{R}}\partial_{x}\widetilde{\chi}\Big{(}(1-2\eta)v^{2}+\widetilde{F}(\eta)+\dfrac{(3-2\eta)(\partial_{x}\eta)^{2}}{4(1-\eta)^{2}}\Big{)}$
		$\displaystyle+\dfrac{1}{2}\int_{\mathbb{R}}\partial_{x}^{3}\widetilde{\chi}\big{(}\eta+\ln(1-\eta)\big{)},$		(6.2)

where $\widetilde{F}(\rho)=\rho f(1-\rho)-F(1-\rho)$ .

We notice that applying Lemma 6.1 to $\widetilde{\chi}\equiv 1$ implies the conservation of the momentum. In this sense, we also mention that the case $k=1$ in (39) is a consequence of Remark 1.15. For $k\in\{2,...,N\}$ , we apply Lemma 6.1 to $\widetilde{\chi}(x,t):=\chi\left(\tau_{0}\big{(}x-X_{k}(t)\big{)}\right)$ where $\chi(x)=\frac{1}{2}(1+\tanh(x))$ and $X_{k}(t)=\frac{1}{2}\big{(}a_{k}(t)+a_{k-1}(t)\big{)}$ . Therefore we can differentiate $\widetilde{p}_{k}$ and it leads to

	$\displaystyle\dfrac{d}{dt}\widetilde{p}_{k}(t)=\int_{\mathbb{R}}\bigg{(}\tau_{0}\chi^{\prime}\Big{(}\tau_{0}\big{(}x-$	$\displaystyle X_{k}(t)\big{)}\Big{)}\Big{(}\widetilde{q}\big{(}(\eta(t,x),v(t,x))\big{)}-X_{k}^{\prime}(t)\eta v\Big{)}$
		$\displaystyle+\dfrac{\tau_{0}^{3}}{2}\chi^{\prime\prime\prime}\Big{(}\tau_{0}\big{(}x-X_{k}(t)\big{)}\Big{)}\big{(}\eta(t,x)+\ln(1-\eta(t,x)\big{)}\bigg{)}dx,$

where

\widetilde{q}(\eta,v)=(1-2\eta)v^{2}+\widetilde{F}(\eta)+\dfrac{(3-2\eta)(\partial_{x}\eta)^{2}}{4(1-\eta)^{2}}.

According to the arguments in [14], we decompose this integral into two complementary parts. Namely, we decompose the line as an interval $I$ focusing on the area where $\chi$ varies and a second part which takes into account the whereabouts of the solitons in which $\chi^{\prime}$ is small. This study can be summarized as follows.

Lemma 6.2.

Given an interval $I$ , there exist two positive constants $C$ and $C_{\ln}$ that only depends on $\mathfrak{c}^{*}$ such that we have

\dfrac{d}{dt}\widetilde{p}_{k}(t)\geq-C\sup_{x\in I^{c}}\chi^{\prime}\Big{(}\tau_{0}\big{(}x-X_{k}(t)\big{)}\Big{)}+\tau_{0}\int_{I}q(\eta,v)\chi^{\prime}\Big{(}\tau_{0}\big{(}x-X_{k}(t)\big{)}\Big{)}dx,

(6.3)

where

q(\eta,v)=(1-2\eta)v^{2}-\sqrt{c_{s}^{2}-\dfrac{\nu_{\mathfrak{c}^{*}}^{2}}{4}}|\eta v|+\left(-\int_{0}^{1}rf^{\prime}(1-r\eta)dr-\tau_{0}^{2}C_{\ln}\right)\eta^{2}.

Proof.

To obtain the second term in the right-hand side of (6.3), we first use (38) to get $X_{k}^{\prime}(t)^{2}\leq c_{s}^{2}-\frac{\nu_{\mathfrak{c}^{*}}^{2}}{4}$ and also write a first order Taylor expansion between $\eta$ and 0 which reads

\widetilde{F}(\eta)=-\eta^{2}\int_{0}^{1}rf^{\prime}(1-r\eta)dr.

(6.4)

Moreover, by (31), $\eta\leq\frac{1+\beta^{*}}{2}<1$ , so that $\frac{(3-2\eta)(\partial_{x}\eta)^{2}}{4(1-\eta)^{2}}\geq 0$ . Some elementary analysis on the function $\xi\mapsto\xi+\ln(1-\xi)$ provides the constant $C_{\ln}$ such that for any $\xi\in[0,\frac{1+\beta^{*}}{2}]\subset[0,1)$ , we have $\big{|}\xi+\ln(1-\xi)|\leq\frac{C_{\ln}}{4}\xi^{2}$ . Combining this and the fact that $|\chi^{\prime\prime\prime}|\leq 8\chi^{\prime}$ yields to the right-hand side term in (6.3).
Concerning the complementary set $I^{c}$ , we just bound, up to a constant, the integrand of $\frac{d}{dt}\widetilde{p}_{k}(t)$ by the energy density $e(\eta,v)$ . This can be dealt as in [4], apart from the terms involving $\eta^{2}$ . For these terms, we use (H1) and (6.4), to get

|\widetilde{F}(\eta)+\eta+\ln(1-\eta)|\leq\bigg{(}2\|f^{\prime}\|_{L^{\infty}([0,2])}+C_{\ln}\bigg{)}\dfrac{F(1-\eta)}{c_{s}^{2}},

and the conclusion follows. ∎

By continuity of $f^{\prime}$ , there exists $\mathfrak{d}\in(0,\frac{1}{2}]$ such that

m_{\mathfrak{d}}:=\min_{|x|\leq\mathfrak{d}}\big{(}-f^{\prime}(1+x)\big{)}>0,

(6.5)

because of (2). We can now conclude the proof of Proposition 1.17.

Proof of Proposition 1.17.

We apply Lemma 6.2 to

I=I_{k}(L):=\left[X_{k}(t)-\dfrac{L-1+\sigma^{*}t}{4},X_{k}(t)+\dfrac{L-1+\sigma^{*}t}{4}\right],

with some suitable choice of $(\alpha,L)$ that is exhibited later.

Step 1: Nonnegativity on $I_{k}(L)$ . According to the orthogonal decomposition of $\eta$ , we write for any $t\in[0,T]$ , $\eta(t,x)=\eta_{\mathfrak{c}(t),\mathfrak{a}(t)}(x)+\varepsilon_{\eta}(t,x)$ . By (28) and the one-dimensional Sobolev embedding, there exists $\alpha_{4}$ small enough such that for $\alpha\leq\alpha_{4}$ and $(t,x)\in[0,T]\times I_{k}(L)$ ,

\big{|}\eta(t,x)\big{|}\leq\sum_{j=1}^{N}\big{|}\eta_{c_{j}(t),a_{j}(t)}(x)\big{|}+\dfrac{\mathfrak{d}}{2}.

By exponential decay, we also have for any $j\in\{1,...,N\}$ , $\big{|}\eta_{c_{j}(t),a_{j}(t)}(x)\big{|}\leq K_{d}e^{-a_{d}\nu_{c_{j}(t)}|x-a_{j}(t)|}$ . By virtue of (37), we have for any $j$ , and $(t,x)\in[0,T]\times I_{k}(L)$ ,

\displaystyle|x-a_{j}(t)|

\displaystyle\geq\big{|}X_{k}(t)-a_{j}(t)\big{|}-\big{|}X_{k}(t)-x|\geq\dfrac{L-1}{4}.

Therefore, using in addition (33), there exists $L_{4}$ large enough such that for $L\geq L_{4}$ , for any $(t,x)\in[0,T]\times I_{k}(L)$ , $\big{|}\eta(t,x)\big{|}\leq\mathfrak{d}$ . Now labelling $\widetilde{\sigma}_{0}(\eta):=-\int_{0}^{1}rf^{\prime}(1-r\eta)dr-\tau_{0}^{2}C_{\ln}$ , we first notice that $\widetilde{\sigma}_{0}(\eta)\geq\frac{m_{\mathfrak{d}}}{2}+\mathcal{O}\left(\tau_{0}^{2}\right)$ . We impose that $\tau_{0}$ is small enough so that $\widetilde{\sigma}_{0}(\eta)>0$ . We apply Gauss’ reduction method to compute

q(\eta,v)=\widetilde{\sigma}_{0}(\eta)\left(\eta-\dfrac{\sigma_{0}}{2\widetilde{\sigma}_{0}(\eta)}v\right)^{2}+\left(1-2\eta-\dfrac{\sigma_{0}^{2}}{4\widetilde{\sigma}_{0}(\eta)}\right)v^{2},

where $\sigma_{0}:=\sqrt{c_{s}^{2}-\frac{\nu_{\mathfrak{c}^{*}}^{2}}{4}}$ . Now, we claim the following, and conclude the proof.

Claim 6.3.

Up to taking smaller parameters $\mathfrak{d}$ and $\tau_{0}$ , we have for any $(t,x)\in[0,T]\times I_{k}(L)$ ,

\eta(t,x)\leq\mathfrak{d}\text{ and }1-2\eta-\dfrac{\sigma_{0}^{2}}{4\widetilde{\sigma}_{0}(\eta)}\geq 0.

We then deduce that the integral in (6.3) is positive.

Step 2: Conclusion. Since $\chi^{\prime}(y)\leq 2e^{-2|y|}$ for any $y\in\mathbb{R}$ , we infer that for $x\notin I_{k}(L)$ , $\chi^{\prime}\big{(}\tau_{0}(x-X(t))\big{)}\leq 2e^{-\tau_{0}\frac{L-1+\sigma^{*}t}{2}}$ . Combining the estimate on $I_{k}(L)^{c}$ with Step 1, we conclude that up to reducing the value of $\mathfrak{d}$ defined in (6.5) and the value of $\tau_{0}$ , there exists suitable $(\alpha_{4},L_{4})$ such that for any $(\alpha,L)\prec(\alpha_{4},L_{4})$ , and any $t\in[0,T]$ , the monotonicity formula (39) can be derived from (6.3). To finish, we just state that (40) is a straightforward corollary of the monotonicity formula on $\widetilde{p}_{k}$ . ∎

To finish this section, we prove Claim 6.3.

Proof of Claim 6.3.

The way we have constructed $(\alpha_{4},L_{4})$ only depends on $\mathfrak{d}$ . Then we always can take $(\alpha_{4},L_{4})$ such that $|\eta|\leq\mathfrak{d}$ . Secondly,

1-2\eta-\dfrac{\sigma_{0}^{2}}{4\widetilde{\sigma}_{0}(\eta)}\geq 0\Longleftrightarrow\eta\leq\dfrac{1}{2}-\dfrac{\sigma_{0}^{2}}{8\widetilde{\sigma}_{0}(\eta)}.

Since $\eta$ is bounded and $f^{\prime\prime}$ is continuous, and using (3), we have

\dfrac{1}{2}-\dfrac{\sigma_{0}^{2}}{8\widetilde{\sigma}_{0}(\eta)}=\dfrac{1}{2}-\dfrac{c_{s}^{2}-\frac{\nu_{\mathfrak{c}^{*}}}{4}}{8\Big{(}\frac{-f^{\prime}(1)}{2}+\mathcal{O}\left(\eta\right)+\mathcal{O}\left(\tau_{0}^{2}\right)\Big{)}}=\dfrac{\nu_{\mathfrak{c}^{*}}^{2}}{8c_{s}^{2}}+\mathcal{O}\left(\mathfrak{d}^{2}\right)+\mathcal{O}\left(\tau_{0}^{2}\right).

Taking possibly smaller $\mathfrak{d}$ and $\tau_{0}$ , we obtain

\eta\leq\mathfrak{d}\leq\dfrac{1}{2}-\dfrac{\sigma_{0}^{2}}{8\widetilde{\sigma}_{0}(\eta)}.

∎

Here we give the proof of Proposition 1.18.

Proof.

Assertion (41) is a straightforward deduction from (37). Now we prove simultaneously the remaining assertions. Integrating (40), yields the upper bound

\mathcal{G}(t)-\mathcal{G}(0)\leq\mathcal{O}\left(Le^{-a_{d}\tau_{0}L}\right).

(6.6)

Combining (6.6), the upper bound on $\mathcal{G}(0)$ and the lower bound on $\mathcal{G}(t)$ in Corollary 1.13 then provides, for any $t\in[0,T]$ ,

\left\|\varepsilon(t)\right\|_{\mathcal{X}}^{2}\leq\mathcal{O}\left(\left\|\varepsilon(0)\right\|_{\mathcal{X}}^{2}\right)+\mathcal{O}\left(\left|\mathfrak{c}(0)-\mathfrak{c}^{*}\right|^{2}\right)+\mathcal{O}\left(\left\|\varepsilon(t)\right\|_{\mathcal{X}}^{3}\right)+\mathcal{O}\left(\left|\mathfrak{c}(t)-\mathfrak{c}^{*}\right|^{2}\right)+\mathcal{O}\left(Le^{-a_{d}\tau_{0}L}\right).

(6.7)

Now, we write

\left|\mathfrak{c}(t)-\mathfrak{c}^{*}\right|\leq\left|\mathfrak{c}(t)-\mathfrak{c}(0)\right|+\left|\mathfrak{c}(0)-\mathfrak{c}^{*}\right|.

(6.8)

Arguing the same way than in the proof of Corollary 1.12 (see (4.1)), we obtain that $\frac{d^{2}}{dc^{2}}\big{(}p(Q_{c})\big{)}$ is uniformly bounded on $[c_{k}(t),c_{k}(0)]$ (or $[c_{k}(0),c_{k}(t)]$ ). Then, a Taylor expansion and (33) provides $\alpha_{5}>0$ small enough such that if $\alpha\leq\alpha_{5}$ , we have uniformly in $t$ ,

\left|p(Q_{c_{k}(0)})-p(Q_{c_{k}(t)})\right|\geq\dfrac{\kappa_{\mathfrak{c}^{*}}}{2}|c_{k}(t)-c_{k}(0)|+\mathcal{O}\left(\left|c_{k}(t)-c_{k}(0)\right|^{2}\right),

(6.9)

hence

\left|\mathfrak{c}(t)-\mathfrak{c}^{*}\right|\leq\mathcal{O}\left(\left|p(Q_{c_{k}(0)})-p(Q_{c_{k}(t)})\right|\right)+\left|\mathfrak{c}(0)-\mathfrak{c}^{*}\right|.

(6.10)

From the definition of $\widetilde{p}$ and Proposition 1.9 and the choice of $\tau_{0}$ made in the proof of Corollary 1.13, we have for any $k\in\{1,...,N\}$ and any $t\in[0,T]$ ,

\widetilde{p}_{k}(t)-\widetilde{p}_{k+1}(t)=p_{k}(t)=p(Q_{c_{k}(t)})+\mathcal{O}\left(\left\|\varepsilon(t)\right\|_{\mathcal{X}}^{2}\right)+\mathcal{O}\left(Le^{-a_{d}\tau_{0}L}\right),

(6.11)

hence

\big{|}p(Q_{c_{k}(0)})-p(Q_{c_{k}(t)})\big{|}\leq\sum_{l=k}^{k+1}\big{|}\widetilde{p}_{l}(0)-\widetilde{p}_{l}(t)\big{|}+\mathcal{O}\left(\left\|\varepsilon(t)\right\|_{\mathcal{X}}^{2}\right)+\mathcal{O}\left(\left\|\varepsilon(0)\right\|_{\mathcal{X}}^{2}\right)+\mathcal{O}\left(Le^{-a_{d}\tau_{0}L}\right).

(6.12)

By (39),

\big{|}\widetilde{p}_{l}(0)-\widetilde{p}_{l}(t)\big{|}=\mathcal{O}\left(Le^{-a_{d}\tau_{0}L}\right),

therefore, by plugging the previous estimate in (6.12) and by using (6.10), we derive (42) . Furthermore, taking possibly $\alpha_{5}$ smaller, there also exists $L_{5}$ large enough such that if $L\geq L_{5}$ , (6.7) provides (43). ∎

Appendix A Useful estimates

In the following lemmas, we shall make a crucial use of the exponential decay of the solitons.

Lemma A.1.

Set $\mathfrak{c}^{*}:=(c_{1}^{*},...,c_{N}^{*})$ . Then there exists $K_{lip}>0$ only depending on $\mathfrak{c}^{*}$ such that for any $\mathfrak{a}:=(a_{1},...,a_{N}),(a_{1}^{*},...,a_{N}^{*})\in\mathbb{R}^{N}$ and $\mathfrak{c}\in\mathcal{B}(\mathfrak{c}^{*},\delta^{*})$ with $\delta^{*}=\min(\frac{\mu_{\mathfrak{c}^{*}}}{2},\frac{\nu_{\mathfrak{c}^{*}}}{2})$ , we have

\left\|R_{\mathfrak{c}^{*},\mathfrak{a}^{*}}-R_{\mathfrak{c},\mathfrak{a}}\right\|_{\mathcal{X}}\leq K_{lip}\big{(}\left|\mathfrak{c}^{*}-\mathfrak{c}\right|+\left|\mathfrak{a}^{*}-\mathfrak{a}\right|\big{)}.

Proof.

By invariance under translation, we can assume $\mathfrak{a}=0$ . In order to obtain some Lipschitz estimate, we infer the following control, using the assumptions on $\delta^{*}$ and (10). For $l\in\{0,1\}$ , we have

	$\displaystyle\left\|\nabla_{(a,c)}\partial_{x}^{l}\eta_{c_{k},a_{k}}(x)\right\|+\left\|\nabla_{(a,c)}v_{c_{k},a_{k}}(x)\right\|$	$\displaystyle\leq K_{d}(1+\nu_{c_{k}}^{2})\Big{(}1+\dfrac{1}{c_{k}^{3}}\Big{)}e^{-a_{d}\nu_{c_{k}}\|x-a_{k}\|}$
		$\displaystyle\leq K_{d}(1+\nu_{c_{k}^{}-\delta^{}}^{2})\Big{(}1+\dfrac{1}{\mu_{\mathfrak{c}^{}}^{3}}\Big{)}e^{-a_{d}\nu_{c_{k}^{}+\delta^{*}}\|x-a_{k}\|}$

Therefore, $\left\|\nabla_{(c,a)}Q_{c_{k},a_{k}}\right\|_{\mathcal{X}}$ is bounded by a constant $K_{lip}$ , depending only on $\mathfrak{c}^{*}$ , for any $(c,a)$ and this leads to the desired estimate. ∎

Remark A.2.

We can use similarly the exponential decay of Subsection 2.2 in order to show that Lemma A.1 still holds if we take the norms $\|.\|_{\mathcal{X}^{k}}$ with $k\in\{1,...,4\}$ .

Here we give a proof of Lemma 1.11. We define

\mathfrak{e}_{\mathfrak{c}}:=\max\left\{\left\|\eta_{c_{k}}\right\|_{L^{\infty}}\big{|}k\in\{1,...,N\}\right\}<1,

by (8).

Proof of Lemma 1.11.

We define for $k\in\{1,...,N\}$ , $L>0$ and $\mathfrak{a}=(a_{1},...,a_{N})\in\mathrm{Pos}_{N}(L)$ , the sets

J_{k}:=\left\{x\in\mathbb{R}\Big{|}|x-a_{k}|<\frac{L}{3}\right\}.

By construction $J_{k}\cap J_{j}=\emptyset$ . Set $J=\cup_{k=1}^{N}J_{k}$ . For all $k\in\{1,...,N\}$ and $x\in J_{k}$ , $\left|\eta_{c_{k}^{*},a_{k}}(x)\right|\leq\mathfrak{e}_{\mathfrak{c}^{*}}<1$ . Then

\left|\sum_{k=1}^{N}\eta_{c_{k}^{*},a_{k}}(x)\mathds{1}_{J_{k}}(x)\right|\leq\mathfrak{e}_{\mathfrak{c}^{*}}.

By exponential decay (10), we write for any $x\in\mathbb{R}$ ,

	$\displaystyle\|\eta_{\mathfrak{c}^{*},\mathfrak{a}}(x)\|$	$\displaystyle\leq\left\|\sum_{k=1}^{N}\eta_{c_{k}^{},a_{k}}(x)\mathds{1}_{J_{k}}(x)\right\|+\left\|\sum_{k=1}^{N}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}\eta_{c_{j}^{},a_{j}}(x)\mathds{1}_{J_{k}}(x)\right\|+\left\|\sum_{k=1}^{N}\eta_{c_{k}^{*},a_{k}}(x)\mathds{1}_{J^{c}}(x)\right\|$
		$\displaystyle\leq\mathfrak{e}_{\mathfrak{c}^{}}+K_{d}\sum_{k=1}^{N}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}e^{-\nu_{\mathfrak{c}^{}}\|x-a_{j}\|}\mathds{1}_{J_{k}}(x)+K_{d}\sum_{k=1}^{N}e^{-\nu_{\mathfrak{c}^{*}}\|x-a_{k}\|}\mathds{1}_{J^{c}}(x).$

Since $\mathfrak{a}\in\mathrm{Pos}_{N}(L)$ , for $x\in J_{j}$ with $j\neq k$ ,

e^{-\nu_{\mathfrak{c}^{*}}|x-a_{k}|}\leq e^{-\frac{2\nu_{\mathfrak{c}^{*}}L}{3}}.

Moreover, for $x\in J^{c}$ , and any $k\in\{1,...,N\}$ ,

e^{-\nu_{\mathfrak{c}^{*}}|x-a_{k}|}\leq e^{-\frac{\nu_{\mathfrak{c}^{*}}L}{3}}.

We conclude the proof by taking $L_{6}$ large enough such that

\beta^{*}:=\mathfrak{e}_{\mathfrak{c}^{*}}+N(N-1)K_{d}e^{-\frac{2\nu_{\mathfrak{c}^{*}}L_{6}}{3}}+K_{d}Ne^{-\frac{\nu_{\mathfrak{c}^{*}}L_{6}}{3}}<1,

and choose $\alpha_{6}<1-\beta^{*}$ . ∎

We deduce from (4) and (5) the following result that provides a control on the operator $\mathcal{X}$ -norm of the first derivative of the energy and the momentum.

Lemma A.3.

For $Q,\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})$ , we have

\left|\nabla p(Q).\varepsilon\right|\leq\left\|Q\right\|_{L^{2}\times L^{2}}\left\|\varepsilon\right\|_{\mathcal{X}}\quad\text{and}\quad\left|\nabla^{2}p(Q).(\varepsilon,\varepsilon)\right|\leq\left\|\varepsilon\right\|_{\mathcal{X}}^{2}.

Moreover there exists a constant $K_{E}>0$ , and integers $i_{E},\widetilde{i}_{E}$ such that for $Q=(\eta,v)\in\mathcal{NX}_{hy}(\mathbb{R})$ and $\varepsilon\in\mathcal{X}_{hy}(\mathbb{R})$ , we have for $l\in\{1,2,3\}$ ,

\left|\nabla^{l}E(Q).(\varepsilon_{l})\right|\leq K_{E}\left(1+\dfrac{1}{\inf_{\mathbb{R}}(1-\eta)^{i_{E}}}+\left\|Q\right\|_{\mathcal{X}}^{\widetilde{i}_{E}}+\left\|f^{(l-1)}(1-\eta)\right\|_{L^{\infty}}\right)\left\|\varepsilon\right\|_{\mathcal{X}}^{l},

where $\varepsilon_{l}$ designates $\varepsilon,(\varepsilon,\varepsilon)$ or $(\varepsilon,\varepsilon,\varepsilon)$ according to the value of $l$ .

Proof.

The two first derivatives of the energy can be directly deduced from the expressions (3.6), (3). Regarding the third one, we compute

\nabla^{3}E(Q)(\varepsilon,\varepsilon,\varepsilon)=\int_{\mathbb{R}}\left(\dfrac{3}{4(1-\eta)^{2}}+\dfrac{3}{2(1-\eta)^{3}}+\dfrac{3}{8(1-\eta)^{4}}+\dfrac{f^{\prime\prime}(1-\eta)}{2}\right)\varepsilon_{\eta}^{3}-\int_{\mathbb{R}}\varepsilon_{\eta}^{2}\varepsilon_{v},

and deduce the desired estimate. ∎

Appendix B Exponential decay for multivariate polynomials

The following lemma gives an explicit control on the $L^{p}$ -norm of a product of decaying and translated exponential functions. Its proof can be found in [4].

Lemma B.1.

Let $(a,b)\in\mathbb{R}^{2}$ with $a<b$ , $(\nu_{a},\nu_{b})\in(\mathbb{R}_{+}^{*})^{2}$ and set $y^{\pm}=\max(\pm y,0)$ , then

\left\|e^{-\nu_{a}(.-a)^{+}}e^{-\nu_{b}(.-b)^{-}}\right\|_{L^{p}}\leq\bigg{(}\dfrac{2}{p\min(\nu_{a},\nu_{b})}+b-a\bigg{)}^{\frac{1}{p}}e^{-\min(\nu_{a},\nu_{b})(b-a)},

for all $p\in[1,...,+\infty]$ and with the convention $\frac{1}{p}=0$ if $p=+\infty$ .

For $M\geq 2$ , we recall the following linear subspace of the set of multivariate real polynomials:

\displaystyle\mathcal{P}_{M}=\mathcal{P}[X_{1},...,X_{M}]:=\Big{\{}\sum_{|\alpha|\leq m}p_{\alpha}X^{\alpha}\Big{|}m\in\mathbb{N},p_{ke_{i}}=0\ ,\forall(k,i)\in\{0,...,m\}\times\{1,...,M\}\Big{\}}.

In light of the previous lemma, plugging functions that decay exponentially into such polynomials, we can give the following proof of Lemma 3.1.

Proof of Lemma 3.1.

Writing $P=\sum_{|\alpha|\leq m}p_{\alpha}X^{\alpha}\in\mathcal{P}[X_{1},...,X_{M}]$ , we have for every $\alpha=(\alpha_{1},...,\alpha_{M})$ at least two indexes $i\neq j$ such that $\alpha_{i},\alpha_{j}\geq 1$ . Since the remaining functions $(\tau_{a_{k}}f_{k})^{\alpha_{k}}$ ( $k\notin\{i,j\}$ ) are bounded by a constant $K>0$ , we obtain

	$\displaystyle\left\\|P\big{(}\tau_{a_{1}}f_{1},...,\tau_{a_{M}}f_{M}\big{)}\right\\|_{L^{p}}$	$\displaystyle\leq\sum_{\|\alpha\|\leq m}\|p_{\alpha}\|\left\\|(\tau_{a_{1}}f_{1})^{\alpha_{1}}...(\tau_{a_{M}}f_{M})^{\alpha_{M}}\right\\|_{L^{p}}$
		$\displaystyle\leq K\left\\|(\tau_{a_{i}}f_{i})^{\alpha_{i}}(\tau_{a_{j}}f_{j})^{\alpha_{j}}\right\\|_{L^{p}}$
		$\displaystyle=\mathcal{O}\big{(}\left\\|e^{-\alpha_{i}b_{i}\|x-a_{i}\|}e^{-\alpha_{j}b_{j}\|x-a_{j}\|}\right\\|_{L^{p}}\big{)}.$

Then, since $\alpha_{i},\alpha_{j}\geq 1$ and $y^{\pm}\leq|y|$ for any $y\in\mathbb{R}$ , according to Lemma B.1, we have

\left\|P\big{(}\tau_{a_{1}}(f_{1}),...,\tau_{a_{M}}(f_{M})\big{)}\right\|_{L^{p}}=\mathcal{O}\left(\Big{(}\frac{2}{p\min_{k}(b_{k})}+L\Big{)}^{\frac{1}{p}}e^{-\min_{k}(b_{k})L}\right).

(B.1)

∎

We now give some asymptotic developments of nonlinear quantities for the chain of solitons, in terms of polynomials of $\mathcal{P}_{M}$ .

Lemma B.2.

F(1-\eta_{\mathfrak{c},\mathfrak{a}})=\sum_{k=1}^{N}F(1-\eta_{c_{k},a_{k}})+\mathcal{O}\big{(}S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}S^{3,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}B^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}C^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}

f(1-\eta_{\mathfrak{c},\mathfrak{a}})=\sum_{k=1}^{N}f(1-\eta_{c_{k},a_{k}})+\mathcal{O}\big{(}(S^{2,N}+B^{N})(\eta_{\mathfrak{c},\mathfrak{a}})\big{)},

and for any $k\in\{1,...,N\}$

f^{\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}})=f^{\prime}(1-\eta_{c_{k},a_{k}})+\mathcal{O}\Big{(}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}\eta_{c_{j},a_{j}}\Big{)}.

Proof.

We write for any $x\in\mathbb{R}$

F(1+x)=-\dfrac{x^{2}}{2}f^{\prime}(1)-\dfrac{x^{3}}{2}\int_{0}^{1}(1-t)^{2}f^{\prime\prime}(1+tx)dt,

(B.2)

hence, by the multinomial formulae,

	$\displaystyle F(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{k=1}^{N}F(1-$	$\displaystyle\eta_{c_{k},a_{k}})=-\dfrac{f^{\prime}(1)}{2}\Big{(}\eta_{\mathfrak{c},\mathfrak{a}}^{2}-\sum_{k=1}^{N}\eta_{c_{k},a_{k}}^{2}\Big{)}$
		$\displaystyle-\Big{(}\eta_{\mathfrak{c},\mathfrak{a}}^{3}-\sum_{k=1}^{N}\eta_{c_{k},a_{k}}^{3}\Big{)}\int_{0}^{1}\dfrac{(1-t)^{2}}{2}f^{\prime\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}}t)dt$
		$\displaystyle-\sum_{k=1}^{N}\eta_{c_{k},a_{k}}^{3}\int_{0}^{1}\dfrac{(1-t)^{2}}{2}\big{(}f^{\prime\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}}t)-f^{\prime\prime}(1-\eta_{c_{k},a_{k}}t)\big{)}dt$
		$\displaystyle=-f^{\prime}(1)S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})-(3B^{N}+6S^{3,N})(\eta_{\mathfrak{c},\mathfrak{a}})\int_{0}^{1}\dfrac{(1-t)^{2}}{2}f^{\prime\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}}t)dt$
		$\displaystyle-\sum_{k=1}^{N}\eta_{c_{k},a_{k}}^{3}\int_{0}^{1}\dfrac{(1-t)^{2}}{2}\big{(}f^{\prime\prime}(1-\eta_{\mathfrak{c},\mathfrak{a}}t)-f^{\prime\prime}(1-\eta_{c_{k},a_{k}}t)\big{)}dt.$

On the other hand, we have for $x\in\mathbb{R}$

f(1+x)=xf^{\prime}(1)+x^{2}\int_{0}^{1}(1-t)f^{\prime\prime}(1+tx)dt,

(B.3)

hence

	$\displaystyle f(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{k=1}^{N}f(1-\eta_{c_{k},a_{k}})$	$\displaystyle=\Big{(}\eta_{\mathfrak{c},\mathfrak{a}}^{2}-\sum_{k=1}^{N}\eta_{c_{k},a_{k}}^{2}\Big{)}\int_{0}^{1}(1-t)f^{\prime\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})dt$
		$\displaystyle\ +\sum_{k=1}^{N}\eta_{c_{k},a_{k}}^{2}\int_{0}^{1}(1-t)\big{(}f^{\prime\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})-f^{\prime\prime}(1-t\eta_{c_{k},a_{k}})\big{)}dt.$

By Lemma 2.8, for any $k\in\{1,...,N\}$ , $\eta_{c_{k},a_{k}}$ is bounded uniformly with respect to $\mathfrak{c},\mathfrak{a}$ and $x$ . Then by continuity of $f^{\prime\prime}$ and $f^{\prime\prime\prime}$ , there exists $M$ independent of $\mathfrak{c},\mathfrak{a}$ and $x,t$ such that for $i=1,2$ , $\big{|}\int_{0}^{1}\frac{(1-t)^{i}}{i}f^{\prime\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})dt\big{|}\leq M$ and

\big{|}f^{\prime\prime}(1-t\eta_{\mathfrak{c},\mathfrak{a}})-f^{\prime\prime}(1-t\eta_{c_{k},a_{k}})\big{|}\leq Mt|\eta_{\mathfrak{c},\mathfrak{a}}-\eta_{c_{k},a_{k}}|\leq Mt\Big{|}\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{N}\eta_{c_{j},a_{j}}\Big{|},

then we obtain

F(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{k=1}^{N}F(1-\eta_{c_{k},a_{k}})=\mathcal{O}\big{(}S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}S^{3,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}B^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}C^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)},

(B.4)

whereas

f(1-\eta_{\mathfrak{c},\mathfrak{a}})-\sum_{k=1}^{N}f(1-\eta_{c_{k},a_{k}})=\mathcal{O}\big{(}S^{2,N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}+\mathcal{O}\big{(}B^{N}(\eta_{\mathfrak{c},\mathfrak{a}})\big{)}.

(B.5)

Dealing the same way than previously with a Taylor expansion of order $1$ , we also get the last expression in Lemma B.2. ∎

References

[1] M. Abid, C. Huepe, S. Metens, C. Nore, C. T. Pham, L. S. Tuckerman, and M. E. Brachet. Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence. Fluid Dynamics Research, 33(5-6):509–544, December 2003.
[2] H. Berestycki and P.-L. Lions. Nonlinear scalar fields equations, I Existence of a ground state. Arch. Rational Mech. Anal., 82:313–345, 1983.
[3] J. Berthoumieu. Minimizing travelling waves for the one-dimensional nonlinear Schrödinger equations with non-zero condition at infinity. Pre-print ArXiV, arXiv:2305.17516, 2024.
[4] F. Bethuel, P. Gravejat, and D. Smets. Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation. Ann. Inst. Fourier, 64(1), 2014.
[5] D. Chiron. Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one. Nonlinearity, 25(3):813–850, 2012.
[6] D. Chiron. Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one. Anal. PDE, 6(6):1327–1420, 2013.
[7] C. Coste. Nonlinear Schrödinger equation and superfluid hydrodynamics. Eur. Phys. J. B, 1(2):245–253, 1998.
[8] C. Gallo. Schrödinger group on Zhidkov spaces. Adv. Differential Equations, 9(5-6):509–538, 2004.
[9] M. Grillakis, J. Shatah, and W.A. Strauss. Stability theory of solitary waves in the presence of symmetry I. J. Funct. Anal., 74(1):160–197, 1987.
[10] Y.S. Kivshar and B. Luther-Davies. Dark optical solitons: physics and applications. Phys. Rep., 298(2-3):81–197, 1998.
[11] Y.S. Kivshar, D.E. Pelinovsky, and Y.A. Stepanyants. Self-focusing of plane dark solitons in nonlinear defocusing media. Phys. Rev. E, 51(5):5016–5026, 1995.
[12] Zhiwu Lin. Stability and instability of traveling solitonic bubbles. Adv. Differential Equations, 7(8):897–918, 2002.
[13] M. Maris. Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity. SIAM J. Math. Anal., 40(3):1076–1103, 2008.
[14] Y. Martel, F. Merle, and T.-P. Tsai. Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations. Comm. Math. Phys., 231(2):347–373, 2002.
[15] H. Queffélec and C. Zuily. Analyse pour l’agrégation. Sciences Sup. Dunod, Paris, Quatrième edition, 2013.
[16] M.I. Weinstein. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal., 16(3):472–491, 1985.

$\displaystyle\left\\|Q(t)-R_{\mathfrak{c}^{*},\mathfrak{a}(t)}\right\\|_{\mathcal{X}}$	$\displaystyle\leq\left\\|\varepsilon(t)\right\\|_{\mathcal{X}}+\left\\|R_{\mathfrak{c}(t),\mathfrak{a}(t)}-R_{\mathfrak{c}^{*},\mathfrak{a}(t)}\right\\|_{\mathcal{X}}$
	$\displaystyle\leq\mathcal{O}\left(\left\\|\varepsilon(0)\right\\|_{\mathcal{X}}+\left\|\mathfrak{c}(0)-\mathfrak{c}^{}\right\|+L_{0}e^{-a_{d}\tau_{0}L_{0}}+\big{\|}\mathfrak{c}(t)-\mathfrak{c}^{}\big{\|}\right)$
	$\displaystyle\leq K_{6}\left(\alpha_{0}+e^{-a_{d}\frac{\tau_{0}}{2}L_{0}}\right).$	(44)

	$\displaystyle\inf_{\mathfrak{a}\in\mathrm{Pos}_{N}(L_{0}-2)}\left\\|Q(T^{})-R_{\mathfrak{c}^{},\mathfrak{a}}\right\\|_{\mathcal{X}}$	$\displaystyle\leq\left\\|Q(T^{})-R_{\mathfrak{c}^{},\mathfrak{a}(t^{n})}\right\\|_{\mathcal{X}}$
		$\displaystyle\leq\left\\|Q(T^{})-Q(t^{n})\right\\|_{\mathcal{X}}+\left\\|Q(t^{n})-R_{\mathfrak{c}^{},\mathfrak{a}(t^{n})}\right\\|_{\mathcal{X}}$
		$\displaystyle\leq\left\\|Q(T^{*})-Q(t^{n})\right\\|_{\mathcal{X}}+K_{6}\left(\alpha_{0}+e^{-a_{d}\frac{\tau_{0}}{2}L_{0}}\right).$

$\displaystyle H_{c_{k}}(\varepsilon)$	$\displaystyle=\lambda^{2}H_{c_{k}}(\partial_{x}Q_{c_{k}^{}})+\mu^{2}H_{c_{k}}\big{(}\nabla p(Q_{c_{k}^{}})\big{)}+H_{c_{k}}(\varepsilon^{*})$
	$\displaystyle+2\lambda\left\langle\mathcal{H}_{c_{k}}(\partial_{x}Q_{c_{k}^{}}),\varepsilon^{}\right\rangle_{L^{2}\times L^{2}}+2\mu\left\langle\mathcal{H}_{c_{k}}\big{(}\nabla p(Q_{c_{k}^{}})\big{)},\varepsilon^{}\right\rangle_{L^{2}\times L^{2}}$	(4.4)
	$\displaystyle+2\lambda\mu\left\langle\mathcal{H}_{c_{k}}(\partial_{x}Q_{c_{k}^{}}),\nabla p(Q_{c_{k}^{}})\right\rangle_{L^{2}\times L^{2}}.$

	$\displaystyle G(Q)=$	$\displaystyle\sum_{k=1}^{N}\big{(}E(Q_{c_{k}^{}})-c_{k}^{}p(Q_{c_{k}^{*}})\big{)}+\dfrac{1}{2}\bigg{(}\sum_{k=1}^{N}H_{c_{k}}(\varepsilon_{k})+\sum_{k=0}^{N}H_{0}^{k}(\varepsilon_{k,k+1},\varepsilon_{k,k+1})\bigg{)}$
		$\displaystyle+\mathcal{O}\big{(}\left\|\mathfrak{c}-\mathfrak{c}^{*}\right\|^{2}\big{)}+\mathcal{O}\Big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}^{2}\big{(}\tau+e^{-\frac{a_{d}\tau L}{2}}\big{)}\Big{)}+\mathcal{R}_{\mathfrak{c},\mathfrak{a}}(\varepsilon)+\mathcal{O}\left(\left\\|\varepsilon\right\\|_{\mathcal{X}}^{4}\right)$
		$\displaystyle+\mathcal{O}\Big{(}\Lambda(L,\mathfrak{c})(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\Big{)}+\mathcal{O}\big{(}\left\\|\varepsilon\right\\|_{\mathcal{X}}\Lambda(L,\mathfrak{c})^{\frac{1}{2}}(e^{-a_{d}\nu_{\mathfrak{c}}L}+e^{-a_{d}\tau_{0}L})\big{)},$

	$\displaystyle\left\\|P\big{(}\tau_{a_{1}}f_{1},...,\tau_{a_{M}}f_{M}\big{)}\right\\|_{L^{p}}$	$\displaystyle\leq\sum_{\|\alpha\|\leq m}\|p_{\alpha}\|\left\\|(\tau_{a_{1}}f_{1})^{\alpha_{1}}...(\tau_{a_{M}}f_{M})^{\alpha_{M}}\right\\|_{L^{p}}$
		$\displaystyle\leq K\left\\|(\tau_{a_{i}}f_{i})^{\alpha_{i}}(\tau_{a_{j}}f_{j})^{\alpha_{j}}\right\\|_{L^{p}}$
		$\displaystyle=\mathcal{O}\big{(}\left\\|e^{-\alpha_{i}b_{i}\|x-a_{i}\|}e^{-\alpha_{j}b_{j}\|x-a_{j}\|}\right\\|_{L^{p}}\big{)}.$

Orbital stability of a chain of dark solitons for general nonintegrable Schrödinger equations with non-zero condition at infinity.

Abstract

1 Introduction

Theorem 1.1 (Gallo [8]).

1.1 Travelling wave solutions and minimizing property

1.2 Chain of solitons in the transonic regime

Theorem 1.2.

Remark 1.3.

Remark 1.4.

Theorem 1.5.

Remark 1.6.

1.3 Sketch of the proof

1.3.1 Coercivity around the soliton QcQ_{c}

Proposition 1.7.

1.3.2 Almost minimizing property of a sum of solitons

Proposition 1.8.

Proposition 1.9.

1.3.3 Orthogonal decomposition

Proposition 1.10.

Lemma 1.11.

Corollary 1.12.

Corollary 1.13.

1.3.4 Evolution in time

Proposition 1.14.

Remark 1.15.

Remark 1.16.

Proposition 1.17.

1.3.5 Proof of the orbital stability

Proposition 1.18.

Proof of Theorem 1.5.

2 Existence of a branch of smooth travelling waves

Theorem 2.1 (Theorem 4 in [5]).

2.1 A priori existence of a branch of travelling waves

2.2 Decay at infinity

Proposition 2.2 ([12],[5]).

Remark 2.3.

Proposition 2.4.

Proof.

Remark 2.5.

Lemma 2.6.

Proof.

Claim 2.7.

Proof.

Lemma 2.8.

Proof.

Lemma 2.9.

Proof.

2.3 Proof of Theorem 1.2

3 Perturbation of a sum of solitons and minimizing properties

Lemma 3.1.

Remark 3.2.

Proof of Proposition 1.8.

Claim 3.3.

Proof.

Claim 3.4.

Proof.

4 Orthogonal decomposition and control on the functional GG

Proof of Corollary 1.12.

Corollary 4.1.

Proof.

Corollary 4.2.

Proof of Corollary 1.13.

5 Dynamics of the modulation parameters

Proof of Proposition 1.14.

Claim 5.1.

Proof.

6 Monotonicity and final estimates

Lemma 6.1.

Lemma 6.2.

Proof.

Proof of Proposition 1.17.

Claim 6.3.

Proof of Claim 6.3.

Proof.

Appendix A Useful estimates

Lemma A.1.

Proof.

Remark A.2.

Proof of Lemma 1.11.

Lemma A.3.

1.3.1 Coercivity around the soliton $Q_{c}$

4 Orthogonal decomposition and control on the functional $G$