Joackim Bernier
email: joackim.bernier@ens-rennes.fr

Erwan Faou
email: erwan.faou@inria.fr

EXISTENCE AND STABILITY OF TRAVELING WAVES FOR DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS OVER LONG TIMES

We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schrödinger equation (DNLS) with cubic nonlinearity, near the continuous limit. We construct a family of solutions close to the continuous traveling waves and prove their stability over long times. Applying a modulation method, we also show that we can describe the dynamics near these discrete traveling waves over long times.

1. Introduction 1.1. Motivations and main results. We study existence and stability of solitary traveling waves for the discrete nonlinear Schrödinger equation (DNLS) on a grid hZ of stepsize h ą 0 and with a cubic focusing non linearity. This equation is a differential equation on C hZ defined by (see [START_REF] Kevrekidis | The discrete nonlinear Schrödinger equation[END_REF] for details about its derivation) [START_REF] Bambusi | Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation[END_REF] @g P hZ, iB t u g " u g`h ´2 u g `ug´h h 2 `| u g | 2 u g .

We focus on this equation near its continuous limit (as h goes to 0), called non linear Schrödinger equation (NLS), defined as the following partial differential equation (2) @x P R, iB t upxq " B 2 x upxq `|upxq| 2 upxq.

We study solutions of DNLS [START_REF] Bambusi | Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation[END_REF] with a behavior close to the continuous traveling waves of NLS [START_REF] Bambusi | Continuous approximation of breathers in one-and two-dimensional DNLS lattices[END_REF]. Such solitons u are global solutions of NLS with speed of oscillation ξ 1 and speed of advection ξ 2 , satisfying

(3) @t 0 P R, @t P R, @x P R, upt 0 `t, xq " e iξ1t upt 0 , x ´ξ2 tq.

1

The parameter ξ " pξ 1 , ξ 2 q characterizes travelling waves up to gauge transform upxq Þ Ñ e iγ upxq and advection upxq Þ Ñ upx ´yq. For NLS they are given explicitly by their values at time t " 0 (4) @x P R, ψ ξ pxq " e ˙2.

for speed of oscillation ξ 1 and speed of advection ξ 2 satisfying

(5)

ξ 1 ą ˆξ2 2 ˙2 .
On a grid, the notion of traveling wave is not as clear as on a line, and we cannot define traveling waves for DNLS as easily as those of NLS by [START_REF] Bernier | Optimality and resonances in a class of compact finite difference schemes of high order[END_REF]. The difficulty comes from the definition of the advection. Indeed, the canonical advection on a grid is only defined when the distance to cross is a multiple of the stepsize h. Of course, we could find some reasonable extensions of (3) in the discrete case. For example, a possible definition of discrete traveling waves could be for solution u to DNLS to satisfy [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] @t 0 P R, @n P Z, @g P hZ, u g pt 0 `nτ q " e iξ1nτ u g´nh pt 0 q with ξ 2 τ " h, for some speeds ξ 1 , ξ 2 P R. Even if this definition seems to be the most natural, it is not the only one possible.

For example, we could replace h by 2h in this definition or to do things even more complicated, and no canonical choice appears obvious. There is at least one class of solutions that can be defined without ambiguity, the standing waves (i.e. when ξ 2 " 0) which are solutions of the form (7) @t 0 P R, @t P R, upt 0 `tq " e iξ1t uptq.

for some speed of oscillation ξ 1 P R.

We define the discrete L 2 and H 1 norms as follows:

for v P C hZ , } v } 2 L 2 phZq " h ÿ gPhZ | v g | 2 and } v } 2 H 1 phZq " h ÿ gPhZ ˇˇˇv g ´vg´h h ˇˇˇ2 `} v } 2 L 2 phZq .
Of course, these norms are equivalents but not uniformly with respect to h. Since we focus on the continuous limit (i.e. when h goes to 0), uniformity with respect to h is crucial. The discrete L 2 norm, } ¨}2 L 2 phZq is a constant of the motion of DNLS associated, through Noether Theorem (see, for example, [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF] for details about this Theorem), to its invariance under gauge transform action. As L 2 phZq is an algebra we can deduce by Cauchy-Lipschitz Theorem that DNLS is globally well-posed in L 2 phZq. Moreover, DNLS is a Hamiltonian system associated with the Hamiltonian [START_REF] Eilbeck | The discrete self-trapping equation[END_REF] H DNLS puq " h 2

ÿ gPhZ ˇˇˇu g`h ´ug h ˇˇˇ2 ´h 4 ÿ gPhZ | u g | 4 .
As we can guess from its expression, this Hamiltonian is very useful to establish some estimates of coercivity with the discrete H 1 norm, uniformly with respect to h. The continuous traveling waves of NLS defined by (4) verify a property of stability called orbital stability. If for a given time a solution of NLS is close enough of a traveling wave, then it stays close of this traveling wave for all times, up to an advection and a gauge transform. This property has been first proven by Cazenave and Lions in 1982 in [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] by a compactness method and in 1986 by Weinstein in [START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] with what we call nowadays the energy-momentum method. This second method is more quantitative than the first one, and the estimates of stability we give in this article are all based on it. It has been developed by Grillakis, Shatah and Strauss in 1987 in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] and [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] (see also [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF] for a very clear presentation of this method).

Theorem 1.1. Cazenave and Lions [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF], Weinstein [START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] For each couple of speed ξ P R 2 , such that ξ 1 ą ´ξ2 2 ¯2, there exists a constant c ą 0, such that for all solutions u of NLS (2) with }up0q ´ψξ } H 1 pRq ă c and }up0q} 2 L 2 pRq " }ψ ξ } 2 L 2 pRq , for all time t P R, there exist y, γ P R such that c}uptq ´eiγ ψ ξ px ´yq} H 1 pRq ď }up0q ´ψξ } H 1 pRq .

This result does not give any information on the exact position of the solution. To remedy this problem, modulational stability methods have been developed, which allows to follow very precisely this solution (see [START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF] or [START_REF] Fröhlich | Solitary wave dynamics in an external potential[END_REF]).

If we try to apply energy-momentum method to construct orbitally stable traveling waves for DNLS, the main difficulty comes from the definition of the advection on the grid. We discuss this problem in detail in section 2. However this problem is easily solved when considering standing waves (i.e. ξ 2 " 0) with symmetric perturbations for which the solution, remaining symmetric for all times, cannot move. In 2010, Bambusi and Penati proved in [START_REF] Bambusi | Continuous approximation of breathers in one-and two-dimensional DNLS lattices[END_REF] the existence of standing waves of DNLS looking like those of NLS. In fact, they constructed two kinds of standing waves. Each ones are real valued and symmetric but the first ones, called Sievers-Takeno modes or onsite , are centered in 0 whereas the second ones, called Page modes or off-site, are centered in h 2 . In 2013, in [START_REF] Bambusi | Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation[END_REF], Bambusi, Faou and Grébert, studying fully discrete approximation in time and space of NLS standing waves, gave some results of their orbital stability. The construction of these standing waves is also realized in a 2016 paper of Jenkinson and Weinstein (see [START_REF] Jenkinson | Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier[END_REF]), with another kind of approximations. If we focus only on the onsite standing waves, we summarized a piece of these results in the following theorem.

Theorem 1.2. Existence and orbital stability of standing waves For all ξ 1 ą 0, there exists h 0 , C, c ą 0 such that for all h ă h 0 , there exists a unique φ h ξ1 P H 1 phZ; Rq symmetric, centered in 0, and ζ 1 P R, such that ' e iζ1t φ h ξ1 is a solution of DNLS, ' |ζ 1 ´ξ1 | `} φ h ξ1 ´ψpξ1,0q |hZ } H 1 phZq ď Ch 2 , ' If u is a solution of DNLS such that up0q is symmetric, centered in 0, and } up0q ´φh ξ1 } H 1 phZq ă c then for all t P R, there exists γ P R such that } uptq ´eiγ φ h ξ1 } H 1 phZq ď C} up0q ´φh ξ1 } H 1 phZq . Note that the same theorem holds, for the off site standing waves. We just need to write "symmetric, centered in h 2 " instead of "symmetric, centered in 0" and "ψ pξ1,0q p. ´h 2 q |hZ " instead of "ψ pξ1,0q |hZ ". Usually, it is enough to prove existence and orbital stability of NLS standing waves to get some orbitally stable traveling wave. Indeed, NLS is invariant by Galilean transformation , defined by

upt, xq Þ Ñ e i v 2 px´vtq`ip v 2 q
2 t upt, x ´vtq.

However, it seems there is no such transformation for DNLS. So we cannot apply the same strategy.

The second reason why existence of orbitally stable traveling waves for DNLS seems very uncertain is more experimental. If we assume that DNLS admits a moving traveling wave (i.e. ξ 2 ‰ 0) that is orbitally stable and looking like a continuous traveling wave, ψ ξ , then the solution of DNLS generated by the discretization of ψ ξ on hZ, should look like ψ ξ for all times, up to an advection and a gauge transform. But there are some reasonable numerical simulations for which it is not what is observed (see [START_REF] Jenkinson | Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier[END_REF]). In fact, the speed of this solution seems going to 0 as t goes to infinity. In the literature, this phenomenon is usually called Peierls-Nabarro barrier (see [START_REF] Jenkinson | Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier[END_REF], [START_REF] Kevrekidis | The discrete nonlinear Schrödinger equation[END_REF] and [START_REF] Oxtoby | Moving solitons in the discrete nonlinear Schrödinger equation[END_REF]). A rigorous proof of this phenomenon seems to be an open problem. However, it is really difficult to observe when h is small enough (in fact, stability for exponentially long times is expected, see [START_REF] Oxtoby | Moving solitons in the discrete nonlinear Schrödinger equation[END_REF]).

Before stating our main results, let us first formulate an easy corollary of them, showing that there exists quasi-traveling waves to DNLS close to the continuous limit, for times of order Oph ´2q, preventing the phenomena described above to appear before this time scale. Theorem 1.3. For all ε ą 0 and for all ξ P R 2 such that ξ 1 ą ´ξ2 2 ¯2, there exist h 0 , C, T 0 ą 0 such that T 0 " 8 when ξ 2 " 0 and T 0 Ñ 8 when the speed ξ 2 Ñ 0, and such that if h ă h 0 , y 0 , γ 0 P R and u is the solution of DNLS such that @g P hZ, u g p0q " e iγ0 ψ ξ pg ´y0 q, then, there exist γ, y P C 1 pRq satisfying γp0q " γ 0 and yp0q " y 0 such that, for all t ě 0,

@t ď T 0 h ´2`ε , sup gPhZ ˇˇugptq ´eiγptq ψ ξ pg ´yptqq ˇˇď Ch 2 and @t ď T 0 h ´2`ε , | 9 γptq ´ξ1 | `| 9 yptq ´ξ2 | ď Ch 2 .
The proof of Theorem 1.3 is a straightforward application of Theorem 1.7 (or Theorem 1.4 if ξ 2 " 0). It would be possible to write the same result with the discrete H 1 norm instead of the L 8 norm.

To obtain this result, the strategy is to construct a function close to the continuous solitary wave ψ ξ for given parameters ξ " pξ 1 , ξ 2 q, which define solitary waves of a modified version of DNLS essentially defined by removing the aliasing terms. This typically gives bound for time scales of order Oph ´1q for orbital stability in H 1 phZq. Moreover as the aliasing terms are small for regular functions, we can combine this analysis with a result of control of discrete Sobolev norms of DNLS to reach the time scale Oph ´2q. We give now the details of our results. The first one is a result of existence and stability in H 1 of discrete traveling waves for times of order h ´1.

Theorem 1.4. Let Ω be a relatively compact open subset of

" ξ P R 2 | ξ 1 ą ´ξ2 2 ¯2* .
There exist h 0 , κ, r, ą 0 such that for all h ă h 0 , for all ξ P Ω, there exists η h ξ P H 8 pRq with (9) }η h ξ ´ψξ } H 1 pRq ď κh 2 , satisfying the following property.

If v P H 1 phZq is an approximation of η h ξ up to a gauge transform or an advection, i.e.

Dγ 0 , y 0 P R, } v ´pe iγ0 η h ξ p¨´y 0 qq |hZ } H 1 phZq ď r,
then there exist γ, y P C 1 pRq with γp0q " γ 0 and yp0q " y 0 such that if T ą 0 and u, the solution of DNLS with up0q " v, satisfy (10) @t P p0, T q, δptq :" } uptq ´pe iγptq η h ξ p¨´yptqqq |hZ } H 1 phZq ď r, then we have for all t P p0, T q,

(11) | 9 γptq ´ξ1 | `| 9 yptq ´ξ2 | ď κ pδp0q `δptq `e´ h q, and (12)
δptq ď κ e h|ξ 2 |t 2 pδp0q `e´ h q.

The functions η h ξ are constructed in the third section and estimates [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] and [START_REF] Jenkinson | Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier[END_REF] are proven in the fourth section. Now, we discuss this result. We focus on inequalities [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] and [START_REF] Jenkinson | Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier[END_REF].

' If we remove the exponential terms, it is a result stronger than the classical inequality of orbital stability (see Theorem 1.1) as it includes a result of modulation. ' The exponential terms "e ´ h " means that any discretization of η h ξ is not exactly a traveling wave of DNLS. ' The time dependent exponential term means that the estimate of stability holds while t|ξ 2 | is smaller than h ´1. In particular, if we focus on standing waves (i.e. ξ 2 " 0), we get an estimate of stability for all times. Since our perturbation does not need to be symmetric, it is an extension of the previous results (see Theorem 1.2). ' If up0q is a discretization of η h ξ (i.e. if δp0q " 0) then the estimate of stability holds longer. Indeed, while t|ξ 2 | is smaller than 3 h 2 (up to a multiplicative constant) , then the bootstrap (10) condition is satisfied. In particular, we deduce of the second inequality that at the end of this time, u has crossed the distance 3 h 2 (up to a multiplicative constant), still looking like η h ξ . Now, we discuss some consequences and applications of the proof of Theorem (1.4). These extensions are linked to the two relevant exponents for h in this theorem.

First, there is a control of η h ξ ´ψξ by Oph 2 q (see ( 9)). This error is a consistency error. It is due to the approximation of the second derivative by a finite difference formula of order 2. Such an estimate depends on the finite difference operator used to approximate second derivative in space. For example, if we consider the generalization of DNLS (1) called Discret Self-Trapping equation (DST, see [START_REF] Eilbeck | The discrete self-trapping equation[END_REF]) [START_REF] Kevrekidis | The discrete nonlinear Schrödinger equation[END_REF] @g P hZ, iB t u g "

1 h 2 ÿ kPZ a k u g´kh `| u g | 2 u g ,
where pa k q kPZ P L 1 pZ; Rq is a symmetric sequence (i.e. a k " a ´k for all k), consistent of order 2n,

n P N ˚, (14)
@u P H 8 pRq, 1 h 2 ÿ kPZ a k uphkq " hÑ0 B 2 x up0q `Oph 2n`2 q,
and satisfying the estimate of stability [START_REF] Peyrière | Convolution, séries et intégrales de Fourier[END_REF] Dα ą 0, @ω P p0, πq, ´ÿ kPZ a k cospkωq ě αω 2

then Theorem (1.4) holds for DST and we can replace (9) by }η h ξ ´ψξ } H 1 pRq ď κh 2n . In particular, this extension includes usual pseudo spectral method and the usual high order discrete second derivatives (see [START_REF] Bernier | Optimality and resonances in a class of compact finite difference schemes of high order[END_REF] for details about these formulas) whose non-zero terms are given by

a ˘k " 2p´1q k`1 k 2 C n´k 2n C n 2n , if 0 ă k ă n, and a 0 " ´2 n ÿ j"1 1 j 2 .
Second, there is the right exponential term e h|ξ 2 |t 2

giving the stability estimates for times of order h ´1. As the error terms come mainly from aliasing effects, the control of stability for times larger than 1 h essentially relies on a control of higher Sobolev norms for long times uniformly with respect to h. More precisely, we define the discrete homogeneous Sobolev norm } ¨} 9

H n phZq by ( 16)

} u } 2 9
H n phZq " xp´∆ h q n u, uy L 2 phZq , with p∆ h uq g "

u g`h ´2 u g `ug´h h 2 ,
and the Sobolev norm by

} u } 2 H n phZq " n ÿ k"0 } u } 2 9
H k phZq .
Then we have the following version of Theorem 1.4 (see Remark 4.3 for its proof).

Theorem 1.5. In Theorem 1.4, the inequality (12) can be replaced by

(17) @n P N ˚, δptq ď κ ˆδp0q `e´ h `at|ξ 2 |h n´1 2 sup 0ăsăt } upsq} 9 H n phZq ˙.
With such an estimate, we see that to obtain stability over exponentially long times, it would be enough to prove a control of the growth of the homogeneous Sobolev norm of the type Ct α , with α independent of n and h and C independent of h. Note that for the continuous case, it is indeed the case for the solutions of NLS for which the H s norms are uniformly bounded in times by using integrability arguments (see for example [START_REF] Sohinger | Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on[END_REF]). Note that such bounds hold for linear Schrödinger equation with a smooth potential in t and x (see [START_REF] Bourgain | On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential[END_REF]).

For DNLS, it is possible to obtain polynomial control of the growth of Sobolev norms by using the higher modified energy method. The following result was obtained in [START_REF] Bernier | Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schödinger equations on hZ[END_REF] by the first author: Theorem 1.6 (Growth of discrete Sobolev norms, see [START_REF] Bernier | Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schödinger equations on hZ[END_REF]). For all n P N ˚, there exists C ą 0, such that for all h ą 0, if u is a solution of DNLS then for all t P R [START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] } uptq} 9

H n phZq ď C " } up0q} 9 H n phZq `M 2n`1 3 up0q `|t| n´1 2 M 4n´1 3 up0q ı ,
where M up0q " } up0q} 9 H 1 phZq `} up0q} 3 L 2 phZq . The exponents of the up0q norms are natural and correspond to an homogeneous estimate preserved by scalings in h. As a corollary of Theorem (1.5) and Theorem 1.6, we get an extension of Theorem 1.3 for smooth perturbations of η h ξ . It is a result of stability for times of order h ´2 for such perturbations.

Theorem 1.7. Let Ω be a relatively compact open subset of

" ξ P R 2 | ξ 1 ą ´ξ2 2 ¯2*
and h 0 , κ, r, ą 0 be the constants given in Theorem 1.4. For all ε, s ą 0, there exists n P N ˚such that for all ρ ą 0, there exist C, T 0 ą 0 with (19) T 0 " 8 when ξ 2 " 0 and T 0 Ñ 8 when the speed ξ 2 Ñ 0, and h 1 P p0, h 0 q, such that for all h ă h 1 , ξ P Ω and for all v P H n pRq, if }v} 9 H n pRq ď ρ and }ψ ξ ´v} H 1 pRq ď r 2p1 `κq then any solution u of DNLS such that D y 0 , γ 0 P R, @g P hZ, u g p0q " e iγ0 vpg ´y0 q satisfies, for all t ě 0 such that ,

(20) @ t ď T 0 h ´2`ε , } uptq ´pe iγptq η h ξ p¨´yptqqq |hZ } H 1 phZq ď C `}η h ξ ´v} H 1 pRq
`hs where γ, y P C 1 pRq satisfy γp0q " γ 0 , yp0q " y 0 and

(21) @ t ď T 0 h ´2`ε , | 9 γptq ´ξ1 | `| 9 yptq ´ξ2 | ď C `}η h ξ ´v} H 1 pRq `hs ˘.
This Theorem is proven in Appendix (see Section 5.1). Note that if we can prove a control on the growth of high Sobolev norms by Opt αpn´1q q with α ă 1 2 , then we would adapt Theorem 1.7 to reach a stability time of order h ´α`ε . 1.2. Notations. Sometimes some notations could be ambiguous, so in this subsection we clarify them.

' In all this paper, we consider C as an R Euclidian space of dimension 2 equipped with the scalar product "¨" defined by

@z 1 , z 2 P C, z 1 ¨z2 " pz 1 z 2 q " z 1 z 2 ` z 1 z 2 .
Consequently, L 2 pR; Cq scalar product is defined by

@u 1 , u 2 P L 2 pR; Cq, xu 1 , u 2 y L 2 pRq " ż u 1 pxq ¨u2 pxq dx .
In particular, we consider all the Fréchet differentials as R linear applications. ' If u : R Ñ C is a real function and h ą 0, we define the discrete seconde derivative of u by

@x P R, ∆ h upxq " upx `hq `upx ´hq ´2upxq h 2 .
' We define the cardinal sine function on R by sincpxq :" sinpxq x .

' As usual when we consider second derivative, we identify the continuous bilinear forms with the operators from the space to its topological dual space. More precisely, if E is a normed vector space and b is a continuous bilinear form on E, we identify b with the operator r b : E Ñ E 1 defined by bpx, yq " r bpx, yq, x, y P E. Consequently, it makes sense to try to invert b.

' If M P M n pRq is a square matrix of length n then }M } p is the matrix norm of M associated to the p norm on R n . Similarly, if ξ P R 2 , |ξ| :" a ξ 2 1 `ξ2 2 is the 2 norm of ξ. ' If E is a set then 1 E is the characteristic function of E .
Acknoledgements. The authors are glad to thank Dario Bambusi, Benoît Grébert and Alberto Maspero for their helpful comments and discussions during the preparation of this work.

Aliasing generating inhomogeneity

In this section, we explain why DNLS can be interpreted as an inhomogeneous equation on R and why we cannot apply directly the energy-momentum method to get stable traveling waves. This section is also an introduction to most of the tools used in the this paper.

The energy-momentum method is a way to construct orbitally stable equilibria of a Hamiltonian system, relatively to a Lie group action. It has been used by Weinstein in [START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] to prove the orbital stability of the traveling waves of NLS. Then it has been developed, in the general context of Hamiltonian systems by Grillakis, Shatah, Strauss in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF]. A clear and rigorous presentation of the method and its formalism in a general setting is given in the paper [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF] of De Bièvre, Genoud, and Rota Nodari.

A crucial part of this method is based on Noether theorem, requiring to identify invariant Lie group actions with Hamiltonian flows. For DNLS, the Lie group actions are defined by gauge transform u Þ Ñ e iγ u and discrete advection u Þ Ñ pu g`a q gPhZ . The gauge transform is clearly the flow of the Hamiltonian } u } 2 L 2 phZq but the discrete advection is only defined for a countable set of values a P hZ and cannot naturally be associated with a Hamiltonian.

First, we need to extend the advection for any values a P R and then try to identify this extension with the flow of an Hamiltonian. Then we are going to see that the Hamiltonian of DNLS (see [START_REF] Eilbeck | The discrete self-trapping equation[END_REF]) is not invariant by this advection, and that the error is driven by aliasing terms.

Shannon's advection.

There are natural ways to define an advection, denoted by τ a , on the grid hZ. For a given interpolation operator I h : L 2 phZq Ñ L 2 pRq we can carry the advection on R to the grid hZ by making the following diagram commute

(22) L 2 phZq I h τa / / L 2 phZq I h L 2 pRq uÞ Ñup ¨´aq / / L 2 pRq
In general, this construction does not work, as the advection of an interpolation is not necessary an interpolation (see, for example with a finite element interpolation). However, there exists a classical interpolation called Shannon interpolation for which this construction can be applied. Let us define the discrete Fourier transform F h and Fourier Plancherel transform

F (23) F h : $ & % L 2 phZq Ñ L 2 pR{ 2π h Zq u Þ Ñ ω Þ Ñ h ÿ gPhZ u g e igω and F : $ & % L 2 pRq Ñ L 2 pRq u Þ Ñ ω Þ Ñ ż R upxqe ixω dx
where the last integral is defined by extending the operator defined on L 1 pRqXL 2 pRq. We also use the notation p u " F u. The Shannon interpolation , denoted by I h , is defined through the following diagram

(24) L 2 phZq F h / / I h 1 1 L 2 pR{ 2π h Zq uÞ Ñ1 p´π h , π h q u / / L 2 pRq F ´1 / / L 2 pRq .
With this construction, this interpolation clearly enjoys some useful properties. In this subsection, we show that the DNLS Hamiltonian is not invariant by Shannon's advection. We recall some classical properties of Shannon interpolation, see for example [START_REF] Peyrière | Convolution, séries et intégrales de Fourier[END_REF] for more details.

Proposition 2.4.

If u P L 2 phZq then I h u | hZ " u.
This proposition is just a corollary of the following decomposition, where the series converges in L 8 pRq X L 2 pRq, @x P R, I h upxq "

ÿ gPhZ u g sincpπ x ´g h q.
Corollary 2.5. The Shannon interpolation of u is the only function in L 2 pRq with Fourier transform support included in r´π h , π h s and whose values on hZ are those of u. Now, we detail a classical property of Shannon interpolation that is crucial in this paper.

Proposition 2.6. If u P H 1 pRq then u :" u | hZ P L 2 phZq and for all ω P p´π h , π h q we have

(29) z I h upωq " ÿ kPZ p upω `2π h kq.
Proof. First observe that the series (29) converges in L 2 p´π h , π h q. Indeed, using Cauchy Schwarz inequality, we have

ÿ kPZzt0u }p upω `2π h kq} L 2 p´π h , π h q ď ÿ kPZzt0u } y B x upω `2π h kq} L 2 p´π h , π h q h |2k ´1|π ď ? 2π}B x u} L 2 pRq g f f e ÿ kPZzt0u h 2 p2k ´1q 2 π 2 .

Now define v P BL 2

h through its Fourier transform

p vpωq " 1 p´π h , π h q ÿ kPZ p upω `2π h kq.
If we prove that the values of v on hZ are the same as the values of u then we conclude the proof with Corollary 2.5. Using inverse Fourier transform formula and continuity of Fourier Plancherel transform, we get for j P Z,

vphjq " 1 2π ż R p vpωqe ´iωhj dω " 1 2π ÿ kPZ ż π h ´2π h k ´π h ´2π h k p upωqe ´ipω´2 π h kqhj dω " 1 2π ÿ kPZ ż π h ´2π h k ´π h ´2π h k p upωqe ´iωhj dω " 1 2π ż R p upωqe ´iωhj dω " uphjq.
We now express the DNLS Hamiltonian in terms of Shannon interpolation:

Lemma 2.7. For all u P L 2 phZq, let u " I h u, then we have

(30) H DNLS puq " 1 2 ż R ˇˇˇu px `hq ´upxq h ˇˇˇ2 dx ´1 4 ż R ˆ1 `2 cosp 2πx h q ˙|upxq| 4 dx .
Proof. Since the Shannon interpolation I h is an isometry between L 2 phZ; Cq and L 2 pR; Cq, we have

h ÿ gPhZ ˇˇˇu g`h ´ug h ˇˇˇ2 " ż R ˇˇˇu px `hq ´upxq h ˇˇˇ2 dx .
Now we calculate the nonlinear part. First, we use the same argument of isometry to prove that

(31) h ÿ gPhZ |u g | 4 " xu, | u | 2 uy L 2 phZq " xu, I h p| u | 2 uqy L 2 pRq .
But we deduce from Proposition 2.6 that for ω P R

F I h p| u | 2 uqpωq " 1 p´π h , π h q pωq ÿ kPZ z |u| 2 upω `2π h kq.
However, since u P BL 2 h , we have

supp z |u| 2 u Ă supp p u `supp p u `supp p ū Ă " ´3π h , 3π h  .
Consequently, if k R t´1, 0, 1u the term in the sum is zero. Furthermore, it is clear that for any v P L 2 pRq, γ P R, p vp¨`γq " z e iγx v. So we have

F I h p| u | 2 uqpωq " 1 p´π h , π h q pωqF "ˆ1 `2 cosp 2πx h q ˙|u| 2 u  pωq.
We conclude by plugging this relation in (31).

We this Lemma 2.7, we can observe that H DNLS is not invariant by advection. This default of invariance is due to an inhomogeneity generated by aliasing errors.

2.3.

The flow of DNLS in the space of the Shannon interpolations. Thanks to Shannon interpolation, we identify functions defined on a grid with functions of BL h 2 . We will now see that it is equivalent to consider the flow of DNLS on a grid, or consider the Hamiltonian flow on BL h 2 associated with the Hamiltonian (32)

@u P BL 2 h , H h DNLS puq :" 1 2 ż R ˇˇˇu px `hq ´upxq h ˇˇˇ2 dx ´1 4 ż R ˆ1 `2 cosp 2πx h q ˙|upxq| 4 dx .
Applying Lemma 2.2, we obtain:

Lemma 2.8. Let h ą 0, u P C 1 pR; L 2 pRqq and u " I h puq. Then u is a solution of DNLS (see (1)) if and only if @t P R, @v P BL 2 h , xiB t uptq, vy L 2 pRq " d H h DNLS puptqqpvq. We conclude with the following result showing that discrete Sobolev norms are equivalent to continuous Sobolev norms on BL h 2 : Lemma 2.9. Let u P L 2 phZq and u " I h u P BL 2 h . Then we have

2 π }u} H 1 pRq ď } u } H 1 phZq ď }u} H 1 pRq .
Proof. By construction, we know that } u } L 2 phZq " }u} L 2 pRq . So we just need to focus on the other part of the H 1 phZq norm. Indeed, applying Shannon isometry and Fourier Plancherel isometry, we have

} u } 2 9 H 1 phZq " ÿ gPhZ ˇˇˇu g`h ´ug h ˇˇˇ2 " ż R ˇˇˇu px `hq ´upxq h ˇˇˇ2 dx " 1 2π ż R 4 h 2 sin 2 ˆωh 2 ˙|p upωq| 2 dω " 1 2π ż π h ´π h sinc 2 ˆωh 2 ˙|ωp upωq| 2 dω P 1 2π ż π h ´π h |ωp upωq| 2 dω rsinc 2 p π 2 q, 1s " }B x u} 2 L 2 pRq « ˆ2 π ˙2 , 1 ff .
Similarly, we can prove that for high order homogeneous Sobolev norms (see ( 16)), we have for all u P L 2 phZ; Cq and u " I h puq,

(33) ˆ2 π ˙n }u} 9 H n pRq ď } u } 9 H n phZq ď }u} 9 H n pRq .

Traveling waves of the homogeneous Hamiltonian

In the previous subsection, we have seen that the Hamiltonian of DNLS is not invariant by Shannon's advection. This default of invariance is due to an inhomogeneity generated by an aliasing error (the highly oscillatory terms in (32)), preventing a faire use of energy-momentum method to get stable traveling waves. Let us introduce the following perturbation of the DNLS Hamiltonian, obtained by removing these aliasing terms:

(34) @u P BL 2 h , H h puq " 1 2 ż R ˇˇˇu px `hq ´upxq h ˇˇˇ2 dx ´1 4 }u} 4 L 4 pRq .
This new Hamiltonian is clearly invariant by gauge and advection transform, and we will be able to apply the energy-momentum method. Moreover, for smooth function, it is very close to the DNLS Hamiltonian.

In the first subsection, we construct, with a perturbative method, critical points of Lagrange functions associated with (34). These critical points are the functions η h ξ of Theorem 1.4. They are traveling waves for the dynamic associated to this homogeneous Hamiltonian. In the second subsection, we focus on their regularity and their orbital stability.

In all this section, we only consider speeds ξ in Ω, a relatively compact open subset of

" ξ P R 2 | ξ 1 ą ´ξ2 2 ¯2* .

Construction of the traveling waves. Let us introduce the Lagrange function

L h ξ : BL 2 h Ñ R defined by (35) @u P BL 2 h , L h ξ puq " H h puq `ξ1 2 }u} 2 L 2 pRq `ξ2 2 xiB x u, uy L 2 pRq .
We prove in the following lemma that traveling waves generated by H h are critical points of L h ξ . Proposition 3.1. Let ξ P R 2 , h ą 0 and u P C 1 pR; BL 2 h q be such that @t P R, @x P R, upt, xq " e iξ1t up0, x ´ξ2 tq.

Then the following properties are equivalents

(36) @t P R, @v P BL 2 h , xiB t uptq, vy L 2 pRq " d H h puptqqpvq,

and

(37) d L h ξ pup0qq " 0. Proof. By a straightforward calculation, we have, for all t, x P R,

B t upt, xq " iξ 1 upt, xq ´ξ2 B x upt, xq.
Consequently, testing this relation against v P BL 2 h , we get for all t, x P R,

xiB t uptq, vy L 2 pRq " ´d ˆξ1 2 } ¨}2 L 2 pRq `ξ2 2 xiB x ¨, ¨yL 2 pRq ˙puptqqpvq.
So (36) is clearly equivalent to (38) @t P R, d L h ξ puptqq " 0. In particular (36) ñ (37) is obvious.

Conversely, to prove (37) ñ (36), we just need to prove that if

u 0 P BL 2 h is a critical point of L h ξ and γ, y P R then e iγ u 0 p. ´yq is also a critical point of L h ξ . Define T γ,y : BL 2 h Ñ BL 2 h by @v P BL 2 h , T γ,y v " e iγ vp. ´yq. Since L h
ξ is invariant by gauge transform and advection, we have @v P BL 2 h , L h ξ pT γ,y vq " L h ξ pvq. Calculating the derivative with respect to v in u 0 , we get

@v P BL 2 h , d L h ξ pT γ,y u 0 qpT γ,y vq " d L h ξ pu 0 qpvq " 0.
Since T γ,y is an invertible operator on BL 2 h (because T ´1 γ,y " T ´γ,´y ), T γ,y u 0 is also a critical point L h ξ .

In the following Theorem, we construct critical points of the Lagrange functions L h ξ as perturbations of the continuous traveling waves ψ ξ embedded in BL 2 h . Theorem 3.2. There exist h 0 , C, ρ, α ą 0 such that for all h ă h 0 and for all ξ P Ω, there exists η h

ξ P BL 2 h satisfying a) d L h ξ pη h ξ q " 0, b) }η h ξ ´ψξ } H 1 pRq ď Ch 2 , c) @x P R, η h ξ p´xq " η h ξ pxq, d) if u P BL 2 h is such that }u ´ηh ξ } H 1 pRq ă ρ, up´xq " upxq for all x P R and d L h ξ puq " 0 then u " η h ξ , e) if v P BL 2 h X Spanpη h ξ , iη h ξ , B x η h ξ q K L 2 , then we have d 2 L h ξ pη h ξ qpv, vq ě α}v} 2 H 1 pRq . Furthermore, ξ Þ Ñ η h ξ is C 1
and for all h ă h 0 , for all ξ P Ω, we have

@ζ P R 2 , } d ξ η h ξ pξqpζq ´dξ ψ ξ pξqpζq} H 1 pRq ď C|ζ|h 2 .
The remainder of this section is devoted to the proof of this Theorem. It is divided in three steps. The idea of the proof is to apply, for each value of ξ, the inverse function Theorem to solve d L h ξ puq " 0. We give an adapted version of this result, see Theorem 5.3, proven in Appendix. Moreover, we have to pay attention to symmetries and establish estimates uniform with respect to ξ P Ω and h small enough.

Step 1: Identify the function to invert First, we need a point around which apply the inverse function Theorem. To do this, we consider the orthogonal projection of the continuous traveling wave ψ ξ on BL 2 h (for the L 2 pRq norm) denoted by ψ h ξ . Using Fourier Plancherel transform we observe that ψ h ξ and ψ ξ are linked by their Fourier transform through the relation

(39) x ψ h ξ " 1 p´π h , π h q x ψ ξ .
Sometimes it is useful to extend this notation for h " 0 with ψ 0 ξ " ψ ξ . Now, we have to take care about the symmetries of the problem. Indeed, since the set of the critical points of L h ξ is stable under advection and gauge transform, we expect that the differential of d L h ξ is not invertible in this critical point. However, there is a classical trick to avoid the problem generated by these symmetries.

To explain this trick we need to introduce an operator on BL 2 h S h :

" BL 2 h Ñ BL 2 h u Þ Ñ px Þ Ñ up´xqq.
This symmetry is natural for our problem because L h ξ is invariant under its action.

Lemma 3.3. For all h ą 0, for all ξ P R 2 , for all u P BL 2 h , we have L h ξ pS h puqq " L h ξ puq. Proof. It can be proven by a straightforward calculation.

This operator induces a decomposition of BL 2

h very well adapted to our problem BL 2

h " Kerpid ´Sh q ' Kerpid `Sh q. This decomposition is also a topological decomposition because these subspaces are closed for the } ¨}H 1 pRq norm. In all the paper, these spaces are always implicitly equipped with this norm.

The continuous traveling waves is invariant under this symmetry. Indeed, we can verify (see (4)) that @x P R, ψ ξ p´xq " ψ ξ pxq.

Consequently, we expect η h ξ to be invariant under the action of S h . The space Kerpid ´Sh q is not invariant under advection or gauge transform, so we avoid the previous difficulty. Moreover, we have the following result Lemma 3.4. For all h ą 0, for all ξ P R 2 , for all u P Kerpid ´Sh q, for all v P Kerpid `Sh q, we have

d L h ξ puqpvq " 0.
Proof. Applying Lemma 3.3, we get

L h ξ pu ´vq " L h ξ pu `vq.
Then, if we compute the derivative with respect to v P Kerpid `Sh q, we get

d L h ξ puqpvq " ´d L h ξ puqpvq.
With this lemma, we see that a critical point of d L h ξ |Kerpid ´Sh q is a critical point of L h ξ . Hence we will apply the inverse function Theorem 5.3 in the point ψ h ξ which is in Kerpid ´Sh q (it is a straightforward calculation), and to the function d L h ξ |Kerpid ´Sh q .

Step 2: Invertibility of the derivative Now, we want to prove that d 2 L h ξ |Kerpid ´Sh q pψ h ξ q is invertible and to estimate the norm of its invert uniformly with respect to ξ P Ω and h small enough. The strategy of the proof is to establish that d 2 L h ξ pψ h ξ q is negative in the direction of ψ h ξ and positive in the direction L 2 -orthogonal to ψ h ξ in Kerpid ´Sh q. Then it will be possible to conclude using a classical lemma of functional analysis (see Lemma 5.7 ).

We are going to establish most of our estimates from the continuous limit. So we need to introduce the continuous Lagrange function associated to NLS, defined on H 1 pRq by

L ξ puq " 1 2 }B x u} 2 L 2 pRq ´1 4 }u} 4 L 4 pRq `ξ1 2 }u} 2 L 2 pRq `ξ2 2 xiB x u, uy L 2 pRq .
Of course, as expected, we can verify that ψ ξ is a critical point of L ξ . We will have to compare precisely ψ h ξ and ψ ξ . So we need a precise control of the regularity of ψ ξ .

Lemma 3.5. There exist C ą 0 and ε ą 0 such that for all ξ P Ω and all ω P R

| x ψ ξ pωq| ď Ce ´ε|ω| .
Proof. It is a classical result of elliptic regularity. Here we can see it directly through formula (4). We also could prove it directly with the same ideas as in Theorem 3.15 below.

First, we prove, through the following lemma, that d 2 L h ξ pψ h ξ q is negative in the direction of ψ h ξ .

Lemma 3.6. There exist α ą 0 and h 0 ą 0 such that for all h ă h 0 and all ξ P Ω we have

d 2 L h ξ pψ h ξ qpψ h ξ , ψ h ξ q ď ´α}ψ h ξ } 2 H 1 pRq .
Proof. If u P H 1 pRq we have

d 2 L ξ puqpu, uq " d L ξ puqpuq ´2}u} 4 L 4 pRq . Consequently, since ψ ξ is a critical point of L ξ , we have d 2 L ξ pψ ξ qpψ ξ , ψ ξ q " ´2}ψ ξ } 4 L 4 pRq . However, ξ Þ Ñ }ψ ξ } 4 L 4 pRq and ξ Þ Ñ }ψ ξ } 2
H 1 pRq are continuous positive maps on Ω. So, there exists α ą 0 such that, for all ξ P Ω,

d 2 L ξ pψ ξ qpψ ξ , ψ ξ q " ´2}ψ ξ } 4 L 4 pRq ď ´α}ψ ξ } 2 H 1 pRq . Since }ψ h ξ } 2 H 1 pRq ď }ψ ξ } 2 H 1 pRq (see (39)
), to conclude this proof it is enough to prove that L h ξ pψ h ξ qpψ h ξ , ψ h ξ q goes to d 2 L ξ pψ ξ qpψ ξ , ψ ξ q when h goes to 0, uniformly with respect to ξ P Ω. We can write

d 2 L h ξ pψ h ξ qpψ h ξ , ψ h ξ q " d 2 L ξ pψ ξ qpψ ξ , ψ ξ q `żR ˇˇˇˇψ h ξ px `hq ´ψh ξ pxq h ˇˇˇˇ2 ´|B x ψ h ξ | 2 dx `d2 L ξ pψ h ξ qpψ h ξ , ψ h ξ q ´d2 L ξ pψ ξ qpψ ξ , ψ ξ q. (40)
First, with Fourier Plancherel isometry, we control by the classical estimate of consistency, the term generated by the discretization of the second derivative ˇˇˇˇˇż

R |B x ψ h ξ | 2 ´ˇˇˇˇψh ξ px `hq ´ψh ξ pxq h ˇˇˇˇ2 dx ˇˇˇˇˇ" 1 2π ż π h ´π h " ω 2 ´4 h 2 sin 2 ˆωh 2 ˙ | x ψ ξ pωq| 2 dω ď 1 2π ż π h ´π h 1 ´sinc 2 `ωh 2 ω2 ω 4 | x ψ ξ pωq| 2 dω ď sup ωPR 1 ´sinc 2 `ωh 2 ω2 }B 2 x ψ ξ } 2 L 2 pRq " ˆh 2 ˙2 sup ωPR 1 ´sinc 2 pωq ω 2 }B 2 x ψ ξ } 2 L 2 pRq .
Furthermore, we deduce from Lemma 3.5 that }B 2 x ψ ξ } 2 L 2 pRq can be estimated uniformly with respect to ξ P Ω. The convergence of the second term in (40) is easier. Indeed, we deduce from Lemma 3.5 that ψ h ξ goes to ψ ξ when h goes to 0, uniformly with respect to ξ P Ω. We conclude because it is clear that the map u Þ Ñ d 2 L ξ puqpu, uq is Lipschitz on bounded subsets of H 1 pRq, uniformly with respect to ξ P Ω. Now, we give the most important lemma of this proof, establishing the coercivity property of the discrete Lagrange functions uniformly with respect to the parameters. Lemma 3.7. There exist α ą 0 and h 0 ą 0 such that for all ξ P Ω and all h ă h 0 we have

(41) @v P BL 2 h X Spanpiψ h ξ , B x ψ h ξ , ψ h ξ q K L 2 , d 2 L h ξ pψ h ξ qpv, vq ě α}v} 2 H 1 pRq . Proof.
We are going to establish this estimate by a perturbation of the continuous case. Indeed, for the continuous Lagrangian this result has been proved by Weinstein in [START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF]. There exists α ą 0 such that for all

ξ P Ω @u P H 1 pRq X Spanpiψ ξ , B x ψ ξ , ψ ξ q K L 2 , d 2 L ξ pψ ξ qpv, vq ě α}v} 2 H 1 pRq
. Literally, it is not exactly the result of Weinstein. We explain, in Lemma 5.4 of the Appendix how to get this estimate from the original result. Moreover, this result can be slightly extended to obtain the existence of two constants c 1 , c 2 ą 0 such that for all ξ P Ω,

if }u ´ψξ } H 1 pRq ă c 1 and max `|xψ ξ , vy L 2 pRq |, |xiψ ξ , vy L 2 pRq |, |xB x ψ ξ , vy L 2 pRq | ˘ă c 2 }v} H 1 then d 2 L ξ puqpv, vq ě α 8 }v} 2 H 1 pRq . ( 42
)
This result is a consequence of Lemma 5.6 given in Appendix. With its formalism we take E " H 1 pRq, b " d 2 L ξ and X " Spanpiψ ξ , B x ψ ξ , ψ ξ q. This last family is free because ψ ξ is not a plane wave. Consequently, the associated Gram matrix is invertible. Finally, we just need to verify that the constants c 1 and c 2 given by the lemma can be controlled uniformly with respect to ξ P Ω. But it is a direct consequence of the estimate proven in Lemma 5.6 since the Gram matrix is a continuous function of ξ P Ω. Now, we focus on estimate (41) of Lemma 3.7. Let h 0 ą 0 be small enough to get that for all h ă h 0 and all ξ P Ω, we have

}ψ h ξ ´ψξ } H 1 pRq ă c 1 . Let us fix h ă h 0 , ξ P Ω and consider a direction v P BL 2 h X Spanpψ h ξ , iψ h ξ , B x ψ h ξ q K L 2 . We decompose v as v " v `vb with p v " 1 p´ω0,ω0q p v and ω 0 " 2θ h 0 where θ P p0, π
2 q is a constant (independent of h, ξ and h 0 ) that we will determine later. Consider the following decomposition (43)

d 2 L h ξ pψ h ξ qpv, vq " d 2 L h ξ pψ h ξ qpv , v q `d2 L h ξ pψ h ξ qpv b , v b q `2 d 2 L h ξ pψ h ξ qpv b , v q.
We estimate separately each one of these terms as follows:

' For the first one, we deduce from Lemma 3.5 and the constraint on v that there exists ε, C ą 0

(independent of ξ) such that max `|xψ ξ , v y L 2 pRq |, |xiψ ξ , v y L 2 pRq |, |xB x ψ ξ , v y L 2 pRq | ˘ď Ce ´εω0 }v} H 1 pRq .
Consequently, if h 0 is small enough to get Ce ´εω0 ă c 2 , we can apply (42) to get

d 2 L ξ pψ h ξ qpv , v q ě α 8 }v } 2 H 1 pRq .
Hence we have

d 2 L h ξ pψ h ξ qpv , v q ě α 8 }v } 2 H 1 pRq `d2 L h ξ pψ h ξ qpv , v q ´d2 L ξ pψ h ξ qpv , v q " α 8 }v } 2 H 1 pRq `1 2π ż R " 4 h 2 sin 2 ˆωh 2 ˙´ω 2  | p v pωq| 2 dω " α 8 }v } 2 H 1 pRq ´1 2π ż |ω|ăω0 " 1 ´sinc 2 p ωh 2 q  |ω p v pωq| 2 dω ě α 8 }v } 2 H 1 pRq ´"1 ´sinc 2 pθq ‰ }v } 2 H 1 pRq
Choosing θ P p0, π 2 q to have 1 ´sinc 2 pθq ă α 16 , we get

d 2 L h ξ pψ h ξ qpv , v q ě α 16 }v } 2 H 1 pRq .
' For the second term, we use Fourier Plancherel isometry to get

d 2 L h ξ pψ h ξ qpv b , v b q ě 1 2π ż R sinc 2 ˆωh 2 ˙ω2 | p v b pωq| 2 dω ´3}ψ h ξ } 2 L 8 pRq }v b } 2 L 2 pRq ´|ξ 2 | 2 }B x v b } L 2 pRq }v b } L 2 pRq ě sinc 2 pθq}B x v b } 2 L 2 pRq ´3}ψ h ξ } 2 L 8 pRq }v b } 2 L 2 pRq ´|ξ 2 | 2 }B x v b } L 2 pRq }v b } L 2 pRq .
However, applying Fourier Plancherel isometry we get

}v b } 2 L 2 pRq " 1 2π ż |ω|ąω0 |p vpωq| 2 dω ď 1 ω 2 0 1 2π ż |ω|ąω0 |ωp vpωq| 2 dω " 1 ω 2 0 }B x v b } 2 L 2 pRq .
Consequently, we have

d 2 L h ξ pψ h ξ qpv b , v b q ě ˜sinc 2 pθq ´3}ψ h ξ } 2 L 8 pRq ω 2 0 ´|ξ 2 | 2ω 0 ¸}B x v b } 2 L 2 pRq ě ˜sinc 2 pθq ´3}ψ h ξ } 2 L 8 pRq ω 2 0 ´|ξ 2 | 2ω 0 ¸ω2 0 1 `ω2 0 }v b } 2 H 1 pRq
Since these quantities can be controlled uniformly with respect to ξ P Ω, if h 0 is small enough, we have for all ξ P Ω

d 2 L h ξ pψ h ξ qpv b , v b q ě 1 2 sinc 2 pθq}v b } 2 H 1 pRq
. ' For the third term, since the frequency supports of v and v b are disjoint, we get

d 2 L h ξ pψ h ξ qpv b , v q " d 2 } ¨}4 L 4 4 pψ h ξ qpv b , v l q ě ´3}ψ h ξ } 2 L 8 pRq }v b } L 2 pRq }v } L 2 pRq ě ´3}ψ h ξ } 2 L 8 pRq }v } H 1 pRq }v b } H 1 pRq a 1 `ω2 0 ě ´3}ψ h ξ } 2 L 8 pRq 2 a 1 `ω2 0 ´}v } 2 H 1 pRq `}v b } 2 H 1 pRq ¯.
Controlling this quantity uniformly with respect to ξ P Ω, we deduce that if h 0 is small enough then

d 2 L h ξ pψ h ξ qpv b , v q ě ´β 2 ´}v } 2 H 1 pRq `}v b } 2 H 1 pRq ¯,
with β " minp 1 2 sinc 2 pθq, α 16 q. Applying these three estimates, we deduce that there exists an h 0 ą 0 such that if h ă h 0 and ξ P Ω then for all v P BL 2 h X Spanpψ h ξ , iψ h ξ , B x ψ h ξ q K L 2 , we have

d 2 L h ξ pψ h ξ qpv, vq ě β 2 ´}v } 2 H 1 pRq `}v b } 2 H 1 pRq ¯" β 2 }v} 2 H 1 pRq .
Before focusing on the invertibility of d 2 L h ξ |Kerpid ´Sh q pψ h ξ q, we give a small but useful lemma (particularly to control uniformly the norm of the inverse).

Lemma 3.8. For all r ą 0, there exists C ą 0 such that for all h ą 0 and all ξ P Ω, we have for all u, v, w P BL 2 h with }w}

H 1 pRq ă r | d 2 L h ξ pwqpu, vq| ď C}u} H 1 pRq }v} H 1 pRq . Proof. Since | sinpωq| ď |ω|, we observe that, for all u, v P BL 2 h | d 2 L h ξ pwq| ď }B x u} L 2 pRq }B x v} L 2 pRq `3}w} 2 L 8 pRq }u} L 2 pRq }v} L 2 pRq `ξ1 }u} L 2 pRq }v} L 2 pRq `|ξ 2 |}B x u} L 2 pRq }v} L 2 pRq .
The result is thus a simple consequence of the classical Sobolev inequality,

}w} 2 L 8 pRq ď }w} L 2 pRq }B x w} L 2 pRq .
In the following concluding Lemma, we prove the invertibility of d 2 L h ξ |Kerpid ´Sh q pψ h ξ q and control the norm of its inverse uniformly with respect to ξ P Ω and h small enough. Lemma 3.9. There exist h 0 ą 0 and C ą 0 such that for all ξ P Ω and all h ă h 0 , d 2 L h ξ |Kerpid ´Sh q pψ h ξ q is invertible and the norm of its inverse is smaller than C.

Proof. We use Lemma 5.7 of the Appendix, by taking E " Kerpid ´Sh q (equipped with } ¨}H 1 pRq norm), T " d 2 L h ξ |Kerpid ´Sh q pψ h ξ q, E p " Spanpψ h ξ q K L 2 X Kerpid ´Sh q and E m " Spanpψ h ξ q. To get the coercivity estimate on E m we apply Lemma 3.6, while coercivity on E p is obtained from Lemma 3.7 after noticing that Kerpid ´Sh q Ă BL 2 h X Spanpψ h ξ , iψ h ξ , B x ψ h ξ q K L 2 , which is obvious since iψ h ξ , B x ψ h ξ P Kerpid `Sh q Ă Kerpid ´Sh q K L 2 . Applying Lemma 5.7, we obtain the invertibility of d 2 L h ξ |Kerpid ´Sh q pψ h ξ q and an explicit control of the norm of its inverse in terms of α p , α m and }T }. However, with Lemma 3.6 and Lemma 3.7, we have a uniform control of α p and α m with respect to ξ P Ω and h small enough, the uniform control of }T } being given by Lemma 3.8.

Step 3: The resolution and its consequences We now want to apply the inverse function theorem 5.3 to d L h ξ |Kerpid ´Sh q in ψ h ξ . In the following Lemma, we focus on the last assumption required, i.e. d 2 L h ξ is a Lipschitz function. Lemma 3.10. For all R ą 0 there exists k ą 0 such that for all ξ P Ω, h ą 0, u 1 , u 2 , v, w P BL 2 h , with

}u 1 } H 1 pRq ă R and }u 2 } H 1 ppRqq ă R, we have } d 2 L h ξ pu 1 qpv, wq ´d2 L h ξ pu 2 qpv, wq} ď k}u 1 ´u2 } H 1 pRq }v} H 1 pRq }w} H 1 pRq .
Proof. We use mean value inequality. Indeed d 3 L h ξ " ´1 4 d 3 }¨} 4 L 4 pRq is clearly a bounded function on bounded subsets of H 1 pRq.

Applying Lemma 3.10 and Lemma 3.9, we deduce that assumptions of the inverse function Theorem 5.3 are fulfilled. In the following Proposition, we give its conclusion. Proposition 3.11. There exist h 0 , r, λ, C ą 0 such that if h ă h 0 and ξ P Ω then

' d L h
ξ |Kerpid ´Sh q is a C 1 diffeomorphism from tu P Kerpid ´Sh q | }u ´ψh ξ } H 1 pRq ă ru onto its image, ' if u P Kerpid ´Sh q and }u ´ψh ξ } H 1 pRq ă r then } d 2 L h ξ |Kerpid ´Sh q puq ´1} L pKerpid ´Sh q 1 ;Kerpid ´Sh qq ď C, ' if ρ ă r and Φ P Kerpid ´Sh q 1 with }Φ ´d L h ξ |Kerpid ´Sh q pψ h ξ q} Kerpid ´Sh q 1 ă λρ then there exists u P Kerpid ´Sh q such that }u ´ψh ξ } H 1 pRq ă ρ and d L h ξ |Kerpid ´Sh q puq " Φ.

To apply this result to Φ " 0, we will show that the norm of d L h ξ |Kerpid ´Sh q pψ h ξ q is small when h Ñ 0, uniformly in ξ P Ω. It is exactly, what we establish in the following Lemma, which also explains the error term "h 2 " in Theorem 1.4. Lemma 3.12. For all h 0 ą 0 there exists M ą 0 such that if h ă h 0 and ξ P Ω then

@v P BL 2 h , | d L h ξ pψ h ξ qpvq| ď M h 2 }v} H 1 pRq .
Proof. The arguments are very similar to the proof of Lemma 3.6. The key point is the estimate of the consistency error associated to the discretization of the second derivative by finite differences.

sup ωPR ˇˇˇˇs inc 2 `ωh 2 ˘´1 ω 2 ˇˇˇˇ} B 4 x ψ ξ } L 2 pRq }v} L 2 pRq " ˆh 2 ˙2 sup ωPR ˇˇˇs inc 2 pωq ´1 ω 2 ˇˇˇ} B 4 x ψ ξ } L 2 pRq }v} L 2 pRq .
As we can see from Lemma 3.5, }B 4

x ψ ξ } L 2 pRq is clearly bounded uniformly with respect to ξ P Ω. To control the second term in (44), we use mean value inequality and Lemma 3.5 to get some constants M, C ą 0 independent of h and ξ P Ω such that ˇˇˇˇd }.} 4

L 4 pRq 4 pψ ξ qpvq ´d }.} 4 L 4 pRq 4 pψ h ξ qpvq ˇˇˇˇď M }ψ ξ ´ψh ξ } L 2 pRq }v} L 2 pRq ď Ce ´πε h }v} L 2 pRq ,
which shows the result, provided h ă h 0 small enough.

Applying Lemma 3.12, if h 0 is smaller than b λr 2M we can choose Φ " 0 in Proposition 3.11 and we denote by η h ξ the corresponding critical point of L h ξ |Kerpid ´Sh q . As shown in the first step, η h ξ is thus a critical point of L h ξ , and with Proposition 3.11, we have proven the points aq to dq of Theorem 3.2. It remains to show the coercivity estimate eq and the regularity with respect to ξ.

To obtain the coercivity estimate, we just have to perturb the estimate of Lemma 3.7 with Lemma 5.6 presented in Appendix. This is given by the following result Lemma 3.13. There exist α ą 0, h 0 ą 0 and ρ ą 0 such that for all ξ P Ω, h ă h 0 and u P BL 2 h such that }u ´ψh ξ } H 1 pRq ă ρ, we have (45)

@v P BL 2 h X Spanpiu, B x u, uq K L 2 , d 2 L h ξ puqpv, vq ě α}v} 2 H 1 pRq . Proof.
The proof is very similar to the first part of the proof of Lemma 3.7, but we need to track precisely the dependence of the constant with respect to h.

First, applying Lemma 3.7, we know that there exists h 0 ą 0 and α ą 0 such that for all h ă h 0 and all ξ P Ω we have

@v P BL 2 h X Spanpiψ h ξ , B x ψ h ξ , ψ h ξ q K L 2 , d 2 L h ξ pψ h ξ qpv, vq ě α}v} 2 H 1 pRq
. We want to apply Lemma 5.6 in ψ h ξ in order to perturb this estimate and prove that there exist h 0 ą 0, c 1 , c 2 ą 0 such that for all ξ P Ω and all h ă h 0 , if

}u ´ψh ξ } H 1 pRq ď c 1 and max `|xψ h ξ , vy L 2 pRq |, |xiψ h ξ , vy L 2 pRq |, |xB x ψ h ξ , vy L 2 pRq | ˘ď c 2 }v} H 1 pRq , then d 2 L h ξ puqpv, vq ě α}v} 2 H 1 pRq ě α 2 }v} 2 H 1 pRq .
To do this, we apply Lemma 5.6 in

ψ h ξ with E " BL 2 h , X " Spanpiψ h ξ , B x ψ h ξ , ψ h ξ q and b " d 2 L h ξ . The Gram matrix is G h ξ " ¨}ψ h ξ } 2 L 2 pRq xiψ h ξ , B x ψ h ξ y L 2 pRq 0 xiψ h ξ , B x ψ h ξ y L 2 pRq }B x ψ h ξ } 2 L 2 pRq 0 0 0 }ψ h ξ } 2 L 2 pRq ‹ '.
To prove that the constants c 1 , c 2 ą 0 -explicitly given by Lemma 5.6-are independent of ξ P Ω and h small enough, we have to control uniformly the inverse of G h ξ , the norm of ψ h ξ in H 1 pRq, the norm of d 2 L h ξ pψ h ξ q and prove that d 2 L h ξ is uniformly Lipschitz. The control of ψ h ξ in H 1 pRq is obvious, and Lemma 3.10 shows that d 2 L h ξ is uniformly Lipschitz. In Lemma 3.8, we have proven that the norm d 2 L h ξ pψ h ξ q is uniformly bounded with respect to h and ξ P Ω. So we just need to focus on the Gram matrix.

As explained in the proof of Lemma 3.7 G 0 ξ " G ξ is invertible. Furthermore, ph, ξq Þ Ñ G h ξ is a continuous function on R `ˆΩ, so there exists h 0 ą 0 and M ą 0 such that for all h ă h 0 and all ξ P Ω, G h ξ is invertible and }pG h ξ q ´1} 8 ď M . To prove (45), let us set ρ " minpc 1 , c 2 q and consider h ă h 0 and ξ P Ω. Let u, v P BL 2 h be such that }u ´ψh ξ } H 1 pRq ă ρ and v P Spanpiu, B x u, uq which shows the result.

The following Lemma concludes the proof of Theorem 3.2. It shows that ξ Þ Ñ η h ξ is C 1 and that its derivative with respect to ξ is a good approximation of the derivative of ψ ξ with respect to ξ. Lemma 3.14. Let h 0 , r, λ, C ą 0 be the constants given in Proposition 3.11 and M ą 0 be the constant associated with h 0 ą 0 given in Lemma 3.12. Let h 1 :" minph 0 , b λr 2M q and for any h ă h 1 and ξ P Ω, let η h ξ denotes the critical point of L h ξ at a distance smaller than r from ψ h ξ . There exists k ą 0 such that for all

h ă h 1 , for all ξ P Ω, ξ Þ Ñ η h ξ is C 1 and } d ξ ψ ξ pζq ´dξ η h ξ pζq} H 1 pRq ď k|ζ|h 2 .
Proof

Consequently, ζ Þ Ñ η h ζ is C 1 . Now, we have to prove that d ξ η h
ξ is an approximation of d ξ ψ ξ . First, we introduce some constants c, ε ą 0 such that for all ξ P Ω and all ζ P R 2 , we have

(46) @ω P R, | { d ξ ψ ξ pζqpωq| ď c|ζ|e ´ε|ω| .
There are several ways to establish this property. The most direct is probably to deduce it from the explicit formula of ψ ξ (see ( 4)). But it can also be proven with elliptic regularity as in Theorem 3.15 below. Then, we deduce from the definition of ψ h ξ that for all h ą 0, ξ Þ Ñ ψ h ξ is C 1 and there exists k ą 0 such that (47) @h ą 0, @ξ P Ω,

@ζ P R 2 , } d ξ ψ ξ pζq ´dξ ψ h ξ pζq} H 1 pRq ď k|ζ|e ´επ 2h .
So we just need to prove that d ξ η h ξ is an approximation of d ξ ψ h ξ of order 2 in h. To compare these quantities, we are going to prove that they are almost solutions of the same linear equation.

Since η h ξ is a critical point of L h ξ , it satisfies for all v P Kerpid ´Sh q, d L h ξ |Kerpid ´Sh q pη h ξ qpvq " 0. So we can calculate the derivative with respect to ξ to obtain that

@ζ P R 2 , @v P Kerpid ´Sh q, d 2 L h ξ |Kerpid ´Sh q pη h ξ qpv, d ξ η h ξ pζqq `bh ζ rη h ξ spvq " 0,
where b h ζ rus P pKerpid ´Sh qq 1 is defined for u P Kerpid ´Sh q by b h ζ ruspvq :" ζ 1 xu, vy L 2 pRq `ζ2 xiB x u, vy L 2 pRq . Similarly, we define E h ξ,ζ P Kerpid ´Sh q 1 by @ζ P R 2 , @v P Kerpid ´Sh q, d 2 L h ξ |Kerpid ´Sh q pψ h ξ qpv, d ξ ψ h ξ pζqq `bh ζ rψ h ξ spvq " E h ξ,ζ pvq. Then, we get (in Kerpid ´Sh q 1 ), for all ζ P R 2 ,

d 2 L h ξ |Kerpid ´Sh q pη h ξ qpd ξ ψ h ξ pζq ´dξ η h ξ pζqq " " d 2 L h ξ |Kerpid ´Sh q pη h ξ q ´d2 L h ξ |Kerpid ´Sh q pψ h ξ q ı pd ξ ψ h ξ pζqq `bh ζ rη h ξ ´ψh ξ s `Eh ξ,ζ
pvq. However, we have proven in Proposition 3.11 that d 2 L h ξ |Kerpid ´Sh q pη h ξ q is invertible and that the norm of its invert is smaller than C. So we just need to control the three right terms of the last equality.

' Applying ( 46) and ( 47), for all h ą 0 and all ξ P Ω, we have } d ξ ψ h ξ pζq} H 1 pRq ď 2|ζ|k. So applying Lemma 3.10, there exists κ ą 0, such that for all h ă h 1 , all ξ P Ω, all ζ P R 2 and all v P Kerpid ´Sh q, ˇˇ"d 2 L h ξ |Kerpid ´Sh q pη h ξ q ´d2 L h ξ |Kerpid ´Sh q pψ h ξ q ı pd ξ ψ h ξ pζqqpvq ˇď

κ}η h ξ ´ψh ξ } H 1 pRq |ζ|}v} H 1 pRq ď M κ λ h 2 |ζ|}v} H 1 pRq .
' The estimate of the second term is obvious. Indeed, for all h ă h 1 , all ξ P Ω, all ζ P R 2 and all v P Kerpid ´Sh q we have

|b h ζ rη h ξ ´ψh ξ spvq| ď |ζ|p}η h ξ ´ψh ξ } L 2 pRq }v} L 2 pRq `}B x pη h ξ ´ψh ξ q} L 2 pRq }B x v} L 2 pRq q ď 2 M λ h 2 |ζ|}v} H 1 pRq .
' The bound on the term E h ξ,ζ is more difficult to obtain. First, we have to identify it. Since ψ ξ is a critical point of L ξ , it satisfies d L ξ pψ ξ qpvq " 0 for all v P H 1 pRq. By calculating its derivative with respect to ξ, we get for ζ P R 2 ,

d 2 L ξ pψ ξ qpv, d ξ ψ ξ pζqq `ζ1 xψ ξ , vy L 2 pRq `ζ2 xiB x ψ ξ , vy L 2 pRq " 0.
In particular, we can choose v P Kerpid ´Sh q. Consequently, we get

d 2 L h ξ pψ h ξ qpv, d ξ ψ h ξ pζqq `bh ζ rψ h ξ s `xp∆ h ´B2 x q d ξ ψ h ξ pζq, vy L 2 pRq `d2 } ¨}4 L 4 pRq 4 pψ h ξ qpv, d ξ ψ h ξ pζqq ´d2 } ¨}4 L 4 pRq
4 pψ ξ qpv, d ξ ψ ξ pζqq " 0.

So we have

E h ξ,ζ pvq " d 2 } ¨}4 L 4 pRq 4 pψ ξ qpv, d ξ ψ ξ pζqq ´d2 } ¨}4 L 4 pRq 4 pψ h ξ qpv, d ξ ψ h ξ pζqq `xpB 2 x ´∆h q d ξ ψ h ξ pζq, vy L 2 pRq .
To estimate xpB 2 x ´∆h q d ξ ψ h ξ pζq, vy L 2 pRq we use the same method as in Lemma 3.12 and we can find an universal constant C univ ą 0 such that (48) ξ pζq} H 2 pRq are bounded uniformly with respect to ξ P Ω and h ă h 1 . Consequently, by using (47), there exist ą 0, κ ą 0 such that for all h ă h 1 , all ξ P Ω and all ζ P R 2 , we have

ˇˇxpB 2 x ´∆h q d ξ ψ h ξ pζq, vy L 2 pRq ˇˇď C univ h 2 } d ξ ψ h ξ pζq}
|E h ξ,ζ pvq| ď κ ´h2 `e´ h ¯ď κh 2 ˜1 `ˆ2 e ˙2¸,
which concludes the proof of the Lemma.

3.2. Gevrey uniform regularity, Lyapunov stability and some adjustments. The discrete traveling waves constructed in Theorem 3.2 enjoy most of the properties of the continuous traveling waves ψ ξ . In this subsection, we analyse some of these properties useful to prove Theorem 1.4.

First, we study their regularity. Of course, since they belong to BL 2 h they are entire functions but we can give a control of them in Gevrey norms uniformly with respect to h and ξ. Theorem 3.15. There exists h 0 ą 0 such that for all M ą 0, there exist C, ε ą 0 such that for all h ă h 0 and all ξ P Ω, if u P BL h 2 satisfies }u} H 1 pRq ď M then (49) d L h ξ puq " 0 ñ @ω P R, |p upωq| ă Ce ´ε|ω| . Proof. To get this result of elliptic regularity, we prove, in the following lemma, a result of coercivity. Lemma 3.16. Let f : R Ñ R be a function continuous in 0 such that f p0q " 1. Assume that there exists m ą 0 such that f ě m on R. Then there exist α ą 0 and h 0 ą 0 such that for all ξ P Ω and h ă h 0 we have

@ω P R, ω 2 f phωq `ξ2 ω `ξ1 ě α `1 `ω2 ˘.
Proof. First, observe that we have

ω 2 `ξ2 ω `ξ1 " ˆω `ξ2 2 ˙2 `ξ1 ´ˆξ 2 2 ˙2 .
Consequently, there exists β ą 0 such that for all ξ P Ω, we have

ω 2 `ξ2 ω `ξ1 ě β `1 `ω2 ˘.
Second observe that there exists ω 0 ą 0 such that, for all |ω| ą ω 0 we have

mω 2 `ξ2 ω `ξ1 ě m 2 `1 `ω2 ˘.
Consequently, for such ω and for any h ą 0, we have

ω 2 f phωq `ξ2 ω `ξ1 ě m 2 `1 `ω2 ˘.
Now, since f is continuous in 0, there exists δ ą 0 such that if |ω| ă δ then |f pωq ´1| ă β 2 . Consequently, if |ω| ă ω 0 and h ă δ ω0 ": h 0 then we have

ω 2 f phωq `ξ2 ω `ξ1 " ω 2 `ξ2 ω `ξ1 `ω2 pf phωq ´1q ě β `1 `ω2 ˘´β 2 ω 2 ě β 2 `1 `ω2 ˘.
We now prove the elliptic regularity result (49 Applying Lemma 3.16 to f pωq " sinc 2 `ω 2 ˘`1 p´π,πq c pωq, for which m " 4 π 2 , there exist h 0 ą 0 and α ą 0 such that if ξ P Ω and h ă h 0 ,

@ω P ´´π h , π h ¯, 4 h 2 sin 2 ˆωh 2 ˙´ξ 2 ω `ξ1 ě α `1 `ω2 ˘.
Hence, we have using (50)

(51) @ω P ´´π h , π h ¯, α `1 `ω2 ˘|p upωq| ď |p upωq| ˚|p ūpωq| ˚|p upωq|.
Now, we prove by induction (on n) that there exists C ą 0, that only depend of α and M such that

(52) @ 1 ď p ď 8, }ω n p u} L p pRq ď C n n!.
First, we consider the cases n " 1 and n " 0. Since we have assumed that }u} H 1 pRq ď M , we have

}p u} L 1 pRq ď › › › › 1 ? 1 `ω2 › › › › L 2 pRq } a 1 `ω2 p upωq} L 2 pRq " ? 2π}u} H 1 pRq ď ? 2πM.
Then, we get from (51)

}ωp u} L 1 pRq ď }p1 `ω2 qp u} L 1 pRq ď 1 α }p u} 3 L 1 pRq ď ? 8π 3 M 3 α .
Furthermore, we also get from (51),

}p1 `|ω|qp u} L 8 pRq ď 1 α › › › › 1 `|ω| 1 `ω2 › › › › L 8 pRq }p u} 3 L 1 pRq .
We deduce (52) for n " 0 and 1 and for the other values of p using Hölder inequality. Now, we assume that (52) is proved for all 0 ď n ď N `1. We deduce from (50) that for all ω P `´π h , π h ˘, we have

|ω| N `2|p upωq| ď |ω| N p1 `ω2 q|p upωq| ď 1 α ˇˇω N pp u ˚p ū ˚p upωqq ˇˇ" 1 α ˇˇˇˇÿ n1`n2`n3"N N ! n 1 !n 2 !n 3 ! pω n1 p uq ˚pω n2 p ūq ˚pω n3 p uqpωq ˇˇˇˇ.
We deduce from Young convolution inequality that if 3 q " 2 `1 p then

}ω N `2 p u} L p pRq ď 1 α ÿ n1`n2`n3"N N ! n 1 !n 2 !n 3 ! n ź j"1 }ω nj p u} L q pRq .
Using the induction hypothesis, we obtain

}ω N `2 p u} L p pRq ď 1 α C N N ! #tpn 1 , n 2 , n 3 q | n 1 `n2 `n3 " N u " 1 2αC 2 C N `2pN `2q!
So, if C is chosen large enough to ensure 2αC 2 ě 1, we obtain the result by induction.

Choosing p " 8 in (52), we get

@n P N, @ω P R ˚, |p upωq| ď ˆC |ω| ˙n n!.
But using Stirling formula, we get an universal constant c ą 0 such that n! ď ce In the following lemma, we prove that Lagrange functions are Lyapunov functions for the traveling waves of the homogeneous Hamiltonian. These uniform estimates are discrete versions of the continuous case, see for example Proposition 8.8 of [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF]. They are the key estimates for applying the energy-momentum method. Lemma 3.17. Let h 0 , C, ρ, α ą 0 be the constants given by Theorem 3.2. There exist r, β, h 1 ą 0 such that for all h ă h 1 , all ξ P Ω,

all u P BL 2 h X Spanpiη h ξ , B x η h ξ q, if }u ´ηh ξ } H 1 pRq ď r and }u} 2 L 2 pRq " }η h ξ } 2 L 2 pRq then (53) β}u ´ηh ξ } 2 H 1 pRq ď L h ξ puq ´L h ξ pη h ξ q.
Proof. Let h 1 ă h 0 and ε ą 0 be such that

@h ă h 1 , @ξ P Ω, }η h ξ } 2 L 2 pRq ě }ψ ξ } 2 L 2 pRq 2 ě ε 2 .
Let r P p0, 1q be a positive constant that will be determined later. Since η h ξ is bounded in H 1 pRq, uniformly with respect to ξ P Ω and h ă h 0 , there exists a constant M ą 0 such that for all ξ P Ω, h ă h 0 , w 1 , w 2 , w 3 P BL 2 h , we have

}η h ξ } H 1 pRq ď M , | d 2 L h ξ pη h ξ qpw 1 , w 2 q| ď M }w 1 } H 1 pRq }w 2 } H 1 pRq and sup }η h ξ ´w} H 1 pRq ď1 | d 3 L h ξ pwqpw 1 , w 2 , w 3 q| ď M }w 1 } H 1 pRq }w 2 } H 1 pRq }w 3 } H 1 pRq .
Indeed, the first estimate has been establish in Lemma 3.8 and the second is obvious since

d 3 L h ξ " d 3 }¨} 4 L 4 pRq 4 . Consider h ă h 1 , ξ P Ω and u P BL 2 h X Spanpiη h ξ , B x η h ξ q K L 2 such that }u ´ηh ξ } H 1 pRq ď r and }u} 2 L 2 pRq " }η h ξ } 2 L 2 pRq . Then we define v " η h ξ `«pu ´ηh ξ q ´ηh ξ }η h ξ } L 2 pRq xu ´ηh ξ , η h ξ }η h ξ } L 2 pRq y L 2 pRq ff .
By construction, v ´ηh ξ belongs to Spanpiη h ξ , B x η h ξ , η h ξ q K L 2 . Furthermore, v ´ηh ξ is a second order perturbation of u ´ηh ξ because, since }u} 2 L 2 pRq " }η h ξ } 2 L 2 pRq , we have

xη h ξ , u ´ηh ξ y L 2 pRq " ´1 2 }u ´ηh ξ } 2 L 2 pRq .
So, we get

}u ´v} H 1 pRq " }η h ξ } H 1 pRq 2}η h ξ } 2 L 2 pRq }u ´ηh ξ } 2 L 2 pRq ď 2M 2 }u ´ηh ξ } 2 H 1 pRq .
Now, we can establish our estimate through a Taylor expansion of L h ξ puq around η h ξ . The first order term vanishes since η h ξ is a critical point of L h ξ . The second order term is controlled by applying the coercivity estimate of d 2 L h ξ (see ( 53)),

L h ξ puq ´L h ξ pη h ξ q ě d 2 L h ξ pη h ξ qpu ´ηh ξ , u ´ηh ξ q ´M }u ´ηh ξ } 3 H 1 pRq " d 2 L h ξ pη h ξ qpv ´ηh ξ , v ´ηh ξ q ´d2 L h ξ pη h ξ qpu ´v, u ´vq `2 d 2 L h ξ pη h ξ qpu ´ηh ξ , u ´vq ´M }u ´ηh ξ } 3 H 1 pRq ě α}v ´ηh ξ } 2 H 1 pRq ´}u ´ηh ξ } 3 H 1 pRq ˜M ˆ2M 2 ˙2 }u ´ηh ξ } H 1 pRq `2M 2M 2 `M " α}u ´ηh ξ } 2 H 1 pRq `α}v ´u} 2 H 1 pRq ´2αxv ´u, u ´ηh ξ y H 1 pRq ´}u ´ηh ξ } 3 H 1 pRq ˜ˆ2M 2 2 ˙2 }u ´ηh ξ } H 1 pRq `4M 2 2 `M ȩ }u ´ηh ξ } 2 H 1 pRq « α ´}u ´ηh ξ } H 1 pRq ˜2α 2M 2 `ˆ2M 2 2 ˙2 }u ´ηh ξ } H 1 pRq `4M 2 2 `M ¸ff ě }u ´ηh ξ } 2 H 1 pRq « α ´r ˜α 4M 2 `ˆ2M 2 2 ˙2 `4M 2 2 `M ¸ff .
Consequently, to prove the Theorem, we just need to choose

r ă α 2 ˜α 4M 2 `ˆ2M 2 2 ˙2 `4M 2 2 `M ¸´1 .
The previous lemma provides a stability control for the solutions of the homogeneous Hamiltonian system. To apply it, two strong assumptions are required: u P Spanpiη h ξ , B x η h ξ q and }u} 2 L 2 pRq " }η h ξ } 2 L 2 pRq . If u is close enough to η h ξ there are two classical tricks to get these assumptions. To fulfill the first condition, the idea is to apply a small gauge transform and a small advection to u. We focus on this problem in the two following Lemmas. To satisfy the second assumption, the idea is to modify ξ 1 . It is the object of the last Theorem of this section.

When η h ξ is well defined through Theorem 3.2, for any v P BL 2 h , we define the matrix A ξ,h rvs by (54) A ξ,h rvs :"

ˆxiη h ξ , ivy L 2 pRq ´xiη h ξ , B x vy L 2 pRq xB x η h ξ , ivy L 2 pRq ´xB x η h ξ , B x vy L 2 pRq
˙.

We will explain later why this matrix is very useful, but first we give a technical Lemma.

Lemma 3.18. Let h 0 , C, ρ, α ą 0 be the constants given by Theorem 3.2. There exists h 1 ă h 0 , M ą 0 and δ ą 0 such that for all h ă h 1 , all ξ P Ω and all v P BL 2 h with }v ´ηh ξ } H 1 pRq ă δ, A ξ,h rvs is invertible and }pA ξ,h rvsq ´1} 8 ď M .

Proof. Let h ă h 0 , ξ P Ω and v P BL 2 h . Since v Þ Ñ A ξ,h rvs is a linear map we have (55) A ξ,h rvs " A ξ,h rη h ξ s `Aξ,h rv ´ηh ξ s.

However, since }η h ξ ´ψξ } H 1 pRq ď Ch 2 , A ξ,h rη h ξ s converges to G ξ , uniformly with respect to ξ P Ω, as h goes to 0, where

G ξ " ˜}ψ ξ } 2 L 2 pRq ´xiψ ξ , B x ψ ξ y L 2 pRq xiψ ξ , B x ψ ξ y L 2 pRq ´}B x ψ ξ } 2 L 2 pRq ¸.
Applying Cauchy-Schwarz inequality, we have

det G ξ " xiψ ξ , B x ψ ξ y 2 L 2 pRq ´}ψ ξ } 2 L 2 pRq }B x ψ ξ } 2 L 2 pRq ď 0.
But the case of equality is excluded since ψ ξ is not a plane wave (i.e. Spanpiψ ξ , B x ψ ξ q is a free family). So G ξ is an invertible matrix. As ξ Þ Ñ G ξ is a continuous map on Ω, there exists M ą 0 such that for all ξ P Ω

}G ´1 ξ } 8 ď M 2 .
As A ξ,h rη h ξ s converges to G ξ when h Ñ 0, there exists h 1 ă h 0 such that for all h ă h 1 and ξ P Ω, A ξ,h rη h ξ s is invertible and }pA ξ,h rη h ξ sq ´1} 8 ď M. Applying the linear decomposition (55), we have A ξ,h rvs " A ξ,h rη h ξ spI 2 `pA ξ,h rη h ξ sq ´1A ξ,h rv ´ηh ξ sq. However, since η h ξ is bounded in H 1 pRq uniformly with respect to ξ and h, there exists δ ą 0 such that for all ξ P Ω and all h ă h 1 , we have

}pA ξ,h rη h ξ sq ´1A ξ,h rv ´ηh ξ s} 8 ă 1 2δ }v ´ηh ξ } H 1 pRq .
Consequently, if }v ´ηh ξ } H 1 pRq ď δ then A ξ,h rvs is invertible and the norm of its invert is bounded by 2M .

Lemma 3.19. There exists λ, δ ą 0 and h 1 ă h 0 , such that for all ξ P Ω, h ă h 1 , v P BL 2 h , if }v ´ηh ξ } H 1 pRq ă δ then there exists γ, y P R such that maxp|γ|, | y |q ď λ}v ´ηh ξ } H 1 pRq and e iγ vp ¨´yq ´ηh ξ P Spanpiη h ξ , B x η h ξ q K L 2 . Proof. For this proof, we introduce a notation. If γ, y P R and v : R Ñ R then T γ,y v :" e iγ vp ¨´yq Let v P BL 2 h . We are going to apply the inverse function Theorem 5.3 to the following function

g v ξ,h : $ & % R 2 Ñ R 2 ˆγ y ˙Þ Ñ ˆxiη h ξ , T γ,y v ´ηh ξ y L 2 pRq xB x η h ξ , T γ,y v ´ηh ξ y L 2 pRq ˙.
g v ξ,h is clearly a C 1 function whose Jacobian matrix is given by Jg v ξ,h pγ, yq " A ξ,h rT γ,y vs. Applying Lemma 3.18, we can find h 1 ă h 0 , δ ą 0 and M ą 0 such that if h ă h 1 and }v ´ηh ξ } H 1 pRq ă δ then Jg v ξ,h p0, 0q is invertible and its norm is smaller than M . We want to prove that Jg v ξ,h is Lipschitz uniformly with respect to ξ, h, v. In fact, since it is a C 1 function, we just need to control its derivative. Using integration by parts, there exists a constant κ ą 0 such that for all y, γ P R we have

} d Jg v ξ,h pγ, yq} L pR 2 ;M2pR 2 qq ď κ}η h ξ } H 2 pRq }T γ,y v} H 1 pRq " κ}η h ξ } H 2 pRq }v} H 1 pRq .
But, applying the result of elliptic regularity (Theorem 3.15), }η h ξ } H 2 pRq is bounded in H 2 pRq uniformly with respect to ξ P Ω and h ă h 0 . So, there exists k ą 0 such that for all ξ P Ω, h ă h 0 and v P BL 2 h with }v ´ηh ξ } H 1 pRq ă δ, we have } d Jg v ξ,h pγ, yq} L pR 2 ;M2pR 2 qq ď k. Now, we apply the inverse function theorem 5.3 to g v ξ,h and we obtain some constants λ ą 0 and r ą 0, such that for all h ă h 1 , ξ P Ω and v P BL 2 h with }v ´ηh ξ } H 1 pRq ď R, @ν P R 2 , |ν| ď r ñ Dγ, y P R, g v ξ,h pγ, yq " g v ξ,h p0, 0q `ν and maxp|γ|, | y |q ď λ|ν|. To prove the lemma, we would like to choose ν " ´gv ξ,h p0, 0q small enough. But since η h ξ is uniformly bounded in H 1 pRq, there exists a constant K ą 0 such that for all h ă h 0 , v P BL 2 h , ξ P Ω,

|g v ξ,h p0, 0q| ď K}η h ξ ´v} H 1 pRq .
So, if }η h ξ ´v} H 1 pRq ď r K , we can choose ν " ´gv ξ,h p0, 0q and the lemma is proven. In the following Theorem, we focus on a change of variable. Usually, NLS traveling waves are not indexed by ξ but by their L 2 norm and their momentum. It would be possible to do the same here. Here, we prove that it is possible to index them by their L 2 norm and their speed of advection (i.e. ξ 2 ). Theorem 3.20. Let h 0 , C, ρ, α ą 0 be the constants given by Theorem 3.2 and let r Ω be a relatively compact open subset of Ω. Then there exist h 1 ă h 0 , δ ą 0, k ą 0 such that for all h ă h 1 , for all ξ P r Ω and for all u P BL 2 h , if }u ´ηh ξ } H 1 pRq ă δ then there exists ζ P Ω such that

(56) " ξ 2 " ζ 2 }η h ζ } 2 L 2 pRq " }u} 2 L 2 pRq and |ζ ´ξ| ď k|}η h ξ } 2 L 2 pRq ´}u} 2 L 2 pRq |.
Proof. From the definition of ψ ξ (see ( 4)), we observe that for all ξ P Ω,

}ψ ξ } 2 L 2 pRq " m ξ }ψ 1,0 } 2 L 2 pRq " 4m ξ " 4 d ξ 2 1 ´ˆξ 2 2 ˙2.
Consequently, there exists β ą 0 such that for all ξ P Ω,

B ξ1 }ψ ξ } 2 L 2 pRq " 2 m ξ ě 2β.
Let h ă h 0 . Applying Theorem 3.2, we know that ξ Þ Ñ η h ξ is a C 1 approximation of ξ Þ Ñ ψ ξ up to an second order error term. Consequently, we have

|B ξ1 }ψ ξ } 2 L 2 pRq ´Bξ1 }η h ξ } 2 L 2 pRq | " 2|xB ξ1 ψ ξ ´Bξ1 η h ξ , ψ ξ y L 2 pRq `xB ξ1 η h ξ , ψ ξ ´ηh ξ y L 2 pRq | ď 2Ch 2 `}ψ ξ } L 2 pRq `}B ξ1 η h ξ } L 2 pRq ď 2Ch 2 `}ψ ξ } L 2 pRq `Ch 2 `}B ξ1 ψ ξ } L 2 pRq ď 2Ch 2 sup ξPΩ `}ψ ξ } L 2 pRq `Ch 2 0 `}B ξ1 ψ ξ } L 2 pRq ": M h 2 .
Let h 1 " minph 0 , β ? M q. If h ă h 0 and ξ P Ω, we have

B ξ1 }η h ξ } 2 L 2 pRq ě β.
Since r Ω is relatively compact open subset of Ω, there exists r ą 0 such that r Ω `BR 2 p0, rq Ă Ω.

Let ξ P r Ω, h ă h 1 and let g be the following function g :

" rξ 1 ´r, ξ 1 `rs Ñ R ζ 1 Þ Ñ }η h ζ1,ξ2 } 2 L 2 pRq . Since g is a continuous map, we have (57) r}η h ξ1´r,ξ2 } 2 L 2 pRq , }η h ξ1`r,ξ2 } 2 L 2 pRq s Ă gprξ 1 ´r, ξ 1 `rsq.

But applying the mean value equality, we have

}η h ξ1´r,ξ2 } 2 L 2 pRq ă }η h ξ } 2 L 2 pRq ´βr ă }η h ξ } 2 L 2 pRq `βr ă }η h ξ1`r,ξ2 } 2 L 2 pRq . Let u P BL 2
h be such that }u ´ηh ξ } H 1 pRq ă δ, where δ P p0, 1q is a positive constant that will be fixed later. Applying triangle inequality, we get

ˇˇ}u} 2 L 2 pRq ´}η h ξ } 2 L 2 pRq ˇˇď δp}u} L 2 pRq `}η h ξ } L 2 pRq q ď δpδ `2}η h ξ } L 2 pRq q ď δp1 `2 sup ξPΩ, hăh0 }η h ξ } L 2 pRq q ": δκ.
So, choosing δ " βr κ , we deduce from (57) that there exists

ζ 1 P rξ 1 ´r, ξ 1 `rs such that }u} 2 L 2 pRq " gpζ 1 q " }η h ζ } 2 L 2 pRq ,
where ζ 2 :" ξ 2 . Applying the mean value equality, we obtain

|ξ ´ζ| ď β ´1 ˇˇ}u} 2 L 2 pRq ´}η h ζ } 2 L 2 pRq
ˇˇ. which proves the result.

Control of the instabilities and modulation

In the last section we have constructed approximate traveling waves η h ξ . In order to prove Theorem 1.4, we now study the dynamics of DNLS around these approximate traveling waves.

We are going to use many results established in the previous section about η h ξ and its properties. In a first paragraph, we summarize the results that will be useful and fix most of the constants.

Step 1: variational properties around the equilibria Let r Ω be a relatively compact open subset of

" ξ P R 2 | ξ 1 ą ´ξ2 2 ¯2*
and Ω a relatively compact open subset of r Ω. In the previous section, we have proven there exist some constants h 0 , ε, C, ρ ą 0 and, for all ξ P r Ω and all h ă h 0 , a function η h ξ P BL 2 h satisfying the following properties. ' From Theorem 3.2, η h ξ is a critical point of L h ξ and it is an approximation of ψ ξ

}η h ξ ´ψξ } H 1 pRq ď Ch 2 .
' From Theorem 3.15, η h ξ is regular function

(58) @ω P R, | x η h ξ pωq| ď Ce ´ε|ω| .
Consequently, we also have

} x η h ξ } H 3 pRq ď C. ' From Lemma 3.17, if u P BL 2 h X Spanpiη h ξ , B x η h ξ q K L 2 , }u} 2 L 2 pRq " }η h ξ } 2 L 2 pRq and }u ´ηh ξ } H 1 pRq ď ρ then (59) 1 C }u ´ηh ξ } 2 H 1 pRq ď L h ξ puq ´L h ξ pη h ξ q.
' From Theorem 3.20, if u P BL 2 h pRq, ξ P Ω and }u ´ηh ξ } H 1 pRq ď ρ then there exists ζ P r Ω such that (60)

" ξ 2 " ζ 2 }η h ζ } 2 L 2 pRq " }u} 2 L 2 pRq
and (using regularity of ξ Þ Ñ η h ξ uniformly with respect to h, see Theorem

P Spanpiη h ξ , B x η h ξ q K L 2 .
' From Lemma 3.18, for all u P BL 2 h pRq, if }u ´ηh ξ } H 1 pRq ď ρ and A h,ξ rus is the matrix defined in (54) then (63)

A h,ξ rus is is invertible and }pA h,ξ rusq ´1} 1 ď C.

' From Lemma 3.8 and Lemma 3.10, for all u

P BL 2 h pRq, if }u ´ηh ξ } H 1 pRq ď ρ then (64) @v, w P BL 2 h , ˇˇd 2 L h ξ puqpv, wq ˇˇď C}v} H 1 pRq }w} H 1 pRq .
We finish this paragraph by a remark. In Theorem 1.4, we compare a solution u of DNLS with some discretizations of η h ξ using discrete Sobolev norms. However, as we explain in Lemma 2.9, it is equivalent to compare directly the Shannon interpolation u of the discrete solution with η h ξ using continuous Sobolev norms.

Step 2: Lyapunov estimation and modulation Let r ą 0 be a positive constant independent of ξ and h that will be determined at the end of this paragraph. Recall that for v : R Ñ R we have @x P R, T γ,y vpxq :" e iγ vpx ´yq. and note that T ´1 γ,y " T ´γ,´y . Let u 0 P BL 2 h be such that δp0q " }u 0 ´Tγ0,y 0 η h ξ } H 1 pRq ă r where ξ P Ω, y 0 , γ 0 P R. Let u be the solution of DNLS in BL 2 h (see Lemma 2.8) such that up0q " u 0 .

Assume that r ă ρ. Applying (60) and (61), there exists ζ P r Ω such that

" ξ 2 " ζ 2 }η h ζ } 2 L 2 pRq " }u 0 } 2 L 2 pRq
and |ζ ´ξ| `}u 0 ´Tγ0,y 0 η h ξ } H 1 pRq ď Cδp0q.

Consequently, we have }η h ξ ´ηh ζ } H 1 pRq ď p1 `Cqδp0q. Now, assume that Cr ă ρ, then applying (62), there exist δ γ , δ y P R such that (65)

" θ 0 " γ 0 `δγ p 0 " y 0 `δy with maxp|δ γ |, |δ y |q ď C 2 δp0q and T ´1 θ0,p0 u 0 P Spanpiη h ζ , B x η h ζ q K L 2 .
We would like to get some functions θ, p P C 1 pR `q such that as long as uptq is close to the orbit of η h ζ (up to gauge transform and advection), we have T ´1 θptq,pptq uptq ¸.

P Spanpiη h ζ , B x η h ζ q K L 2 .
We would like to solve the Cauchy problem associated with this ordinary differential equation with θp0q " θ 0 and pp0q " p 0 . Note that all the terms depend smoothly on t, pptq, θptq, hence to get the existence of a local solution, we need to invert A ζ,h rT ´1 θptq,pptq uptqs. Using the regularity of η h ζ (see ( 58)), we have

}u 0 ´Tθ0,p0 η h ζ } H 1 pRq ď C 3 δp0q. Assuming that C 3 r ă ρ, we get from (63) that A ζ,h rT ´1 θ0,p0 u 0 s is invertible and }pA ζ,h rT ´1 θ0,p0 u 0 sq ´1} 1 ď C.
So (applying, for example, Cauchy-Lipschitz theorem or the implicit functions theorem), there exist T max P p0, 8s and a solution θ, p P C 1 pr0, T max qq of (66) on r0, T max q such that ' θp0q " θ 0 and pp0q " p 0 , ' for all t P r0, T max q, A ζ,h rT ´1 θptq,pptq uptqs is invertible,

' lim tÑTmax |θptq| `|pptq| `}pA ζ,h rT ´1 θptq,pptq uptqsq ´1} 1 " 8
We would like to prove that while }uptq ´Tγptq,yptq η h ξ } H 1 pRq ă r, with γ " θ ´δγ and y " p ´δy where δ γ and δ y are given in (65), the last condition is not satisfied and so γptq and yptq are well defined. This is done by the following Lemma, whose proof is given in Section 5.2 of the Appendix. Lemma 4.1. There exist γ, y P C 1 pR `q such that γp0q " γ 0 , yp0q " y 0 and if T ą 0 satisfies @t P p0, T q, }uptq ´Tγptq,yptq η h ξ } H 1 pRq ă r, then T ă T max and γ " θ ´δγ , y " p ´δy on p0, T q, where δ γ and δ y are defined in (65).

From now on, we consider the functions γ, y given by Lemma 4.1 and T ą 0 satisfying the bootstrap condition @t P p0, T q, δptq :" }uptq ´Tγptq,yptq η h ξ } H To be rigorous, we can verify our assumptions on r and observe that r " ρ 2`C`C 3 is a possible choice.

Step 3: Estimation of δptq Usually, when we apply the energy-momentum method, the Lagrange function is a constant of the motion of DNLS. An estimate of the form (67) allows to control }uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq by L h ζ pu 0 q ´L h ζ pη h ζ q. This latter quantity can be controlled by using a Taylor expansion

L h ζ pu 0 q ´L h ζ pη h ζ q " L h ζ pT ´1 θ0,p0 u 0 q ´L h ζ pη h ζ q ď 1 2 sup }v´η h ζ } H 1 pRq ďρ ˇˇd 2 L h ζ pvqpT ´1 θ0,p0 u 0 ´ηh ζ q ˇď C 2 }u 0 ´Tθ0,p0 η h ζ } 2 H 1 pRq
, where the last estimate is given by (64).

In our case, because of the aliasing terms, L h ζ puptqq is not a constant of the motion. To control these derivatives, we use the following lemma.

Lemma 4.2. If u 1 , u 2 , u 3 , u 4 P BL 2 h and

M h pu 1 , u 2 , u 3 , u 4 q " ż R e 2iπx h u 1 pxqu 2 pxqu 3 pxqu 4 pxq dx,
then we have

|M h pu 1 , u 2 , u 3 , u 4 q| ď 1 4 ÿ σPS4 }y u σ1 1 ωě π 3h } L 2 pRq }y u σ2 1 ωě π 3h } L 2 pRq }y u σ3 } L 1 pRq }y u σ4 } L 1 pRq .
Proof. We identify M h with a convolution product

M h pu 1 , u 2 , u 3 , u 4 q " x u 1 ˚x u 2 ˚x u 3 ˚x u 4 p 2π h q.
But if the sum of four numbers, all smaller than 1, is equals to 2, then at least 2 of them are larger than Then, we conclude the proof using Young convolution inequalities.

Applying this Lemma to estimate the terms of (69) we obtain four types of contributions.

H 1 pRq ď L h ξ puptqq ´L h ξ pη h ζ q " L h ξ pup0qq ´L h ξ pη h ζ q `L h ξ puptqq ´L h ξ pup0qq " L h ξ pup0qq ´L h ξ pη h ζ q `E1 p0q ´E1 ptq `E2 ptq ď C 2 }up0q ´Tθp0q,pp0q η h ζ } 2 H 1 pRq `M e ´ h `h2 M }uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq `M e ´ h `h2 M }up0q ´Tθp0q,pp0q η h ζ } 2 H 1 pRq `4M π|ξ 2 |h ż t 0 e ´ h h 2 `}upsq ´Tθpsq,ppsq η h ζ } 2 H 1 pRq ds .
So there exist some constants h 1 ă h 0 , c ą 0 and λ ą 0 (depending only of ε, C, ρ, h 0 ) such that, for all h ă h 1 , we have with ξ P Ω. Let h 1 ă h 0 a constant that we will determine later. Now consider h ă h 1 and u a solution of DNLS such that D y 0 , γ 0 P R, @g P hZ, u g p0q " e iγ0 vpg ´y0 q.

}uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq ď ce ´ 2h `c}up0q ´Tθp0q,pp0q η h ζ } 2 H 1 pRq `2λh|ξ 2 | ż t 0 e ´ 2h `
We denote by u the Shannon interpolation of u. Without loss of generality, since DNLS is invariant by gauge transform, we can assume γ 0 " 0.

Lemma 5.1. The following inequality holds:

}u 0 ´ηh ξ p¨´y 0 q} H 1 pRq ď }v ´ηh ξ } H 1 pRq `hn´1 ρ.
This lemma is a classical estimate of aliasing, it will be proven at the end of this subsection. Since u 0 , η h ξ P BL 2 h , we can apply Lemma 2.9 to obtain (75) δp0q :" } up0q ´`η h ξ p¨´y 0 q ˘|hZ } H 1 phZq ď }u 0 ´ηh ξ p¨´y 0 q} H 1 pRq ď }v ´ηh ξ } H 1 pRq `hn´1 ρ.

Applying the triangle inequality, we deduce of Theorem 1.4 that

δp0q ď }v ´ψξ } H 1 pRq `}ψ ξ ´ηh ξ } H 1 pRq `hn´1 ρ ď r 2p1 `κq `κh 2 `hn´1 ρ.
Consequently, if h 1 is small enough then δp0q ď r 1`κ . So we can apply Theorem 1.4 and Theorem 1.5. In particular, we get functions γ, y P C 1 pR `q such that, if for all t P p0, T q (76) δptq :" } uptq ´pe iγptq η h ξ p¨´yptqqq |hZ } H 1 phZq ď r, then we have for all t P p0, T q (77) δptq ď κ ˆδp0q `e´ h `at|ξ H n phZq uniformly with respect to h and ξ. We get these bounds in the following lemma that will be proven at the end of this subsection. Lemma 5.2. There exists a constant K ą 0, depending only of Ω, ρ and n such that for all h ă h 0 , κCM 4n´1 3

up0q ď K and κC ´} up0q} 9

H n phZq `M 2n`1 3 up0q ¯ď K.
With the estimate, (79) becomes

(80) δptq ď κδp0q `κe ´ h `Ka |ξ 2 |t n 2 h n´1 2 `Ka |ξ 2 | ? th n´1 2 .
Now, we overcome the bootstrap condition (76). Let T 0 P p0, 8s be a function of |ξ 2 | that will be fixed later. Consider t P p0, T 0 h ´2`ε q such that for all τ ď t, δpτ q ď r. We deduce from (80) that

δptq ď κδp0q `κe ´ h `KT n 2 0 a |ξ 2 |h nε´1 2 `Ka T 0 |ξ 2 |h n´3 2 ` 2 .
Assuming n 0 ě maxp2, ˘´1 ı ´s, we get δptq ă r. Consequently, proceeding as usual by contradiction, we deduce that it was useless to assume that for all τ ď t, δpτ q ď r.

Finally, to conclude rigorously this proof, we have to explain how to get (20) and ( 21). On the one hand, to get (20), we just have to estimate δp0q by (75) in (81) (and to assume that n 0 ´1 ě s). On the other hand, we have to estimate the terms of (78). We control δp0q as previously, δptq by ( 20) and e ´ h by `hs e ˘s. Proof of Lemma 5.1. Let v h be the L 2 orthogonal projection of v on BL 2 h , i.e.

x v h " 1 p´π h , π h q p v.
We introduce w h " u 0 ´vh p¨´y 0 q. Since the H 1 norm is invariant by advection, we have

}u 0 ´ηh ξ p¨´y 0 q} H 1 pRq ď }v h ´ηh ξ } H 1 pRq `}w h } H 1 pRq . Since η h ξ P BL 2 h , v ´vh is orthogonal to η h ξ in H 1 pRq. Consequently, we have }v h ´ηh ξ } H 1 pRq ď }v ´ηh ξ } H 1 pRq . So we just have to prove that }w h } H 1 pRq ď ρh n´1 .
Applying Proposition 2.6, we have

@ω P ´´π h , π h ¯, x w h pωq " ÿ kPZ ˚e´ipω`2 kπ h q y 0 p vpω `2kπ h q.
Consequently, we have

}w h } H 1 pRq ď 1 ? 2π ÿ kPZ ˚}p vpω `2kπ h q a 1 `ω2 } L 2 p´π h , π h q ď 1 ? 2π ÿ kPZ ˚› › › › › y B x vpω `2kπ h q ? 1 `ω2 ω `2kπ h › › › › › L 2 p´π h , π h q .
Assuming h 1 ď 2π, we have ˇˇ?

}w h } H 1 pRq ď π ? 2 }B x pv ´vh q} L 2 pRq ď ˆh π ˙n´1 π ? 2 }B n x pv ´vh q} L 2 pRq ď h n´1 π 2´n ? 2 ρ ď h n´1 ρ.
Proof of Lemma 5.2. There are two quantities to control, } up0q} 9 H n phZq and M up0q . To control } up0q} 9 H n phZq , it is enough to prove that the restriction to hZ is a continuous map from 9

H n pRq to 9 H n phZq, uniformly with respect to h. Indeed, denote w " vp¨´y 0 q. Then applying Proposition 2.6, we have, for all ω P `´π h , π h ˘,

x u 0 pωq "

ÿ kPZ ˚p wpω `2kπ h q.
Since for k ‰ 0 and ω P `´π h , π h ˘, we have ˇˇˇˇω

ω `2kπ h ˇˇˇˇď 1 2|k|
´1 , applying Cauchy Schwarz inequality (and (33) ), we get

} up0q} 9 H n phZq ď }ω n x u 0 } L 2 p´π h , π h q ď }ω n p wpωq} L 2 p´π h , π h q `ÿ kPZ ˚› › › › › ˜ω ω `2kπ h ¸n y B n x wpω `2kπ h q › › › › › L 2 p´π h , π h q ď }B n x w} L 2 pRq `ÿ kPZ ˚} y B n x wpω `2kπ h q} L 2 p´π h , π h q 1 p2|k| ´1q n ď }B n x w} L 2 pRq `}B n x w} L 2 pRq d ÿ kPZ ˚1 p2|k| ´1q 2n " ˜1 `d2 ˆ1 ´1 4 n ˙ζp2nq ¸}B n x v} L 2 pRq ď ˜1 `d2 ˆ1 ´1 4 n ˙ζp2nq
|θptq ´θ0 | `|pptq ´p0 | ď 1 `c2 t,
where c 2 ą 0 is a real constant that will be determine later. Now we define γ and y as C 1 functions on R `such that (84) @t P p0, T crit q, γptq " θptq ´δγ and yptq " pptq ´δy .

Let T ą 0 be such that for all t ă T , δptq " }uptq ´Tγptq,yptq η h ξ } H 1 pRq ă r. θpTcritq,ppTcritq uptqsq ´1} 1 " 2C or |θpT crit q ´θ0 | `|ppT crit q ´p0 | " 1 `c2 T crit . So, here is the contradiction and we have proven that T ď T crit .

Inverse function Theorem.

In this subsection, we give a version of the inverse function theorem.

Theorem 5.3. Let X, Y be some Banach spaces, Ω be an open convex subset of X such that 0 P Ω.

If g : Ω Ñ Y is a C 1 function such that ' d gp0q is invertible, ' d g is a k-Lipschitz function, then, defining β " } d gp0q ´1} ´1 and r " β k , we have ' g is a C 1 diffeomorphism from B X p0, rq X Ω to gpB X p0, rq X Ωq, ' for all x P B X p0, rq X Ω, } d gpxq ´1} ď r βpr´}x}q , ' for all 0 ă ρ ď r, if B X p0, ρq Ă Ω then B Y pgp0q, β
2 ρq Ă gpB X p0, ρqq. Proof. First, we prove that g is injective on B X p0, rq X Ω. Let y P B X p0, rq X Ω. We introduce the application So we also deduce the second point of the theorem through the classical estimate of the Von Neumann series. Now, applying the classical inverse function theorem, we have proven that g is a C 1 diffeomorphism from B X p0, rq X Ω to gpB X p0, rq X Ωq. Finally, we just need to prove the last assertion of the theorem. Let ρ ą 0 be such that 0 ă ρ ď r, B X p0, ρq Ă Ω. We introduce δ P p0, ρq to prove that B Y pgp0q, β 2 δq Ă gpB X p0, δqq. It is enough to prove the last point because

Φ y : " B X p0, rq X Ω Ñ X x Þ Ñ x ´d
B Y pgp0q, β 2 ρq " ď 0ăδăρ B Y pgp0q, β 2
δq and gpB X p0, ρqq " ď 0ăδăρ gpB X p0, δqq.

Let y P B Y pgp0q, β 2 δq, we want to solve gpxq " y. So, we introduce the application Ψ " Φ y |B X p0,δq . We want to apply the Banach fix point theorem. We have proven in (86) that Ψ is δk β ă 1 Lipschitz, so we just need to prove that it preserves B X p0, δq. Indeed, we have ¯2* then there exists c ą 0 such that for all ξ P Ω we have (87) @v P H 1 pRq X Spanpψ ξ , iψ ξ , B x ψ ξ q K L 2 , d 2 L ξ pψ ξ qpv, vq ě c}v} 2 H 1 pRq .

Proof. Weinstein has proven in [START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF] that there exists c ą 0 such that for all v P H 1 pRq, (88) v P Spanpψ p1,0q , iψ 3 p1,0q , B x pψ 3 p1,0q qq K L 2 ñ d 2 L p1,0q pψ p1,0q qpv, vq ě c}v} 2 H 1 . First, we will deduce from this estimate and Lemma 5.5 that (87) holds true for ξ " p1, 0q. Then we will extend this result applying two transformations: dilatation and boost.

Step 1: The case ξ " p1, 0q. We apply Lemma 5.5 below, with the spaces E " H 1 pRq X Spanpψ p1,0q q K L 2 , G " H 1 pRq X Spanpψ p1,0q , iψ 3 p1,0q , B x pψ 3 p1,0q qq K L 2 , F " H 1 pRq X Spanpψ p1,0q , iψ p1,0q , B x ψ p1,0q q K L 2 and eventually H " Spanpiψ p1,0q , B x ψ p1,0q q. We equipped all these spaces with the H 1 pRq norm for which they are closed. By construction, F and H are obviously complementary spaces. However, we have to prove that G and H are complementary spaces.

First, we prove that HXG " t0u. If g " αiψ p1,0q `βB x ψ p1,0q P G then xg, iψ 3 p1,0q y L 2 pRq " xg, B x pψ 3 p1,0q qy L 2 pRq " 0. However, since ψ p1,0q is a real valued function, we have (89) xB x ψ p1,0q , iψ 3 p1,0q y L 2 pRq " xB x pψ 3 p1,0q q, iψ p1,0q y L 2 pRq " 0. Consequently, we deduce that α}ψ p1,0q } 4 L 4 pRq " βxB x pψ 3 p1,0q q, B x ψ p1,0q y L 2 pRq " 0. So we just need to verify from (4) that xB x pψ 3 p1,0q q, B x ψ p1,0q y L 2 pRq ‰ 0 which yields α " β " 0. Now, we prove that H `G " E. Since, by construction G `Spanpiψ 3 p1,0q , B x pψ 3 p1,0q qq " E, we just need to prove that iψ 3 p1,0q , B x pψ 3 p1,0q q P H `G. Since iψ 3 p1,0q and B x pψ 3 p1,0q q are orthogonal, we can decompose iψ p1,0q and B x ψ p1,0q through the decomposition E " G `Spanpiψ 3 p1,0q , B x pψ 3 p1,0q qq to get (with (89)) # iψ p1,0q }ψ p1,0q } 6 L 6 pRq ´}ψ p1,0q } 4 L 4 pRq iψ 3 p1,0q P G, B x ψ p1,0q }B x pψ 3 p1,0q q} 2 L 2 pRq ´xB x pψ 3 p1,0q q, B x ψ p1,0q y L 2 pRq B x pψ 3 p1,0q q P G.

Since the coefficients associated with iψ 3 p1,0q and B x pψ 3 p1,0q q are not zero, we deduce that iψ 3 p1,0q , B x pψ 3 p1,0q q P H `G.

In order to apply Lemma 5.5, with b " d 2 L p1,0q pψ p1,0q q we have to prove that B x ψ p1,0q and iψ p1,0q belong to the kernel of d 2 L p1,0q pψ p1,0q q. Indeed, since L p1,0q pψ p1,0q q is invariant by gauge transform and dilatation, the set of its critical points are also invariant by these transform, i.e. @t P R, @v P H 1 pRq, d L p1,0q pe it ψ p1,0q qpvq " d L p1,0q pψ p1,0q p. ´tqqpvq " 0.

However, since ψ p1,0q is a very regular function (see Lemma 3.5 or directly (4)), we can compute the derivative in t " 0 to get @t P R, @v P H 1 pRq, d 2 L p1,0q pψ p1,0q qpiψ p1,0q , vq " d 2 L p1,0q pψ p1,0q qpB x ψ p1,0q , vq " 0. Now to apply Lemma 5.5, we observe that the required assumption of coercivity of b on G is the result of Weinstein (88), and we obtain the result.

Step 2: Extension by dilatation and boost Denote by T the dilatation action defined by T m puqpxq " mupmxq for all x P R, u P H 1 pRq and m ą 0, and let B bz the boost action defined by B ν u :" e iνx u for all x P R, u P H 1 pRq and ν P R. These transformations are useful because we have the following relations @m, µ ą 0, @ν P R, L p1,0q ˝Tm " m 3 L pm ´2,0q and L pµ,0q ˝Bν " L pµ`ν 2 ,´2νq

With these relations a straightforward calculation shows that (90) L ξ " m 3 ξ L p1,0q ˝Tm ´1 ξ ˝B´ξ 2 2 with m ξ "

d ξ 1 ´ˆξ 2 2 ˙2.
Furthermore, using the definition of ψ ξ , we have

ψ ξ " B ξ 2 2 ˝Tm ξ ψ p1,0q
.

Consequently, we are able to transport the coercivity property from ξ " p1, 0q to any ξ, provided that ξ 1 ą ´ξ2 2 ¯2. First, we observe that if v P H1 pRq X Spanpψ ξ , iψ ξ , ψ 1 ξ q K L2 then T m ´1 ξ ˝B´ξ 2 2 v P H 1 pRq X Spanpψ p1,0q , iψ p1,0q , ψ 1 p1,0q q K L 2 . This inequality implies Estimate (87) because applying Peetre inequality 1 , we get

Second

› › ›T m ´1 ξ ˝B´ξ 2 2 v › › › 2 H 1 " › › › › › B ´m´1 ξ ξ 2 2 ˝Tm ´1 ξ v › › › › › 2 H 1 ě 1 2 }T m ´1 ξ v} 2 H 1 1 `ˆm ´1 ξ ξ2 2 ˙2 " 1 2 m ´1 ξ }v} 2 L 2 `m´3 ξ }B x v} 2 L 2 1 `ˆm ´1 ξ ξ2 2 ˙2 .
5.5. Functional analysis lemmas.

Lemma 5.5. Let F , G be two closed subspaces of a normed space E. If F and H admit a same finite dimensional complementary space H, denote by Π the projection onto G of kernel H. Then Π |F is a normed space vector isomorphism. Furthermore, if b is a bilinear symmetric form on E, H is a subspace of its kernel and if there exists α ą 0 such that @x P G, bpx, xq ě α}x} 2 then there exists β ą 0 such that @x P F, bpx, xq ě β}x} 2 .

Proof. Let P be the projection onto F of kernel H. If f P F then P Πf " f . Indeed, if f " g `h with g P G and h P H then g " Πf " f ´h. Consequently, we would have f " P g " P Πf . Similarly, we can prove that ΠP g " g, for any g P G. So, we have proven that Π ´1 |F " P |G .

To prove the first part of the lemma, we just have to prove that Π and P are continuous to conclude this proof. This is a very classical exercise of normed space vector, whose proof is based on compactness.

The second part of the lemma is a straightforward calculation. Indeed, if x P F then bpx, xq " bpΠx, Πxq ě α}Πx} 2 ě α}Π ´1 |F } ´2}x} 2 .

Lemma 5.6. Let E be a real vector space whose px j q j"1,...,n is a free family. Define X " Spanpx j q j"1,...,n the subspace generated by this family. Let x¨, ¨y1 ,x¨, ¨y2 be two scalar products on E such that the induced norms satisfy } ¨}1 ď c} ¨}2 . Define G P M n pRq the Gram matrix associated to px j q j"1,...,n for the scalar product x¨, ¨y1 , i.e. For any u P E, let bpuq be a bilinear symmetric form continuous for the } ¨}2 norm. Assume that b is k Lipschitz on a ball of radius R ą 0, i.e.

@u, v P B 2 p0, Rq, @y, z P E, |bpuqpy, zq ´bpvqpy, zq| ď k}u ´v} 2 }y} 2 }z} 2 and that there exists α ą 0 such that @y P X K1 , bp0qpy, yq ě α}y} 2 2 .

To prove the Lemma, we have to solve, (92) @y P E, xT x, yy " φpyq with φ P E 1 .

Let z P E m and denote y " z ´T ´1 pp T pm z. First, we verify that P y " 0. Consequently, we deduce from (92) that φpyq " φpz ´T ´1 pp T pm zq " xSx m , zy. However, we verify that ´S verifies assumptions Lemma 5.8 with α " α m . Consequently, S is invertible and so we have

x m " S ´1φ |Em ´S´1 φT pp T pm . Now if we apply (92) for y " y p P E p , we have φpyq " xT pp P x, yy " xT pp x p , yy `xT pm x m , yy.

Consequently, we have

x p " T ´1 pp φ |Ep ´T ´1 pp T pm x m . Finally, we have solved (92). So T is bijective and we verify (91) using the estimate given by Lemma 5.8.

´2n 3 n

 3 n . Consequently, if |ω| ě C and n " t ω C u, we have |p upωq| ď ce ´|ω| 2C , and this shows the result.

}Ψpxq} ď }Ψp0q} `}Ψpxq ´Ψp0q} ď β 2 5 . 4 .

 254 δ} d gp0q ´1} `} d gp0q ´1}}gpxq ´gp0q ´d gp0qx} A result of coercivity. Lemma 5.4 (A reformulation of a Weinstein result in[START_REF] Weinstein | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF]). If Ω is a relatively compact open subset of the set" ξ P R 2 | ξ 1 ą ´ξ2 2

2 2 2 vq " 0 . 2 v

 2202 , we calculate the derivative of the Lagrange function through the transport relation (90),d L ξ pψ ξ qpvq " m 3 ξ drL p1,0q ˝Tm ´1 ξ ˝B´ξ spψ ξ qpvq " m 3 ξ d L p1,0q pψ 1,0 qpT m ´1 ξ ˝B´ξ 2Then we deduce a property of coercivityd 2 L ξ pψ ξ qpv, vq " m 3 ξ d 2 L p1,0q pψ 1,0 qpT m ´1 ξ ˝B´ξ 2

G " ¨xx 1

 1 , x 1 y 1 . . . xx 1 , x n y 1 . . . . . . xx n , x 1 y 1 . . . xx n , x n y 1 ‹ '.

Cq; Rq. Defining

  Proposition 2.1. I h is an isometry between L 2 phZq and its image in L 2 pRq. This image is denoted BL 2 h . It is the subspace of L 2 pRq whose Fourier transform support is a subset of r´π h , π h s, i.e. Cq, xi., .y L 2 phZ;Cq ˘and `BL 2 h , xi., .y L 2 pR;Cq ˘preserving the Hamiltonian structure. Lemma 2.2. Let I be an open subset of R, u P C 1 pI; L 2 phZ; Cqq and H P C 1 pL 2 phZ; Hpuptqqpvq " xiB t uptq, vy L 2 phZq . However, we have xiB t uptq, vy L 2 pRq " xI h iI h B t uptq, vy L 2 phZq , where I h is the adjoint operator of I h . But I h is C linear so we have iI h B t uptq " I h iB t uptq. Furthermore, it is an isometry so we have I h " I ´1 h . Consequently, we get xiB t uptq, vy L 2 pRq " xiB t uptq, vy L 2 phZq . uptq, vy L 2 pRq " xiB t uptq, ivy L 2 pRq " d Mpuptqqpivq " 2xiB x uptq, ivy L 2 pRq " 2xB x uptq, vy L 2 pRq .

	Lemma 2.3. Let M : BL 2 h Ñ R be the momentum defined by
			@u P BL 2 h ,	Mpuq " xiB x u, uy L 2 pRq .
	If u P C 1 pR; BL 2 h q then the following properties are equivalent
	(27)		@t P R,		upt, xq " up0, x `2tq,
	and
	(28)	@t P R, @v P BL 2 h ,		xiB t uptq, vy L 2 pRq " d Mpuptqqpvq.
	Proof. Assume (28) and let t P R, v P BL 2 h . We have
	xB t So since pBL 2 h , } ¨}L 2 pRq q is a Hilbert space, we have
			@t, x P R, B t upt, xq " 2B x upt, xq.
	Consequently, we have upt, xq " up0, x `2tq. The converse is obvious.
	Applying Lemma 2.2 and Lemma 2.3, we deduce that Shannon's advection the flow of the Hamiltonian
	´1 2 M ˝I´1 h .
	2.2. The aliasing error.	BL 2 h " tu P L 2 pRq | Supp p u Ă r´π h	,	π h	su.
	Moreover, the Shannon advection τ a is well defined through (22).
	Proof. We just need to verify that the advection of a Shannon interpolation is an interpolation. So let u P BL 2 h .
	Since we have
			@ω P R,	{ up¨´aqpωq " e ´iωa	p upωq,
	it is clear that Supp { up¨´aq " Supp p u. Consequently, we have proven that up¨´aq P BL 2 h .
	Since Fourier transform support of Shannon interpolations is bounded, BL h 2 functions are very regular functions (they are entire function). Consequently, when we deal with BL h 2 functions we will not justify the
	algebraic calculations.
	We now check that this advection is generated by a Hamiltonian flow. Introducing some formalism, since
	Shannon interpolation is a C linear isometry, we prove in the following Lemma that it is a symplectomorphism
	between `L2 phZ; u " I h u,
	the following properties are equivalents
	(25)	@t P I, @ v P L 2 phZ; Cq,	xiB t uptq, vy L 2 phZq " d Hpuptqqpvq,
	and
	(26)	@t P I, @v P BL 2 h ,	xiB t uptq, vy L 2 pRq " dpH ˝I´1 h qpuptqqpvq.
	Proof. Assume (25) and v P BL 2 h . Since I h is bijective, there exists v P L 2 phZ; Cq such that v " I h v. So we
	have
	dpH ˝I´1 h qpuptqqpvq " dpH ˝I´1
	So we have proven (26). Conversely, we can prove that (25) is a consequence of (26) using the same equalities.
	Applying Lemma 2.2 to identify Shannon advection with a Hamiltonian flow, we just need to identify the
	canonical advection on BL 2 h .

h qpuptqqpI h vq " d

  Since ψ ξ is a critical point of L ξ , we deduce from the definition of ψ h ξ (see (39)) that d L h ξ pψ h ξ qpvq " d L h ξ pψ h ξ qpvq ´d L ξ pψ ξ qpvq " xpB 2 x ´∆h qψ ξ , vy L 2 pRq `d } ¨}4

	(44)						L 4 pRq 4	pψ ξ qpvq	´d } ¨}4 L 4 pRq 4	pψ h ξ qpvq.
	To estimate the first term, we use Fourier Plancherel isometry to get
	|xpB 2 x ´∆h qψ ξ , vy L 2 pRq | " ˇˇˇ1 2π	ż R	"	4 h 2 sin 2 ˆπωh 2	˙´ω 2		x ψ ξ pωq.p vpωq dω ˇˇď

  K L 2 . Then, we have max `|xψ h ξ , vy L 2 pRq |, |xiψ h ξ , vy L 2 pRq |, |xB x ψ h ξ , vy L 2 pRq | ˘ď }u ´ψh ξ } H 1 pRq }v} H 1 pRq ď c 2 }v} H 1 pRq . Consequently, we can apply the result of Lemma 5.6 to get

	d 2 L h ξ puqpv, vq ě α}v} 2 H 1 pRq ě	α 2	}v} 2 H 1 pRq .

  H 2 pRq }v} L 2 pRq . On the other hand, we haveˇˇd 2 } ¨}4 L 4 pRq pψ ξ qpv, d ξ ψ ξ pζqq ´d2 } ¨}L 4 pRq pψ h ξ qpv, d ξ ψ h ξ pζqq ˇď 12}ψ ξ `ψh ξ } L 4 pRq }ψ ξ ´ψh ξ } L 4 pRq }v} L 4 pRq } d ξ ψ ξ pζq} L 4 pRq `12} d ξ ψ ξ pζq ´dξ ψ h ξ pζq} L 4 pRq }ψ h ξ } 2 L 4 pRq }v} L 4 pRq . Applying Gagliardo-Nirenberg inequality, (46) and Lemma 3.5, it is clear that }ψ ξ `ψh ξ } L 4 pRq , }ψ h ξ } 2 L 4 pRq , |ζ| ´1} d ξ ψ ξ pζq} L 4 pRq and |ζ| ´1} d ξ ψ h

  We are going to construct them by solving a differential equation. Taking a time derivative, if such functions exist they have to satisfy

	(66)	A ζ,h rT ´1 θptq,pptq uptqs	ˆ9 θptq 9 pptq ˙" ˜xT ´1 θptq,pptq B t uptq, iη h ζ y L 2 pRq xT ´1 θptq,pptq B t uptq, B x η h ζ y L 2 pRq

  1 pRq ă r.

	By construction, we have
	}uptq ´Tθptq,pptq η h ζ } H 1 pRq ď }uptq ´Tγptq,yptq η h ξ } H 1 pRq `}η h ξ ´Tδγ,δy η h ξ } H 1 pRq `}η h ζ	´ηh ξ } H 1 pRq
			ď δptq `C3 δp0q `p1 `Cqδp0q
			ă p2 `C `C3 qr.
	We assume that p2 `C `C3 qr ď ρ. Since }u} 2 L 2 pRq is a constant of the motion, we have }uptq} 2 L 2 pRq " }η h ζ } 2 L 2 pRq .
	Furthermore, by construction T ´1 θptq,pptq u P Spanpiη h ζ , B x η h ζ q K L 2 , so we can apply (59) to get the Lyapunov control
	of the stability
	(67)	1 C	}uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq ď L h ζ puptqq ´L h ζ pη h ζ q.

  So we have to control its variations. Let t ă T , since H h DNLS puptqq and }uptq} 2 L 2 pRq are constant of the motion, applying the formula of Lemma 30, we obtain the following decomposition uptq, uptqy L 2 pRq ´xiB x up0q, up0qy L 2 pRq " ptq " ξ 2 `xiB x uptq, uptqy L 2 pRq ´xiB x up0q, up0qy L 2 pRq ˘. Note that we write ξ 2 instead of ζ 2 as these two numbers are equal by construction (see (60)). First, we explain how to bound E 1 ptq. It can be decomposed as follow E 3 pT θptq,pptq η h ζ qpuptq ´Tθptq,pptq η ζ,h , . . . , uptq ´Tθptq,pptq η ζ,h loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon

		L h ζ puptqq ´L h ζ pup0qq " H h DNLS puptqq ´Hh DNLS pup0qq	`ζ1 2	´}uptq} 2 L 2 pRq ´}up0q} 2 L 2 pRq	1
					2	ż R	cos	ˆ2πx h	˙p|upt, xq| 4 ´|up0, xq| 4 q dx
	(68)				`ζ2 2 `xiB x E 1 p0q ´E1 ptq	`1 2	E 2 ptq,
	where				E 1 ptq "	1 2	ż R	cos	ˆ2πx h	˙|upt, xq| 4 dx
	and
		E 2 E 1 ptq " 1 4 ´E3 puptqq `E3 puptqq ¯, with E 3 pvq "	ż	R	e	2iπ h |upt, xq| 4 dx .
	Since E 3 is a 4´homogeneous continuous function, its Taylor expansion is exact. So, we have
	(69)	E 3 puptqq "	4 ÿ j"0	1 j!	d j j times	q.

  Up to an universal constant c ą 0, we have }F rT θptq,pptq η h ζ s} L 1 pRq ď c C. ' Up to an universal constant c ą 0, we have ω} L 2 pRq ď ch}uptq ´Tθptq,pptq η h ζ } H 1 pRq . ' Up to an universal constant c ą 0, we have }F ruptq ´Tθptq,pptq η h ζ s} L 1 pRq ď c}uptq ´Tθptq,pptq η h ζ } H 1 pRq . Sometimes, it is also useful to control it by cρ. With these estimates, we get a constant M ą 0 (depending only of ε, C, ρ, h 0 ) such that (70) |E 3 puptqq| ď 2M e ´ h `2M h 2 }uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq . |E 1 ptq| ď M e ´ h `h2 M }uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq . We show now how to control the term E 2 in (68). It is precisely the error generated by the default of invariance by advection. First, we give a more adapted expression of E 2 : B s xiB x upsq, upsqy L 2 pRq ds " 2ξ 2 ż t 0 xiB x upsq, B s upsqy L 2 pRq ds

	' Applying (58) and defining " πε 3 , we have
		}F rT θptq,pptq η h ζ s1 ωě π 3h } 2 L 2 pRq ď C 2	ż	ωě π 3h	e ´2εω dω "	C 2 ε	e ´2ε π 3h "	C 2 ε	e ´2 h .
	}F ruptq ´Tθptq,pptq η h ζ s1 ωě π 3h } L 2 pRq ď ζ s So we deduce that 3h π }F ruptq ´Tθptq,pptq η h
	(71)
		ż t
	E 2 ptq " ξ 2
		0
		" ´4ξ 2	ż t 0	xB x upsq, cos	ˆ2πx h	˙|upsq| 2 upsqy L 2 pRq ds
		" ´ξ2	2π h	ż t 0	ż R	sin	ˆ2πx h	˙|ups, xq| 4 dx ds " ´ξ2	π h	ż t 0	E 3 pupsqq ´E3 pupsqq ds .
	Applying Estimate of E 3 pupsqq (70), we obtain
		|E 2 ptq| ď 4M π|ξ 2 |h	ż t 0	e ´ h h 2 `}upsq ´Tθpsq,ppsq η h ζ } 2 H 1 pRq ds .
	Finally, we apply estimate (67) and we get
	1 C	}uptq ´Tθptq,pptq η h ζ } 2

'

  }upsq ´Tθpsq,ppsq η h ζ } 2 H 1 pRq ds . We can estimate E 4 as we have estimated E 1 ptq in the previous paragraph. Consequently, we get some constants M, independent of h and ζ such that |E 4 | ď M e ´ h `M }uptq ´ηh ζ } H 1 pRq . Applying these three estimates and the control of the norm of the invert of A ζ,h ruptqs, we get from (74) CM e ´ h `CpM `k `cCq}uptq ´ηh ζ } H 1 pRq . However, we have proven that }uptq ´ηh ζ } H 1 pRq ď δptq `p1 `C `C3 qδp0q and |ξ ´ζ| ď Cδp0q. So, since 9 Proof of Theorem 1.7. Let s ą 0, ε P p0, 2q and n P N ˚be such that n ě n 0 ě 2 where n 0 P N ˚will be determined later to be large enough. Let ρ ą 0 and v P H n pRq be such that }v} 9 H n pRq ď ρ and }ψ ξ ´v} H 1 pRq ď

	' | 9 θptq ´ζ1 | `| 9 pptq ´ζ2 | ď θ " 9 γ
	and 9 p " 9 y, we have proven that
	| 9 γptq ´ξ1 | `| 9 yptq ´ξ2 | ď Kpe ´ h `δptq `δp0qq,
	where K depends only of C, M, c and k.
				5. Appendix
	5.1. r 2p1 `κq	,
	Applying Grönwall's lemma, we obtain the estimate
	}uptq ´Tθptq,pptq η h ζ } 2 H 1 pRq	`e´ 2h ď e 2λ|ξ2|ht	" p1 `cqe ´ 2h `c}up0q ´Tθp0q,pp0q η h ζ } 2 H 1 pRq	ı	.
	Now applying Minkowski inequality, we get
	}uptq ´Tθptq,pptq η h ζ } H 1 pRq ď	?	1 `c e λ|ξ2|ht	" e ´ 4h `}up0q ´Tθp0q,pp0q η h ζ } H 1 pRq	ı	.

  2 |h n´3 2 sup So, to use (79), we have to estimate M up0q and } up0q} 9

							˙,
					0ăsăt	} upsq} 9 H n´1 phZq
	and
	(78)	| 9 γptq ´ξ1 | `| 9 yptq ´ξ2 | ď κ pδp0q `δptq `e´ h q.
	Applying Theorem 1.6, we deduce that if (77) is satisfied then
	(79)	δptq ď κ ´δp0q `e´ h `Ca |ξ 2 |t	n 2 h n´1 2 M	4n´1 3 up0q	`Ca |ξ 2 |	?	th n´1 2 ´} up0q} 9 H n phZq	`M 2n`1 3 up0q ¯¯,
	where
		M up0q " } up0q} 9 H 1 phZq `} up0q} 3 L 2 phZq .

  1 ď 1 and T 0 " minp|ξ 2 | ´1, |ξ 2 | ´1 n q, we deduce

	1`2s ε , s`3´ε 2 q, h (81) δptq ď κδp0q `κe ´ h `2Kh s ď κδp0q `´κ	´s e	¯s `2K ¯hs .
	So assuming h 1 ă	"	r 1`κ	`κ `s e ˘s `2K

  } H 1 pRq ď }B x pv ´vh q} L 2 pRq

	1`ω 2 ω`2 kπ h	ˇˇď	2 2|k|´1 for ω P `´π h , π h ˘. Consequently, applying Cauchy-Schwarz
	inequality, we get
	}w h d kPZ ÿ ˚4 p2|k| ´1q 2 "	π ? 2	}B x pv ´vh q} L 2 pRq .
	Since the Fourier support of v ´vh is localized outsize	"	´π h , π h	‰	and n ě 2, we have

  ¸ρ, where ζ is the Riemann zeta function.Finally, to control M up0q , we just have to control } up0q} H 1 phZq . But since we have proven that δp0q ď r 1`κ , we just need to control }η h ξ } H 1 pRq uniformly with respect to ξ P Ω and h ă h 0 . Such an estimate can be obtained by using the bound }η h ξ ´ψh ξ } H 1 pRq ď κh 2 of Theorem 1.4. 5.2. Proof of Lemma 4.1. We would like to define the functions γ and y from θ and p. So we introduce a new time: T crit . It is the largest time, smaller than T max , such that for all t P p0, T crit q, we have

	(82)	}pA ζ,h rT ´1 θptq,pptq uptqsq ´1} ď 2C
	and
	(83)

  To prove Lemma 4.1, it is enough to prove that T ď T crit . We proceed by contradiction. Assume that T crit ă T . So if t ă T crit , we have rT ´1 θptq,pptq uptqs is invertible and }pA ζ,h rT ´1 θptq,pptq uptqsq ´1} 1 ď C. Furthermore, we can estimate xT ´1 θptq,pptq B t uptq, iη h ζ y L 2 pRq and xT ´1 θptq,pptq B t uptq, B x η h ζ y L 2 pRq . Indeed, since u is a solution of DLNS in BL 2 h (see Lemma 2.3), we have xT ´1 θptq,pptq B t uptq, iη h ζ y L 2 pRq " ´B∆ h uptq `ˆ1 `2 cos We are going to estimate these terms. Since t ă T , by definition, we have }uptq ´Tγptq,yptq η h ξ } H 1 pRq ă r and so }uptq} H 1 pRq ď r `C. Consequently, we have }|uptq| 2 uptq} L 2 pRq ď }uptq} 2 L 8 }uptq} L 2 pRq ď pr `Cq 3 . Furthermore, we have seen in (33) that }∆ h η h ζ } L 2 pRq ď }B 2 C 2 pr `Cqp1 `pr `Cq 2 q. Defining c 2 " 2C 2 pr `Cqp1 `pr `Cq 2 q, we have |θptq ´θ0 | `|pptq ´p0 | ď c 2 t. We can apply this inequality and (85) for t " T crit , so we have }pA ζ,h rT ´1 θpTcritq,ppTcritq uptqsq ´1} 1 ď C and |θpT crit q ´θ0 | `|ppT crit q ´p0 | ď c 2 T crit . But it is impossible because by definition of T crit we should have }pA ζ,h rT ´1

	So, we have proven that
		maxp| 9 θptq|, | 9 pptq|q ď
		}uptq ´Tθptq,pptq η h ζ } H 1 pRq ď p2 `C `C3 qr ď ρ.
	Applying (63), we know that
	(85)	A ζ,h ˆ2πx h	˙˙|uptq| 2 uptq, T ´1 θptq,pptq η h ζ	F	L 2 pRq	.
	Since this operator is symmetric for the L 2 norm, we have
	xT ´1 θptq,pptq B t uptq, iη h ζ y L 2 pRq " ´xuptq, T ´1 θptq,pptq ∆ h η h ζ y L 2 pRq	´Bˆ1	`2 cos	ˆ2πx h	˙˙|uptq| 2 uptq, T ´1 θptq,pptq η h ζ	F	L 2 pRq	.

x η h ζ } L 2 pRq ď C. Consequently, we have |xT ´1 θptq,pptq B t uptq, iη h ζ y L 2 pRq | ď Cpr `Cq 3 `Cpr `Cq. Similarly, we could prove that |xT ´1 θptq,pptq B t uptq, B x η h ζ y L 2 pRq | ď Cpr `Cq 3 `Cpr `Cq.

If x, y P R then 1 `px ´yq

ě 1 2 p1 `x2 qp1 `y2 q ´1.

We want to deduce a bound on δ from this inequality. Applying H n pRq .

Step 4: Control of 9 γ and 9 y The idea to obtain the estimate [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] is that ξ is the solution of a perturbed linear equation whose p 9 γ, 9 yq is a solution (i.e. (66)). We work with a fixed t ă T . To simplify the notation, we assume that θptq " pptq " 0. We introduce a notation : for v P BL 2 h , we define

With this formalism, equation ( 66) becomes (see Lemma 2.8)

By construction η h ζ generates a traveling wave of the perturbation of DNLS whose speed is ζ. It means we can apply Proposition 3.1 with upt, xq :" e iζ1 η h ζ px ´ζ2 tq. However, we have e ´iζ1 upt, ¨`ξ 2 tq " Proof. Let y " y `yK be the decomposition of y associated to the algebraic decomposition E " X ' X K1 . So, we get bp0qpy, yq " bp0qpy `yK , y `yK q " bp0qpy K , y K qbp0q `2bp0qpy , y K q `py , y q

Consequently, we just need to control }y } 2 with }y} 2 to get the result when u " 0. However, using basis linear algebra we can prove that y " n ÿ j"1 a j x j with pa j q j"1,...,n " G ´1pxx j , yy 1 q j"1,...,n .

So, we get

Finally, by definition of c 2 , we get bp0qpy, yq ě α 4 }y} 2 2 . Furthermore, since b is k Lipschitz on Bp0, Rq, we deduce directly that if }u} 1 ď c 1 then bpuqpy, yq ě α 8 }y} 2 2 .

Lemma 5.7. Let E be a Banach space of dual space E 1 . Consider a algebraic decomposition of E, E " E p ' E m , and a continuous linear application T :

Then T is invertible and we have

Proof. In the proof we omit the index E 1 , E for all the duality brackets. We define by restrictions T 1 2 P L pE 2 ; E 1 q for 1 , 2 P tp, mu. Then we use a direct corollary of Riesz Theorem to prove that T pp is invertible. This corollary is the following.

Lemma 5.8. Let E be a Banach space of dual E 1 . Consider a continuous linear application T : E Ñ E 1 such that i) Dα ą 0, @x P E, xT x, xy ě α}x} 2 , ii) @x, y P E, xT x, yy " xT y, xy, then T is invertible and }T ´1} ď α ´1.

Now, decomposing x " x p `xm with x p P E p and x m P E m , we introduce operators P : E Ñ E p and S : E m Ñ E 1 m defined by P x " x p `T ´1 pp T pm x m and S " T mm ´Tmp T ´1 pp T pm . Then we verify by symmetry of T (with the same decomposition for y) that @x, y P E, xT x, yy " xT pp P x, P yy `xSx m , y m y.