Raphaël Côte

Xavier Friederich

SPATIAL DECAY OF MULTI-SOLITONS OF THE GENERALIZED KORTEWEG-DE VRIES AND NONLINEAR SCHRÖDINGER EQUATIONS

Keywords: 2010 Mathematics Subject Classication. Primary 35B40, 35Q53, 35Q55, Secondary 35B65, 37K40 Generalized Korteweg-de Vries equations, nonlinear Schrödinger equations, solitons, multi-solitons, spatial decay. 1

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1. Introduction 1.1. (gKdV) multi-solitons. We consider the generalized Korteweg-de Vries equations

∂ t u + ∂ x (∂ 2 x u + u p ) = 0 (gKdV)
where (t, x) ∈ R × R and p 2 is an integer.

Recall that (gKdV) admits a family of explicit traveling wave solutions R c0,x0 indexed by

(c 0 , x 0 ) ∈ R * + × R.
Let Q be the unique (up to translation) positive solution in H 1 (R) (known also as ground state) to the following stationary elliptic problem associated with (gKdV)

Q + Q p = Q,
given by the explicit formula

Q(x) = p + 1 2 cosh 2 p-1 2 x 1 p-1 .
Then for all c 0 > 0 (velocity parameter) and x 0 ∈ R (translation parameter), (1.1)

R c0,x0 (t, x) = Q c0 (x -c 0 t -x 0 )
is a global traveling wave solution of (gKdV) classically called the soliton solution, where

Q c0 (x) = c 1 p-1 0 Q( √ c 0 x).
We are interested here in qualitative properties of the multi-solitons, which are solutions to (gKdV) built upon solitons, and are dened as follows.

Denition 1.1. Let N 1 and consider N solitons R cj ,xj as in (1.1) with speeds 0 < c 1 < • • • < c N . A multi-soliton in +∞ (resp. in -∞) associated with the R cj ,xj is an H 1 -solution u of (gKdV) dened in a neighborhood of +∞ (resp. -∞) and such that (1.2)

u(t) -
The study of multi-solitons is motivated by the soliton resolution conjecture, which asserts that generic solutions to nonlinear dispersive equations should behave as a sum of decoupled solitons for large times. Such a resolution was obtained for the original Korteweg-de Vries equation (KdV), corresponding to p = 2; and the modied Korteweg-de Vries equations (mKdV), corresponding to p = 3. We refer to [START_REF] Eckhaus | The emergence of solutions for the generalized Korteweg-de Vries equation from arbitrary initial conditions[END_REF][START_REF] Peter | Asymptotic Analysis of Solitons Problems[END_REF][START_REF] Chen | Soliton resolution for the focusing modied KdV equation[END_REF] for instance.

In the context of (gKdV), multi-solitons were rst constructed for (KdV) and (mKdV), via the inverse scattering transform. It provides explicit formulas: for (KdV), it writes (1.3)

u = 6 ∂ 2 ∂x 2 ln det M,
where M (t, x) is the N × N -matrix with generic entry

M (i,j) (t, x) = δ i,j + 2 (c i c j ) 1 4 √ c i + √ c j e 1 2 ( √ ci(x-cit)+xi+ √ cj (x-cj t)+xj )
and δ i,j stands for the Kronecker symbol: see [START_REF] Gardner | Korteweg-de Vries equation and generalizations. VI. Methods for exact solutions[END_REF], [24, section 6], or [START_REF] Hirota | Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[END_REF]. We refer to Schuur [27, chapter 5, (5.5)] and to Lamb [14, chapter 5] for a formula for (mKdV). One can observe from these formulas that such a solution u in (1.3) is a multi-soliton both in ±∞ [START_REF] Miura | The Korteweg-de Vries equation: a survey of results[END_REF], with the same velocity parameters c i in ±∞, but with distinct translation parameters whose shifts can be quantied in terms of the c i .

The construction of multi-solitons was subsequently extended to many non integrable models, and (gKdV) is probably the equation for which their study has been most developed. One important result concerns the complete classication of the multi-solitons, depending on the value of p with respect to 5: recall that for p < 5, (gKdV) is L 2 -subcritical and solitons are stable, p = 5 is the L 2 critical equation, and for p > 5, (gKdV) is L 2 -supercritical and solitons are unstable, linearly and nonlinearly. Let us recall it below. Theorem 1.2 (Martel [16]; Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF]; Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]). Let p > 1 be an integer and let N 1, 0 < c 1 < • • • < c N , and x 1 , . . . , x N ∈ R.

If p 5, there exists T 0 0 and a unique multi-soliton u ∈ C ([T 0 , +∞), H 1 (R)) (in +∞) associated with the R ci,xi , i ∈ {1, . . . , N }.

If p > 5, there exists a one-to-one map Φ from R N to the set of all H 1 -solutions of (gKdV) dened in a neighborhood of +∞ such that u is a multi-soliton in +∞ associated with the R ci,xi if and only if there exist λ ∈ R N and T 0 0 such that u |[T0,+∞) = Φ(λ) |[T0,+∞) .

Moreover, in each case, u belongs to C ([T 0 , +∞), H s (R)) for all s 0, and there exist θ > 0 (depending on the c i but independent of s) and positive constants λ s such that for all s 0, (1.4) ∀t T 0 , u(t) - in the above theorem.

N j=1 R cj ,xj (t)
In this article, we go on studying qualitative properties of the multi-solitons and we are concerned with the behavior in space (at xed time) of the multi-solitons of (gKdV). From formula (1.3), one can show that (KdV) multi-solitons are exponentially localized away from the centers of the involved solitons (as solitons are); this is also the case for (mKdV) multi-solitons. Our goal is to extend this property to multi-solitons of (gKdV), that is, non integrable equations.

Our main result is that there is indeed exponential decay in the solitons region and on the left of the train of solitons, and rapid decay on the right of it. Let us state this more precisely. Theorem 1.4. Fix the parameters 0 < c 1 < • • • < c N and x 1 , . . . , x N ∈ R, and let u be a multi-soliton of (gKdV) associated with the solitons R cj ,xj , as in Theorem 1.2 (u is unique if

p 5).
Let β > c N . Then there exist T 1 > 0 and κ > 0 such that for all s ∈ N,

(1) (Exponential decay in the solitons region and to its left) there exists C s > 0 such that for all t T 1 ,

(1.6)

∀x βt, |∂ s x u(t, x)| C s N j=1
e -κ|x-cj t| ;

(2) (Algebraic decay to the right of the last soliton) for all n ∈ N, there exists C s,n > 0 such that for all t T 1 ,

(1.7)

∀x > βt, |∂ s x u(t, x)| C s,n (x -βt) n .
Remark 1.5. Our bounds give κ = O(1/β) as β → +∞, where the implicit constant depends on the c i ; precise rates are stated in Proposition 2.1 and 2.5. Remark 1.6. Theorem 1.4 can be extended to more general nonlinearities, of the form

∂ t u + ∂ x (∂ xx u + f (u)) = 0 where f : R → R is C ∞ , convex on R + and f (0) = f (0) = 0,
and the velocities c j are such that there is a

C 1 map V j → S (R), c → Q c dened on some neighborhood V j of c j , such that ∂ xx Q c + f (Q c ) = cQ c and (1.8) d dc Q 2 c dx c=cj = 0.
These conditions ensure the existence of multi-solitons as done in [5, Theorem 3] (as noted there, condition (1.8) can probably be avoided).

1.2. (NLS) multi-solitons. We extend the decay properties in Theorem 1.4 to the ddimensional nonlinear Schrödinger equation

i∂ t u + ∆u + |u| p-1 u = 0 (NLS) where (t, x) is taken in R × R d (d 1)
, u is a complex valued function, and the nonlinearity is

H 1 -subcritical, that is 1 < p < 1 + 4 (d-2)+ .
Due to the H1 -subcritical assumption, for all ω > 0, there exists a unique positive radial solution Q ω ∈ H 1 (R d ) to the stationary equation

∆Q ω + Q p ω = ωQ ω . Moreover, if z → |z| p-1 z is C s on C (as an R-dierentiable function), then Q ω is C s+2 on R d
and, for all s ∈ {0, . . . , s + 2}, there exists a constant C s depending on s such that for each multi-index σ ∈ N d with |σ| = s the following exponential decay property holds [1] 1

(1.9)

∀x ∈ R d , |∂ σ Q ω (x)| C s e - √ ω|x| .
We can then ensure the existence of soliton solutions for (NLS): given parameters

ω > 0, γ ∈ R, x 0 ∈ R d , and v ∈ R d , the function R ω,v,γ,x 0 (t, x) = Q ω (x -x 0 -vt)e i 1 2 v•x+ ω- |v| 2 4
t+γ satises (NLS).

Concerning the construction of multi-solitons of (NLS), let us recall Theorem 1.7 (Merle [23]; Martel and Merle [START_REF] Martel | Multisolitary waves for nonlinear Schrödinger equations[END_REF], Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF]; Côte and Friederich [START_REF] Côte | On smoothness and uniqueness of multi-solitons of the non-linear schrödinger equations[END_REF]). Let

1 < p < 1 + 4 (d -2) + . Let N 1 and x for all j ∈ {1, . . . , N } ω j > 0, γ j ∈ R, x 0 j ∈ R d , and v j ∈ R d such that for all j = j , v j = v j .
There exists T 0 0 and a solution u ∈ C ([T 0 , +∞), H s (R)) of (NLS) with s = max(1, p -1 ), and a positive constant λ s such that (1.10)

u(t) - N j=1 R ωj ,vj ,γj ,x 0 j (t) H s λ s e -θt as t → +∞.
Moreover, if p is an odd integer, then u belongs to C ([T 0 , +∞), H s (R)) and (1.10) holds for all s 0.

We now state algebraic decay of the multi-solitons of (NLS) outside the solitons region.

Theorem 1.8. Assume that p is an odd integer such that 1 < p < 1 + 4 (d-2)+ . Let N 1 and x for all j ∈ {1, . . . , N } ω j > 0, γ j ∈ R, x 0 j ∈ R d , and v j ∈ R d such that for all j = j , v j = v j , and let u be a multi-soliton associated with these parameters as in Theorem 1.7.

Let β > max{|v j |, j = 1, . . . , N }. There exists T 1 > 0 such that for all s ∈ N d , for all n ∈ N, there exists C s,n > 0 such that for all t T 1 , (1.11)

∀|x| > βt, |∂ s u(t, x)| C s,n (|x| -βt) n .
Remark 1.9. In the solitons region {x : |x| βt}, one has exponential decay in a similar way as (1.6). Indeed, as a straighforward consequence of (1.10), there exists κ > 0 such that for all s ∈ N d , there exists C s > 0 such that (1.12)

∀|x| βt, |∂ s u(t, x)| N j=1 e -κ|x-vj t| .
Remark 1.10. Theorems 1.7 and 1.8 also apply to nonlinear Schrödinger equations

i∂ t u + ∆u + g(u) = 0,
where g : C → C is a smooth nonlinearity, gauge invariant, that is of the form g(z) = zf (|z| 2 ) and so that:

• there exists p ∈ 1, 1 + 4 (d-2)+ such that for all q p, for all r = 0, . . . , q,

∂ q g ∂ r x ∂ q-r y (z) = O |z| p-q
as |z| → +∞.

• the frequencies ω j are such that there is a C 1 map W j → S (R), ω → Q ω dened on some neighborhood W j of ω j , such that

∆Q ω + g(Q ω ) = ωQ ω ,
We refer to Berestycki and Lions [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] for sucient conditions on g to ensure this condition. • the associated linearized operators around the Q ωj

L ωj : H 1 (R d , C) → H 1 (R d , C) v = v 1 + iv 2 → -∆v + ω j v -(f (Q 2 ωj )v + 2Q 2 ωj f (Q 2 ωj )v 1 )
satisfy suitable coercivity assumptions, as in [4, Hypotheses (H3) and (H4)]. We refer to [4, Proposition 1.7] and reference therein for a sucient conditions in the case when Q ωj is a ground state to satisfy these coercivity conditions. Under these assumptions, the Q ωj are exponentially decaying, along with their derivatives, and there exist T 0 , θ > 0, and a multi-soliton u ∈ C ([T 0 , +∞), H ∞ (R d )) of (NLS) such that for all s 0, there exists C s > 0 such that

(1.13) ∀t T, u(t) - N j=1
R ωj ,vj ,γj ,x 0 j (t)

H s C s e -θt .
1.3. Comments and strategy of the proof. Theorems 1.4 and 1.8 show in particular that for each xed time t T 1 , the multi-soliton u(t) belongs to the Schwartz space S (R). To our knowledge, these are the rst results of quantitative spatial decay in a non integrable setting.

In [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF] and [8, Section 3.1.2, (3.6) and (3.9)], the second author denes non dispersive solution of (gKdV) u at +∞ by the property that for some ρ > 0,

x ρt |u(t, x)| 2 dx → 0 as t → +∞.
(such a notion was rst developed by Martel and Merle [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF][START_REF] Martel | Asymptotic stability of solitons for subcritical gKdV equations[END_REF] in the vicinity of solitons). [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF] showed that non dispersion is a dynamical characterization of multi-solitons: more precisely, a solution of (gKdV) which is non dispersive and remains close to a sum of N decoupled solitary waves for positive times is a multi-soliton in +∞. For (KdV), the result is non perturbative: for any solutions (with suciently smooth initial data), non dispersion is equivalent to being a multi-soliton (for (mKdV), breathers may also occur). The convergence (1.4) shows on the other side that multi-solitons are non dispersive indeed. The decay obtained in Theorem 1.4 provides a quantitative version of this non dispersion. As far as we can tell, the classication of non dispersive solutions of (NLS) is not known.

Throughout the proofs, u is as in the statement of Theorem 1.4 or Theorem 1.8, depending on the equation we are studying, and it will be convenient to denote (1.14)

R j := R cj ,xj , R := N j=1 R j , and z(t) := u(t) -R(t),
in the case of the (gKdV) equation and to use analogous notations when considering (NLS).

To prove Theorem 1.4, we actually split space into three regions: the region to the left of the solitons, that is for x αt for some α < c 1 ; the solitons region αt x βt; and the region to the right of the solitons x βt.

Exponential decay of the multi-solitons of (gKdV) on the left of the solitons (for x αt) follows from revisiting a monotonicity argument set up in [7, section 2] (strengthened from Laurent and Martel [START_REF] Laurent | Smoothness and exponential decay of L 2 -compact solutions of the generalized KdV equations[END_REF]) and originally developed by Martel and Merle [START_REF] Martel | A Liouville theorem for the critical generalized Korteweg-de Vries equation[END_REF]. We take advantage here of the convergence (1.2) and the decay of the solitons, instead of a non dispersion assumption, as it is done in the mentioned references.

In the solitons region αt x βt, estimate (1.6) is a direct consequence of the exponental convergence in (1.4).

The main novelty (and where most of our eorts are focused) concerns the region to the right of the solitons x βt. The monotonicity argument, linked to the dynamic of the ow of (gKdV), does not apply anymore: indeed, it would require some knowledge (non dispersion) at t → -∞ (or at least near the minimal existence time, as multi-solitons might blow up for the L 2 -supercritical (gKdV)). From this perspective, the point of Theorem 1.4 is actually to obtain some information of the behavior of multi-solitons for large decreasing times. Also notice that it would be sucient to prove pointwise decay on the region x βt 0 for one time t 0 , and then this information would easily be propagated for t t 0 . This is in line with general statements linked with persistence of regularity and decay of solutions to (gKdV), like Kato smoothing in [START_REF] Kato | On the Cauchy problem for the (generalized) Korteweg-de Vries equation[END_REF] or Isaza, Linares and Ponce [START_REF] Isaza | On decay properties of solutions of the k-generalized KdV equation[END_REF][START_REF] Isaza | On the propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation[END_REF]. Let us also mention [START_REF] Muñoz | Soliton dynamics for generalized KdV equations in a slowly varying medium[END_REF] 2 , where some polynomial decay was obtained (see Lemma 7.4).

Our strategy in the region x βt is as follows. We consider families of integrals of the form

I ϕ,s,x0 (t) := x βt (∂ s x z) 2 (t, x)ϕ(x -x 0 -βt) dx
where ϕ is a suitable weight function. We show that variations of I ϕ,s,x0 are essentially controlled by the I ϕ ,s ,x0 for s ∈ {0, . . . , s + 1}, under the induction hypothesis of an exponential decay in time. Then, by integrations in t and then in x 0 , together with (1.4) (which provides the base case) and a triangular induction process, we show that we can bound I ϕ,s,0 for ϕ(y) = y n for all n ∈ N.

For Theorem 1.8, we develop a similar analysis using integrals of the form

I ϕ,s,x0 (t) := σ∈N d ,|σ|=s |∂ σ z| 2 (t, x)ϕ(|x| -x 0 -βt) dx.
When the nonlinearity is not smooth but merely C s , the multi-solitons still enjoy polynomial decay for the rst derivatives, as it is clear from the proof (see Propositions 3.3 and 4.2). The interested reader may compute the precise rates.

We expect that (gKdV) multi-solitons decay exponentially on the right as well, that is (1.6) holds without the restriction x βt: this seems a natural conjecture as solitons are exponentially localized on both ends. Still, for the time being, estimate (1.7) is meaningful; and similarly we conjecture that (NLS) multi-solitons decay exponentially fast in space. In any case, we believe that our strategy is robust and extends to prove rapid algebraic decay for the multi-solitons of other non linear dispersive models.

The article is organized as follows. We rst study (gKdV) multi-solitons: in section 2, we consider the left region x αt and the solitons region αt x βt; and in section 3, we focus on the right region x βt. Then, in section 4, we turn to the case of the (NLS) multi-solitons and prove Theorem 1.8. In the appendix, we provide some bound on the H s norm of (gKdV) solitons and multi-solitons, and in particular, track the constant λ s in (1.4).

2. Decay of the (gKdV) multi-solitons on the left 2.1. Decay of the (gKdV) multi-solitons on the left of the rst soliton. The goal of this paragraph is to prove Proposition 2.1 (Exponential decay in large time on the left of the rst soliton). Let 0 < α < c 1 and κ α ∈ 0,

√ α 2
. There exists T 1 T 0 such that for all s ∈ N, there exists C s > 0 such that for all t T 1 , (2.1)

∀x αt, |∂ s x u(t, x)| C s e -κα|x-αt| .
Remark 2.2. Using (1.4), one can easily see that the decay (2.1) implies the one stated in (1.6) in the region x αt, with κ = κ α .

Proof. The proof follows the ideas of [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF] and [START_REF] Laurent | Smoothness and exponential decay of L 2 -compact solutions of the generalized KdV equations[END_REF]. To reach the conclusion, we show the existence of T 1 ∈ R such that for each s ∈ N, there exists

K s > 0 such that, with κ := 2κ α , (2.2)
∀t T 1 ,

x αt

(∂ s x u(t, x)) 2 e κ(αt-x) dx K s .
The rst (and main) step is to obtain (2.2) for s = 0. For this, we claim a strong monotonicity property which is the purpose of Lemma 2.3 and Lemma 2.4 below.

Let us introduce, for some κ > 0 to be determined later, the function ϕ dened by

ϕ(x) = 1 2 - 1 π arctan(e κx ).
It satises the following properties

∃λ 0 > 0, ∀x ∈ R, λ 0 e -κ|x| < -ϕ (x) < 1 λ 0 e -κ|x| , (2.3)
∀x ∈ R, |ϕ (3) (x)| -κ 2 ϕ (x).
(2.4)

∃λ 1 > 0, ∀x 0, λ 1 e -κx ϕ(x).
(2.5)

Moreover, let us observe that

(2.6) x<αt u 2 (t, x)e κ(αt-x) dx = x<0 u 2 (t, x + αt) e -κx dx,
and that, for all x 0 < 0,

x0 x<0 u 2 (t, x + αt) e -κx dx e -κx0 x x0 u 2 (t, x + αt) e -κ(x-x0) dx 1 λ 1 e -κx0 R u 2 (t, x + αt) ϕ(x -x 0 ) dx.
(2.7)

Since κ 2 < α, one can choose δ ∈ (0, α -κ 2 )
. We consider T 1 ∈ R to be determined later. Then, for xed t 0 T 1 and x 0 ∈ R, we dene

I (t0,x0) : [T 1 , +∞) → R + t → R u 2 (t, x + αt)ϕ x -x 0 + δ(t -t 0 ) dx.
We have

(2.8) ∀t T 1 , I (t0,x0) (t) = R u 2 (t, x)ϕ x -x 0 + δ(t -t 0 ) -αt dx,
so that by derivation with respect to t, we obtain (2.9)

dI (t0,x0) dt (t) = -3 R u 2 x (t, x)ϕ (x) dx -(α -δ) R u 2 (t, x)ϕ (x) dx + R u 2 (t, x)ϕ (3) (x) dx + 2p p + 1 R u p+1 (t, x)ϕ (x) dx,
where x := x -x 0 + δ(t -t 0 ) -αt. We then claim Lemma 2.3. There exists C 0 > 0 such that (2.10)

∀x 0 ∈ R, ∀t 0 , t T 1 , dI (t0,x0) dt (t) -C 0 e -κ(-x0+δ(t-t0)) .
Proof. Due to property (2.4) of ϕ, we have

(2.11) R u 2 (t, x)ϕ (3) (x) dx -κ 2 R u 2 (t, x)ϕ (x) dx.
Furthermore we control the nonlinear part by considering

I 1 (t) := |x|>-x0+δ(t-t0) u p+1 (t, x)ϕ (x) dx
and

I 2 (t) := |x| -x0+δ(t-t0) u p+1 (t, x)ϕ (x) dx.
On the one hand, we have due to (

(2.12)

I 1 (t) 1 λ 0 e -κ -x0+δ(t-t0) R |u| p+1 (t, x) dx Ce -κ -x0+δ(t-t0) ,
where we have used the Sobolev embedding H 1 (R) → L p+1 (R) and the fact that u belongs to

L ∞ ([T 1 , +∞), H 1 (R)).
Note that C > 0 is independent of x 0 , t 0 , and t.

On the other, we observe that (2.13)

I 2 (t) u(t) p-1 L ∞ x αt x αt u 2 (t, x)|ϕ (x)| dx √ 2 p-1 u(t) p-1 2 L 2 x αt u x (t) p-1 2 L 2 x αt R u 2 (t, x)|ϕ (x)| dx √ 2 p-1 u(t) p-1 2 L 2 x αt sup t T1 u(t) p-1 2 H 1 R u 2 (t, x)|ϕ (x)| dx.
Since u is a multi-soliton, we can choose T 1 0 such that for all t T 1 ,

(2.14)

√ 2 p-1 u(t) p-1 2 L 2 x αt sup t T1 u(t ) p-1 2 H 1 p + 1 2p (α -δ -κ 2 ).
Let us justify it briey (here lies the main change with respect to previous proofs based on non dispersion [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF] or L 2 -compactness [START_REF] Laurent | Smoothness and exponential decay of L 2 -compact solutions of the generalized KdV equations[END_REF]): we have

x αt u 2 (t, x) dx 2 x αt   u - N j=1 R j   2 (t, x) dx + 2 x αt   N j=1 R j   2 (t, x) dx 2C 2 0 e -2θt + 2N N j=1 x αt R 2 j (t, x) dx
and for all j = 1, . . . , N , since α < c j , we have for t 0:

x αt R 2 j (t, x) dx C x αt e -√ cj |x-cj t-xj | e -√ cj |x-cj t-xj | dx C x αt e -√ cj (cj -α)t e -√ cj |x-cj t-xj | dx Ce -√ cj (cj -α)t R e -√ cj |x-cj t-xj | dx Ce -√ cj (cj -α)t .
where C denotes a positive constant which can change from one line to the other and which only depends on c j (see expression (1.1)). Thus, we can pick up C 0 such that for all t 0,

x αt u 2 (t, x) dx C   e -2θt + N j=1 e -√ cj (cj -α)t   ,
and then T 1 0 satisfying (2.14). Taking into account (2.12), this eventually leads to the following estimate

2p p + 1 R u p+1 (t, x)ϕ (x) dx -(α -δ -κ 2 ) R u 2 (t, x)ϕ (x) dx + C 0 e -κ -x0+δ(t-t0) , (2.15)
where C 0 := 2p p+1 C is independent of x 0 , t 0 , and t. Gathering (2.11) and (2.15) in (2.9), we nally deduce

dI (t0,x0) dt (t) -3 R u 2 x (t, x)ϕ (x) dx -C 0 e -κ -x0+δ(t-t0) .
This establishes Lemma 2.3.

As a consequence of the above lemma, (2.16)

∃C 1 > 0, ∀x 0 ∈ R, ∀t t 0 , I (t0,x0) (t 0 ) I (t0,x0) (t) + C 1 e κx0 ,
with C 1 independent of the parameters x 0 and t 0 . Next, we claim the following:

Lemma 2.4. For xed x 0 ∈ R and t 0 T 1 ,

I (t0,x0) (t) → 0 as t → +∞.
Proof. This lemma is shown by adapting the proof in [15, paragraph 2.1, Step 2] and in [START_REF] Friederich | Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons[END_REF]. Let ε be a positive real number. As in the previous proof, because u is a multi-soliton, we can nd

T 1 T 0 large such that for all t T 1 , x<αt u 2 (t, x) dx ε 2 .
Since 0 ϕ 1, this enables us to see that

x<0 u 2 (t, x + αt) ϕ x -x 0 + δ(t -t 0 ) dx x<αt u 2 (t, x) dx ε 2 .
(2.17)

Now, recall that ϕ is decreasing so that

x 0 u 2 (t, x + αt)ϕ x -x 0 + δ(t -t 0 ) dx ϕ -x 0 + δ(t -t 0 ) u(t) 2 L 2 Cϕ -x 0 + δ(t -t 0 ) , (2.18)
with C = u(t) 2 L 2 for all t ∈ J. Moreover, since ϕ(x) → 0 as x → +∞, there exists T 2 ∈ R such that for all t T 2 ,

Cϕ -x 0 + δ(t -t 0 ) ε 2 .
Then, for all t max{T 1 , T 2 },

I (t0,x0) (t) ε 2 + ε 2 = ε.
Hence, we have nished proving Lemma 2.4.

At this stage, we deduce from (2.16) and Lemma 2.4 that (2.19)

∀t 0 T 1 , ∀x 0 ∈ R, I (t0,x0) (t 0 ) C 1 e κx0 .
Thus, (2.7) leads to

∀t T 1 , x0 x<0 u 2 (t, x + αt) e -κx dx C 1 λ 1 .
Letting x 0 → -∞, we infer that

∀t T 1 , x<0 u 2 (t, x + αt) e -κx dx C 1 λ 1 .
which proves (2.2) with s = 0. Now, to conclude to (2.2) for all s ∈ N, one actually proves by induction on s ∈ N the existence of Ks 0 such that for all t T 1 , (2.20)

R (∂ s x u) 2 (t, x + αt) e -κx dx + t+1 t R (∂ s x u) 2 (τ, x + ατ ) e -κx dx dτ Ks .
For s = 0, this is in fact a consequence of (2.2) and of the following estimate: for all t t 0 T 1 ,

I (t0,x0) (t 0 ) -I (t0,x0) (t) C 1 λ 1 e κx0 + 3 t t0 R u 2 x (τ, x + ατ )ϕ (x -x 0 + δ(τ -t 0 )) dx dτ
(which follows from the proof of Lemma 2.3). Indeed, we notice that by (2.3) and since ϕ is decreasing, for τ ∈ [t 0 , t],

λ 0 e -κ|x-x0| < -ϕ (x -x 0 ) -ϕ (x -x 0 + δ(τ -t 0 ))
so that for t = t 0 + 1 in particular, we have

t0+1 t0 x0<x u 2 x (τ, x + ατ )e -κx dx dτ Ce -κx0 I (t0,x0) (t) -I (t0,x0) (t 0 ) Ce -κx0 I (t0,x0) (t 0 + 1) Ce -κx0 I (t0+1,x0) (t 0 + 1) C.
where the last inequality results from (2.19). Taking the limit when x 0 → -∞, we obtain the desired inequality (2.20).

The rest of the induction argument closely follows [15, paragraph 2.3 and paragraph 2.2

Step 2]. Since it does not depend on the properties of the multi-soliton and for the sake of brevity, we will not detail the proof (2.20) for higher values of s.

2.2. Decay of the (gKdV) multi-solitons in the solitons region. Proposition 2.5 (Exponential decay in the solitons region). Let 0 < α < c 1 and β > c N , and dene (2.21)

κ α,β := min √ c 1 , θ c 1 -α , θ β -c N , min j=1,...,N -1 θ c j+1 -c j > 0.
Then for all s ∈ N, there exists C s > 0 such that for all t T 0 ,

(2.22) ∀x ∈ [αt, βt], |∂ s x u(t, x)| C s N j=1 e -κ α,β |x-cj t| .
Proof. Recall the notation z given in (1.14). For all s ∈ N, for all t T 0 , we have by (1.4) and the Sobolev embedding

H 1 (R) → L ∞ (R), ∂ s x z(t) L ∞ C ∂ s x z(t) H 1 C z(t) H s+1 λ s+1 e -θt .
Fix t T 0 . For all j = 1, . . . , N -1 and c j t x c j+1 t, e -θt e -κ α,β (x-cj t) if and only if

x c j + θ κ α,β t,
which is indeed satised since c j+1 c j + θ κ α,β by the choice of κ α,β (2.21). Similarly, for all αt x c 1 t, we have e -θt e -κ α,β (c1t-x) because c 1 α + θ κ α,β . And for all c N t x βt, we have e -θt e -κ α,β (x-c N t) because β c N + θ κ α,β . Thus, we obtain that for all t T 0 and for all αt x βt,

|∂ s x z(t, x)| ∂ s x z(t) L ∞ λ s+1 e -θt λ s+1 N j=1 e -κ α,β |x-cj t| .
Moreover, for all j = 1, . . . , N , |∂ s x R j (t, x)| C j,s e -√ cj |x-cj t| for some C j,s > 0 depending on j and s. Hence, we conclude to (2.22) by the triangular inequality and the fact that κ α,β √ c 1 .

Decay of the (gKdV) multi-solitons on the right of the last soliton

In this subsection, we analyze the behavior of the multi-solitons on the right of the solitons region. We will focus on the proof of the following Proposition 3.1 (Polynomial decay in large time on the right of the last soliton). Let β > c N .

For all s ∈ N and for all n ∈ N, there exists C s,n > 0 such that for all t T 0 , for all x > βt, (3.1)

(∂ s x u(t, x)) 2 C s,n (x -βt) n .
Notice that this statement will appear as a corollary of a more general result, which has its own interest (see Proposition 3.4 below) and which relies on a triangular induction process.

By Theorem 1.2, there exists θ > 0 such that for all s ∈ N, there exists λ s > 0 such that for all t T 0 , (3.2)

z(t) H s λ s e -θt .
One can estimate the growth of λ s with respect to s, and in fact, we will keep for the sequel (3.3)

λ s C.2 µ s 0 , with µ 0 > max √ p, p+1 2
dened in Claim 3.2. The proof of estimate (3.3) is postponed to paragraph A.2 in the Appendix.

3.1. The key ingredient: stability by integration of a well-chosen set of weight functions. Fix η ∈ (0, c 1 ).

The choice of η is made in order to obtain the interaction estimate of Claim 3.2 below, which roughly expresses that the growth of x → e √ ηx is weaker than the decay of the solitons.

Claim 3.2. For all s ∈ N, there exists C > 0 and µ > √ p such that for all j = 1, . . . , N and for all t T 0 ,

(3.4) R |∂ s x R j (t, x)| e √ η(x-βt) dx C2 µ s .
We will consider weight functions ϕ ∈ C 3 (R, R) which satisfy the following assumptions:

(i) lim x→-∞ ϕ(x) = 0 (A) (ii) ∃κ 1 > 0, ∀x ∈ R, 0 ϕ (x) κ 1 e √ ηx (iii) ∃κ 2 > 0, ∀x ∈ R, |ϕ (3) (x)| κ 2 ϕ (x)
and for given s ∈ N,

∃C(s, ϕ) > 0, ∀s ∈ {0, . . . , s}, ∀t T 0 , R (∂ s x z) 2 (t, x)ϕ(x -βt) dx C(s, ϕ)e -θt .

(B(s))

Let us dene the following property which depends on s and ϕ: P (s, ϕ) : ϕ satises (A) and (B(s)). Note that under assumptions (A) (i) and (ii), ϕ is integrable in the neighborhood of -∞ (see also (3.6) below); we can thus dene on R the antiderivative ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] of ϕ:

ϕ [1] : x → x -∞ ϕ(r) dr.
Next, we state the key ingredient, reecting the triangular way of obtaining Proposition 3.1, Proposition 3.3. For s ∈ N, P (s+1, ϕ) ⇒ P (s, ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] ) and moreover, given µ 0 > max √ p, p+1 2 , there exists a constant c(η, κ 1 , κ 2 ) > 0 (which depends only on η, κ 1 and κ 2 ) such that

(3.5) C(s, ϕ [1] ) c(η, κ 1 , κ 2 )2 µ s 0 C(s + 1, ϕ).
As a corollary of the previous proposition, dening the set

E := ϕ ∈ C 3 (R, R) | ϕ satises (A)
and (B(s)) for all s ∈ N , we immediately obtain step by step, in a triangular way, the following Proposition 3.4 (Stability by integration). If ϕ ∈ E, then ϕ [1] ∈ E. Remark 3.5. Proposition 3.4 is enough to prove that the multi-soliton and its derivatives have polynomial decay (see subsection 3.3). Note that if one could improve (3.3) and (3.5) by proving the existence of C > 0 and c(η, κ 1 , κ 2 ) > 0 such that

∀s ∈ N, λ s C s and ∀s ∈ N, ∀ϕ ∈ E, C(s, ϕ [1] ) c(η, κ 1 , κ 2 )C(s + 1, ϕ),
we would deduce that the multi-soliton and all its derivatives decay exponentially on the domain x > βt (see paragraph A.3 in the Appendix).

Proof of Proposition 3.3.

Proof of Claim 3.2. On the one hand, we have

x βt e -√ cj |x-cj t| e √ η(x-βt) dx

x βt

e -√ cj |x-cj t| dx R e -√ cj |x| dx 2 √ c j .
On the other hand, since β > c N and β

√ η < c j √ c j , x>βt e -√ cj |x-cj t| e √ η(x-βt) dx e (cj √ cj -β √ η)t x>βt e ( √ η-√ cj )x dx x>βt e ( √ η-√ cj )x dx e ( √ η-√ cj )βt √ c j - √ η .
Hence, noticing that we have also

∂ k x R j (t, x)
C j,k e -√ cj |x-cj t| , where C j,k is a constant depending on the parameters of the soliton R j and on k only, Claim 3.2 holds.

Proof of Proposition 3.3. First of all, let us check that the properties gathered in (A) are stable by integration, that is, if we assume that ϕ satises (A), then so does ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] . Assumption (ii) shows that ϕ is integrable in the neighborhood of -∞ and

∀x ∈ R, 0 x -∞ ϕ (r) dr κ 1 x -∞ e √ ηr dr.
By (i), we thus obtain

(3.6) ∀x ∈ R, 0 ϕ(x) κ 1 √ η e √ ηx .
Then (iii) implies that ϕ (3) is integrable in the neighborhood of -∞ and so ϕ admits a limit in -∞, which is necessarily 0 (since ϕ (x) → 0 as x → -∞). Finally, by integration, one obtains

∀x ∈ R, |ϕ (x)| κ 2 ϕ(x).
Hence ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] indeed satises (A). Now take s ∈ N and let us show that ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] veries B(s) if one assumes that ϕ satises B(s + 1).

We dene for all s ∈ N, for all x 0 0 and for all t T 0 :

J s,x0 (t) := R (∂ s x z) 2 (t, x)ϕ(x -x 0 -βt) dx.
For ease of reading, we will denote x = x(t) := x -x 0 -βt (for x 0 > 0 and t T 0 ). We rst show the following induction formula which makes the link between the functions J s,x0 , s ∈ N. Lemma 3.6. For all s ∈ N, there exists C s 0 (independent of x 0 ) such that for all t T 0 :

(3.7) d dt J s,x0 (t) C s R s+1 k=0 ∂ k x z 2 (t, x)ϕ (x) dx + C s e -θt s-1 k=0 J k,x0 (t) + C s e -√ ηx0 e -θt .
In addition, for all µ 1 > max

√ p, p+1 2
, there exists γ 1 > 0 independent of s such that for all s, C s γ 1 2 µ s 1 .

(3.8)

Proof. Let us compute

d dt J s,x0 (t) = -3 R ∂ s+1 x z 2 ϕ (x) dx + R (∂ s x z) 2 ϕ (3) (x) dx -β R (∂ s x z) 2 ϕ (x) dx + 2 R ∂ s x   (z + R) p - N j=1 R p j   (∂ s x zϕ) x dx.
By (A) (iii), we have

(3.9) R (∂ s x z) 2 ϕ (3) (x) dx -β R (∂ s x z) 2 ϕ (x) dx (κ 2 + β) R (∂ s x z) 2 ϕ (x) dx.
We now control the nonlinear term

R ∂ s x (z p ) (∂ s
x zϕ) x dx which does not contain any soliton. If s = 0, we observe that

R z p (zϕ) dx = p p + 1 R z p+1 ϕ (x) dx; thus (3.10) R z p (zϕ) dx p p + 1 z(t) p-1 L ∞ R z 2 ϕ (x) dx C R z 2 ϕ (x) dx.
If s 1, we can write

R ∂ s x (z p ) (∂ s x zϕ) x dx = R ∂ s x (z p ) ∂ s x zϕ (x) dx - R ∂ s-1 x (z p ) ∂ s+2 x zϕ + ∂ s+1 x zϕ dx.
We have

∂ k x (z p ) = i1+•••+ip=k k i 1 , . . . , i p ∂ (i1) x z . . . ∂ (ip) x z so that R ∂ s-1 x (z p ) ∂ s+2 x zϕ dx ∂ s+2 x z(t) L ∞ i1+•••+ip=s-1 R ∂ (i1) x z . . . ∂ (ip) x z ϕ(x) dx C z(t) H s+3 1 p i1+•••+ip=s-1 p k=1 R ∂ (i k ) x z p ϕ(x) dx C z(t) H s+3 z(t) p-2 H s i1+•••+ip=s-1 p k=1 R ∂ (i k ) x z 2 ϕ(x) dx Cp s λ s+3 λ p-2 s e -(p-1)θt s-1 k=0 R ∂ k x z 2 ϕ(x) dx, (3.11)
where we used (3.2). Similarly we obtain (3.12)

R ∂ s-1 x (z p ) ∂ s+1 x zϕ dx Cp s λ s+2 λ p-2 s e -(p-1)θt R s-1 k=0 ∂ k x z 2 ϕ (x) dx and (3.13) R ∂ s x (z p ) ∂ s x zϕ dx Cp s λ s+2 λ p-2 s+1 e -(p-1)θt R s k=0 ∂ k x z 2 ϕ (x) dx.
Hence we can take

C s Cp s λ p-1 s+3 2 µ s 1 for all µ 1 > µ 0 > max √ p, p+1 2
and s large enough.

Moreover,

R ∂ s x   (z + R) p - N j=1 R p j -z p   (∂ s x zϕ) x dx = I 1 + I 2 ,
with

I 1 = R ∂ s x   R p - N j=1 R p j   ∂ s+1 x zϕ + ∂ s x zϕ dx
and

I 2 = p-1 k=1 p k i1+•••+ip=s s i 1 , . . . , i p R ∂ i1 x z . . . ∂ i k x z∂ i k+1 x R . . . ∂ ip x R ∂ s+1 x zϕ + ∂ s x zϕ dx.
We have

|I 1 | R ∂ s+1 x z 2 dx 1 2    R   ∂ s x   R p - N j=1 R p j   ϕ   2 dx    1 2 + R (∂ s x z) 2 dx 1 2    R   ∂ s x   R p - N j=1 R p j   ϕ   2 dx    1 2
C z H s+1 p s+1 2 µ s e -2θt e -√ ηx0 .

Moreover (3.14)

|I 2 | p-1 k=1 p k i1+•••+ip=s s i 1 , . . . , i p z H s+2 z k H s+1 R p-k-1 H s+1 R ∂ ip x Rϕ + ∂ ip x Rϕ dx C2 p p s z k+1 H s+2 R p-k-1 H s+1 p s+1 2 µ s e -2θt e -√ ηx0 ,
where the second inequality is a consequence of Claim 3.2. Indeed, Claim 3.2 rewrites as follows: for all x 0 0,

R ∂ k x R j (t, x) e √ η(x-x0-βt) dx Ce -√ ηx0 .
Thus, by property (A) (ii) satised by ϕ, we infer

R ∂ k x R j (t, x) (ϕ(x) + ϕ (x)) dx Ce -√ ηx0 .
Now, we obtain Lemma 3.6 by gathering the above estimates. We can nd a constant γ 1 independent of s and depending only on η, κ 1 and κ 2 such that for s suciently large,

C s γ 1 2 µ s 1 .
Even if it means taking γ 1 greater, we can assume that the above estimate holds for all s.

Remark 3.7. Let us observe that we could obtain sharper estimates than (3.11), (3.12), (3.13), and (3.14) due to integrations by parts. But this would have only little impact on the growth rate in s at this stage, and in the end, it would not improve (3.5).

Then, we obtain the following control of J s,x0 (t):

Lemma 3.8. For all s ∈ N, there exists a constant K s 1 such that for all t T 0 :

(3.15)

J s,x0 (t) K s +∞ t R s+1 k=0 ∂ k x z 2 (t , x)ϕ (x(t )) dx dt + K s e -√ ηx0 e -θt .
In addition, given

µ 2 > max √ p, p+1 2
, there exists γ 2 > 0 such that for all s,

K s γ 2 2 µ s 2 .
Proof. It follows from (3.7) and an induction argument. Notice that for all s ∈ N, J s,x0 (t) → 0 as t → +∞. Thus, for s = 0, (3.15) follows by integration of (3.7) between t and +∞. Now assume that (3.15) is proved for 0, . . . , s -1 for some particular s 1. Then, by integration of (3.7) between t and +∞ (for t T 0 ), it results:

J s,x0 (t) C s +∞ t R s+1 k=0 ∂ k x z 2 (t , x)ϕ (x) dx dt + C s e -√ ηx0
+∞ t e -θt dt

+ C s s-1 s =0 K s +∞ t e -θt +∞ t R s +1 k=0 ∂ k x z 2 (t , x)ϕ (x) dx dt dt + C s s-1 s =0 K s e -√ ηx0 +∞ t e -θt dt C s +∞ t R s+1 k=0 ∂ k x z 2 (t , x)ϕ (x) dx dt + s-1 s =0 C s K s   +∞ t R s +1 k=0 ∂ k x z 2 (t , x)ϕ (x) dx dt   +∞ t e -θt dt + C s θ max{K s , s = 0, . . . , s -1}e -√ ηx0 e -θt .
Hence there exists K s 1 such that

K s C s + s-1 s =0 C s K s 2C s s-1 s =0 K s (3.16)
and for which

J s,x0 (t) K s +∞ t R s+1 k=0 ∂ k x z 2 (t , x)ϕ (x) dx dt + K s e -√ ηx0 e -θt .
From the inequality (3.16) and an induction argument, we can bound

K s 2C s s-2 s =0 K s + K s-1 2C s (1 + 2C s-1 ) s-2 s =0 K s 2C s (1 + 2C s-1 ) . . . (1 + 2C 1 )K 0 2C s × 4C s-1 × • • • × 4C 1 C 0 2 2s-1 s i=0 C i 2 µ s 2 ,
for all µ 2 > µ 1 and s suciently large (see (3.8) in Lemma 3.6).

Let us now conclude the proof of P (s, ϕ [1] ). We integrate estimate (3.15) provided by Lemma 3.8 on [0, +∞) with respect to x 0 . We obtain by Fubini theorem:

for t T 0 , R ∂ s x z 2 (t, x) +∞ 0 ϕ(x -x 0 -βt) dx 0 dx K s +∞ t R s+1 k=0 ∂ k x z 2 (t , x) +∞ 0 ϕ (x -x 0 -βt ) dx 0 dx dt + K s √ η e -θt
and then by an ane change of variable

R ∂ s x z 2 (t, x)ϕ [1] (x -βt) dx K s +∞ t R s+1 k=0 ∂ k x z 2 (t , x)ϕ(x -βt ) dx dt + K s √ η e -θt .
Considering that ϕ satises (B(s + 1)), this nally shows that

R ∂ s x z 2 (t, x)ϕ [1] (x -βt) dx K s θ s+1 k=0 C(k, ϕ)e -θt + K s √ η e -θt
Hence, ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] satises (B(s)) and one can take

C(s, ϕ [1] ) C(s + 2)K s C(s + 1, ϕ).
Thus we obtain (3.5), which nishes proving Proposition 3.3.

3.3.

Rapid decrease on the right: proof of Proposition 3.1.

Proof. Now, we show polynomial decay of z and its derivatives. This consists in an application of Proposition 3.4 and is the object of Claim 3.9 and Claim 3.10 below. Set η ∈ (0, c 1 ) and introduce the function ϕ : R → R dened by

ϕ(x) := 2 π arctan e √ ηx .
The precise form of ϕ is not that important, but this expression is convenient. Observe that ϕ ∈ E, in view of (3.2) and due to ϕ being bounded. We dene a sequence ϕ [n] n∈N of functions R → R as follows: ϕ [0] := ϕ and for all n ∈ N * , for all x ∈ R,

ϕ [n] (x) := x -∞ ϕ [n-1] (y) dy.
By Proposition 3.4, we have

∀n ∈ N, ϕ [n] ∈ E.
The following claim motivates the introduction of this sequence ϕ [n] n .

Claim 3.9 (Polynomial growth of ϕ [n]

). We have for all n ∈ N (3.17)

∀x 0, 0 ϕ [n] (x) 1 √ η n e √ ηx and (3.18) ∀x 0, 1 2
x n n! ϕ [n] (x) n k=0 1 √ η n-k x k k! .
Proof. We argue by induction on n. Note that ϕ [0] = ϕ is an increasing function and that ∀t 0, arctan t t.

Thus ∀x 0, 0 ϕ [0] (x) 2 π e √ ηx e √ ηx ∀x 0, 1 2 = ϕ(0) ϕ [0] (x) 1.
Now assume that (3.17) and (3.18) hold for some n ∈ N being xed.

By denition of ϕ [n+1] and by the induction assumption, we have for all x 0

0 ϕ [n+1] (x) x -∞ 1 √ η n e √ ηt dt 1 √ η n+1 e √ ηx .
In particular,

0 ϕ [n+1] (0) 1 √ η n+1 .
By the induction assumption, we then infer that for all x 0,

ϕ [n+1] (x) = ϕ [n+1] (0) + x 0 ϕ [n] (t) dt satises 0 + x 0 1 2 t n n! dt ϕ [n+1] (x) 1 √ η n+1 + x 0 n k=0 1 √ η n-k t k k! dt.
Thus for all x 0, 1 2

x n+1 (n + 1)! ϕ [n+1] (x) 1 √ η n+1 + n k=0 1 √ η n-k x k+1 (k + 1)! = n+1 k=0 1 √ η n+1-k x k k! .
This nishes the induction argument, hence the proof of Claim 3.9.

Claim 3.10. For all n ∈ N, for all s ∈ N, there exists K s,n 0 such that for all t T 0 ,

x βt

(∂ s x z) 2 (x -βt) n dx K s,n e -θt .
Proof. We have already observed that ϕ [n] belongs to E. Using Claim 3.9, we deduce that for all s ∈ N,

x βt

(∂ s x z) 2 (x -βt) n dx 2n! R (∂ s x z) 2 ϕ [n] (x -βt) dx K s,n e -θt ,
where

K s,n := 2n!C(s, ϕ [n] ).
Proposition 3.1 follows now from Claim 3.10, from the Sobolev embedding H 1 (R) → L ∞ (R), and the decay of all derivatives of the R j . Remark 3.11. For all s ∈ N, there exists ϕ satisfying (A) and (B(s)) and which grows faster than each polynomial function as x → +∞. This follows from Proposition 3.3:

given μ0 > max √ p, p+1 2
, the sum of the series of functions n 0

x n 2 μ0 s+n ∈ E.

See paragraph A.3 in the Appendix for details.

Rapid decay of the (NLS) multi-solitons

In this section, we turn to the case of the nonlinear Schrödinger equation (NLS) (without computing explicitly the dependence of the constants with respect to the dierential parameter s) and, by adapting the technique exposed in the previous section, we prove Theorem 1.8 (which focuses on the region far away the solitons). We do it for general smooth nonlinearities g as in Remark 1.10.

Concerning the exponential decay of (NLS) multi-solitons inside the solitons region (1.12), the proof is completely similar to that of Proposition 2.5 and is left to the reader.

Let us take β > max j {|v j |} and 0 < η min j {ω j }. Note that with this choice of parameters β and η, the following interaction property (analogous to (3.4)) holds: Claim 4.1. For all s ∈ N d , there exists C(s) > 0 such that for all j = 1, . . . , N and for all t T 0 , (4.1)

R d |∂ s R j (t, x)| e √ η(|x|-βt) dx C(s).
Proof. Let us use the exponential decay of the ground states (1.9) and observe that

|x| βt e -√ ωj |x-vj t-x 0 j | e √ η(|x|-βt) dx |x| βt e -√ ωj |x-vj t-x 0 j | dx R d e -√ ωj |x| dx C(j).
On the other hand, we obtain

|x|>βt e -√ ωj |x-vj t-x 0 j | e √ η(|x|-βt) dx Ce ( √ ωj |vj |-√ ηβ)t |x|>βt e -( √ ωj -√ η)|x| dx Ce ( √ ωj |vj |-√ ηβ)t C d e -( √ ωj -√ η)βt Ce √ ωj (|vj |-β)t .
We recall the notation z := u -N j=1 R j and the decay rate θ > 0 such that

∀t T 0 , z(t) H s C s e -θt .
In the spirit of the previous subsection, we consider weight functions ϕ ∈ C 1 (R, R) such that the following assumptions are satised:

(i) lim x→-∞ ϕ(x) = 0 (A ) (ii) ∃κ 1 > 0, ∀x ∈ R, 0 ϕ (x) κ 1 e √ ηx
and for given s ∈ N,

∃C(s, ϕ) > 0, ∀σ ∈ N d , |σ| s ⇒ ∀t T 0 , R |∂ σ z(t, x)| 2 ϕ(|x| -βt) dx C(s, ϕ)e -θt .

(B (s))

Similarly to the case of the (gKdV) equation, we dene the following property which depends on s and ϕ: P (s, ϕ) : ϕ satises (A ) and (B (s)). We state Proposition 4.2. For any s ∈ N, P (s + 1, ϕ) ⇒ P (s, ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] ).

Proof. The proof follows the same scheme as that of Proposition 3.3. We introduce the family of integrals

J s,x0 (t) := σ∈N d , |σ|=s R d |∂ σ z| 2 (t, x)ϕ(|x| -x 0 -βt) dx,
for all s ∈ N and x 0 > 0.

We then show the following induction formula which makes the link between the functions J s,x0 , s ∈ N. Here again we denote x = x(t) := |x| -x 0 -βt (for x 0 > 0 and t T 0 ). Lemma 4.3. For all s ∈ N, there exists C s 0 (independent of x 0 ) such that for all t T 0 : (4.2)

d dt J s,x0 (t) C s R d |σ| s+1 |∂ σ z| 2 (t, x)ϕ (x) dx + C s e -θt s k=0 J k,x0 (t) + C s e -√ ηx0 e -θt .
Remark 4.4. Notice that estimate (4.2) of d dt J s,x0 obtained in Lemma 4.3 contains one additional term (namely e -θt J s,x0 (t)) with respect to the corresponding estimate (3.7) obtained in the case of the (gKdV) equation. This is due to the algebra linked with the structure of the (NLS) equation; for the gKdV case, we manage to eliminate this term by means of one integration by parts.

Proof. Let us take s ∈ N and σ ∈ N d such that |σ| = s. Denoting e l the d-tuple (0, . . . , 1, . . . , 0) for which all components except the l-th one are zero, we compute:

d dt R d |∂ σ z(t, x)| 2 ϕ(|x| -x 0 -βt) dx =2Re R d ∂ t ∂ σ zϕ(x) dx -β R d |∂ σ z(t, x)| 2 ϕ (x) dx = -2Im R d d l=1 ∂ σ+2e l z∂ σ zϕ(x) dx -β R d |∂ σ z(t, x)| 2 ϕ (x) dx -2Im R d ∂ σ   g(z + R) - N j=1 g(R j )   ∂ σ zϕ(x) dx =2Im d l=1 R d ∂ σ+e l z∂ σ z x l |x| ϕ (x) dx -β R d |∂ σ z(t, x)| 2 ϕ (x) dx -2Im R d ∂ σ   g(z + R) - N j=1 g(R j )   ∂ σ zϕ(x) dx.
Notice that the last line results from one integration by parts.

Let us explain how to estimate the integrals appearing in the last equality. For all l = 1, . . . , d, (4.3)

Im R d ∂ σ+e l z∂ σ z x l |x| ϕ (x) dx 1 2 R d |∂ σ+e l z(t, x)| 2 ϕ (x) dx + 1 2 R d |∂ σ z(t, x)| 2 ϕ (x) dx.
For the last term, let us decompose g(z + R) -N j=1 g(R j ) as follows:

g(z + R) - N j=1 g(R j ) = [g(z + R) -g(z) -g(R)] + [g(z)] +   g(R) - N j=1 g(R j )   .
The control of

R d ∂ σ   g(R) - N j=1 g(R j )   ∂ σ zϕ(x) dx
uses Claim 4.1 (we refer to the similar control of I 1 in the proof of Lemma 3.6). In this way, we obtain (4.4)

R d ∂ σ   g(R) - N j=1 g(R j )   ∂ σ zϕ(x) dx Ce -√ ηx0 e -θt .
By the Faà di Bruno formula, ∂ σ (g(z)) rewrites as a linear combination of the following terms

∂ σ1 z j1 . . . ∂ σq z jq ∂ q g ∂ r x ∂ q-r y (z),
where q ∈ {0, . . . , |σ|}, r ∈ {0, . . . , q}, q m=1 |σ m | = |σ|, j l ∈ {1, 2} for m = 1, . . . , q, and z 1 := Re(z), z 2 := Im(z).

We observe that

∂ σ l z jm ∈ L ∞ (R d ) since z(t) ∈ H ∞ (R d
) and due to the Sobolev embedding

H s (R d ) → L ∞ (R d ) available for all s > d 2 .
Now, by means of

∂ q g ∂ r x ∂ q-r y (z) C|z| p-q if q p and ∂ q g ∂ r x ∂ q-r y (z) C if q > p,
one can bound (4.5)

R d ∂ σ (g(z))∂ σ (z)ϕ(x) dx Ce -θt |σ | s R d |∂ σ z| 2 ϕ(x) dx.
We nally deal with the integral R d ∂ σ (g(z + R) -g(z) -g(R)) ∂ σ (z)ϕ(x) dx. By the Faà di Bruno formula applied to ∂ σ (g(z + R) -g(z) -g(R)), it suces to consider each quantity of the form

∂ σ1 (z+R) j1 . . . ∂ σq (z+R) jq ∂ q g ∂ r x ∂ q-r y (z+R)-∂ σ1 z j1 . . . ∂ σq z jq ∂ q g ∂ r x ∂ q-r y (z)-∂ σ1 R j1 . . . ∂ σq R jq ∂ q g ∂ r x ∂ q-r y (R).
(We keep the same notations for indices and dierential parameters as above.) Now the desired estimation is based on the decomposition

∂ σ1 (z + R) j1 . . . ∂ σq (z + R) jq -∂ σ1 z j1 . . . ∂ σq z jq -∂ σ1 R j1 . . . ∂ σq R jq ∂ q g ∂ r x ∂ q-r y (z + R) +∂ σ1 z j1 . . . ∂ σq z jq ∂ q g ∂ r x ∂ q-r y (z + R) - ∂ q g ∂ r x ∂ q-r y (z) +∂ σ1 R j1 . . . ∂ σq R jq ∂ q g ∂ r x ∂ q-r y (z + R) - ∂ q g ∂ r x ∂ q-r y (R)
and on Claim 4.1. We have

(4.6) R d ∂ σ (g(z + R) -g(z) -g(R)) ∂ σ (z)ϕ(x) dx Ce -2θt e -√ ηx0 .
We obtain the Lemma by gathering the previous estimates (4.3), (4.4), (4.5), and (4.6).

Let us now go on with the proof of Proposition 4.2. By integration of estimate (4.7)

d dt J s,x0 (t) C s R d |σ| s+1 |∂ σ z| 2 (t, x)ϕ (x) dx + C s e -θt s k=0 J k,x0 (t) + C s e -√ ηx0 e -θt
(which directly follows from (4.2) and the denition of J s,x0 ) between t and +∞, the term e -θt sup t t J s,x0 (t ) can be absorbed for large values of t and we are lead to a similar result to that of Lemma 3.8: for all t suciently large, (4.8)

J s,x0 (t) K s +∞ t R s+1 k=0 ∂ k x z 2 (t , x)ϕ (x(t )) dx dt + K s e -√ ηx0 e -θt .
for some constant K s independent of x 0 .

By this means, we can complete the proof of Proposition 4.2, and then the algebraic decay of z, as it was done for the (gKdV) multi-solitons in Section 3.3.

A. Appendix

A.1. Growth of the H s norms of the 1-D solitons. The purpose of this appendix is to make the constants more explicit as stated in Section 3. We restrict to the space dimension 1 and monomial nonlinearity.

Proposition A.1. For all µ > √ p, there exists s 0 such that for all s s 0 , (A.1)

Q H s 2 µ s .
Proof. Dierentiating the fundamental equation satised by Q, that is Q + Q p = Q, we obtain the following induction formula:

∀s ∈ N, Q (s+2) = Q (s) - i1+•••+ip=s s i 1 , . . . , i p Q (i1) . . . Q (ip) .
Let us observe that for i 1 , . . . , i p ∈ N such that i

1 + • • • + i p = s,
• if there exists j ∈ {1, . . . , p} such that i

j = s, then R Q (i1) . . . Q (ip) 2 dx = R Q (s) 2 Q 2(p-1) dx Q 2(p-1) L ∞ R Q (s) 2 dx C Q 2(p-1) H 1 Q 2 H s (C being a constant depending only on p); • if for all j ∈ {1, . . . , p} such that i j s -1, then R Q (i1) . . . Q (ip) 2 dx p k=2 Q (i k ) 2 L ∞ R Q (i1) 2 dx C p k=2 Q 2 H i k +1 Q 2 H i 1 C Q 2(p-1) H s Q 2 H s-1 ,
(C being again a constant depending only on p, which can change from one line to the other).

Thus for s ∈ N * ,

Q (s+2) L 2 C Q (s) L 2 + p s Q p H s ,
which implies

Q H s+2 Cp s Q p H s ,
with a constant C depending only on p.

We nally obtain

Q H s C p s 2 -1 p-1 p p s 2 s Q H 1 2 µ s ,
with µ > √ p and s suciently large.

Remark A.2. As a corollary of Proposition A.1, we also obtain the existence of a constant C depending on p and the soliton parameters such that for all s ∈ N,

(A.2)   N j=1 R j (t)   p - N j=1 R j (t) p H s Cp s max j=1,...,N R j p H s e -2θt .
A.2. Proof of estimate (3.3). The goal is here to make explicit the constants appearing in the computations done by Martel [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]Section 3.4] in the proof of smoothness of the multi-solitons. We thus repeat the arguments developed by Martel (presented slightly dierently), keeping track of the growth of the constant λ s (with respect to s) such that

∀t T 0 , z(t) H s λ s e -θt .
Proof. We consider regularity indices s 5, as that case makes the argument easier: the point being that the exponential decay rate θ does not change when we go from the estimation of z H s-1 to that of z H s ; there is a loss for s = 2 (treated in detail in [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]), which can be avoided for s = 3, 4 using an extra argument, see the footnote below in the proof.

The starting point is to study the variations of

d dt R (∂ s x z) 2 dx = 2 R ∂ s x   (z + R) p - N j=1 R p j   ∂ s+1 x z dx.
Thus, the terms which have to be controlled are the source term (involving z only linearly)

2 R ∂ s+1 x   R p - N j=1 R p j   ∂ s x z dx. and -2 R ∂ s+1 x p k=1 p k z k R p-k ∂ s x z dx = -2 p k=1 p k i1+•••+ip=s+1 s + 1 i 1 , . . . , i p I k,i1,...,ip ,
where

I k,i1,...,ip = R ∂ i1 x z . . . ∂ i k x z∂ i k+1 x R . . . ∂ ip x R∂ s x z dx.
(i) For k = 1, integrating by parts, we have

i1+•••+ip=s+1 s + 1 i 1 , . . . , i p R ∂ i1 x z∂ i2 x R . . . ∂ ip x R∂ s x z dx = R ∂ s+1 x zR p-1 ∂ s x z dx + (s + 1)(p -1) R ∂ s x z∂ x RR p-2 ∂ s x z dx + i1+•••+ip=s+1 ∀j,i j s-1 R ∂ i1 x z∂ i2 x R . . . ∂ ip x R∂ s x z dx = 2s + 1 2 R (∂ s x z) 2 ∂ x (R p-1 ) dx + i1+•••+ip=s+1 ∀j,i j s-1 R ∂ i1 x z∂ i2 x R . . . ∂ ip x R∂ s x z dx.
(A.

3)

The rst integral (A.4)

2s + 1 2 R (∂ s x z) 2 ∂ x (R p-1 ) dx
can not be bounded directly in a suitable way. The key idea in [START_REF] Martel | Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] is to add a lower order term

2s + 1 3 p R ∂ s-1 x z 2 R p-1 dx
whose variation at leading order will precisely cancel (A.4). Thus we are lead to consider

F s (t) := R (∂ s x z) 2 dx - 2s + 1 3 p R ∂ s-1 x z 2 R p-1 dx.
Going back to (A.3), we bound (A.5) (ii) Let us now consider the case where 2 k p. If there exists j ∈ {1, . . . , k} such that i j = s + 1, then integrating by parts,

|I k,i1,...,ip | = 1 2 R (∂ s x z) 2 ∂ x z k-1 R p-k dx 1 2 ∂ x R p-k z k-1 L ∞ z 2 H s 1 2 R p-k H 2 z k-1 H 2 z 2 H s .
(A.6)

If there exists j ∈ {1, . . . , k} such that i j = s, then (A.7)

|I k,i1,...,ip | C z k-1 H 2 R p-k H 2 R (∂ s x z) 2 dx.
If there exists j ∈ {1, . . . , k} such that i j = s -1, then for all j ∈ {1, . . . , k} such that j = j, i j p l=1 i l -s + 1 2 s -3 (by the choice of s 5) 3 . Hence, integrating by parts, 3 If s = 3 or 4, a term which is cubic in ∂ s-1

x z, of the type

(∂ s-1
x z) 3 P (R j , z, ∂xz) dx can occur, where P is some function (but there are no terms with higher power of ∂ s-1 x z). Via the Gagliardo-Nirenberg inequality, it can be bounded by (A.9) (iii) For the source term, by integration by parts, Gathering the bounds (A.5), (A.6), (A.7), (A.8), (A.9), it results that the point being that the decay rate in θ is greater than 2: one can then complete the estimates as written here for s 5.

and thus

z 2 H s
Cp s 2 µ s p λ p+1 s-1 e -2θt .

We can therefore take (A.11) 2 e (x-βt) dx is nite for some > 0, it suces by (3.17) that

λ s Cp
+∞ n=0 R (∂ s x z) 2 n ϕ [n] dx
is nite. Thus it suces that the series n 0 C(s, ϕ [n] ) n converges. This condition is satised under the following assumptions:

(A.13) C(s, ϕ [START_REF] Berestycki | Non linear scalar eld equations, I. Existence of a ground state[END_REF] ) c 0 C(s + 1, ϕ) and λ s c0 s .

Indeed, if we assume (A.13), we obtain

C(s, ϕ [n] ) c n 0 c0 2(s+n) c0 2s (c 0 c0 2 ) n ,
which guarantees the existence of > 0 such that the series n 0 C(s, ϕ [n] ) n converges.

From Proposition 3.3, we also deduce, proceeding step by step, that:

C(s, ϕ [n] ) c(η, κ 1 , κ 2 )2 µ s 0 C(s + 1, ϕ) c(η, κ 1 , κ 2 ) n 2 µ s 0 +•••+µ s+n-1 0 C(s + n, ϕ) c(η, κ 1 , κ 2 ) n 2 nµ s+n-1 0 λ 2 s+n .
Taking μ0 > µ 0 , this shows that the integral

- 1 {c j+1 -c j } 3 2

 12

2 R

 2 Cλ s-1 e -θt R pp-1 dx, as mentioned (and by construction), one term cancels (A.4) and the others are bounded as in (i) and (ii).

A. 3 .

 3 Details concerning Remarks 3.5 and 3.11. Let s ∈ N. In order to ensure that x>βt (∂ s x z)

  P (R j , z, ∂xz) L ∞ C(1 + z H 2 ) p-2 e -5θ/2t z (A.8)where C is a universal constant depending only on p.In the other cases, i 1 , . . . , i k s -2 and so |I k,i1,...,ip | -

	|I k,i1,...,ip | = -	1 2 R	∂ s-1 x	z	2 ∂ x ∂ x z . . . ∂ i 1 x i k-1	z∂ i k+1 x	R . . . ∂ ip x R dx
			1 2	z 2 H s-1 ∂ x ∂ x z . . . ∂ i 1	i k-1 x	z∂ i k+1 x	R . . . ∂ ip x R	L ∞
			C z k+1 H s-1 R p-k H s+1 ,
			∂ s-1 x	z∂ x ∂ i1 x z . . . ∂ i k x z∂ i k+1 x	R . . . ∂ ip x R dx	z k+1 H s-1 R p-k H s-1 .
			R
	z	1/6 H s z	5/6 H s-1	3			1/2 H s ,

  Cλ s-1 e -θt R p -Cλ s-1 e -θt p s+2 2 µ s+2 p e -2θt + C z 2 H s e -2θt p s+1 2 p 2 µ s (p-2) + Ce -3θt p s+1 2 p 2 µ s (p-2) λ p+1 s-1 C p s 2 µ s p e -2θt z 2 H s + p s 2 µ s p λ p+1 s-1 e -3θt . |F s | + Csλ 2 s-1 e -2θt . Cp s 2 µ s p e -2θt |F s (t)| + sλ 2 s-1 e -2θt + Cp s 2 µ s p λ p+1 s-1 e -3θt .By integration we nally obtain|F s (t)| Cp s 2 µ s p λ p+1 s-1 e -2θt

	d dt	F s (t)					N j=1	R p j	H s+1	+ C	p k=2	p k	z 2 H s p s+1 z k-1 H 2 R p-k H s
			+ C	p k=2	p k	p s+1 z k+1 H s-1 R p-k H s
	(A.10)
	Moreover, we have by denition of F s ,
			(∂ s x z)	2 dx |F s | + Cs R p-1 H 2 ∂ s-1 x	z 2 L 2
		R
	Hence,
		d dt	F s (t)

We denote |x| the canonical euclidian norm of x ∈ R d (or the modulus in C), and |σ| the length of the multi-index σ ∈ N d ; the context makes it unambiguous.

We thank Y. Martel for pointing to us this reference, upon completion of this work.

Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg, Strasbourg, France Email address: cote@math.unistra.fr Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg, Strasbourg, FranceEmail address: friederich@math.unistra.fr