Refined asymptotics around solitons for gKdV equations

We consider the generalized Korteweg-de Vries equation $$ \partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}$$ with general $C^2$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. Stability theory for $Q_c$ is well-known. In previous works, we have proved that for $f(u)=u^p$, $p=2,3,4$, the family of solitons is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\geq 0$), then $u(t,.+\rho(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+\sim c$. Then, the asymptotic stability result could be extended to the case of general assumptions on $f$ and $Q_c$. The objective of this paper is twofold. The main objective is to prove that in the case $f(u)=u^p$, $p=2,3,4$, $\rho(t)-c_+ t$ has limit as $t\to +\infty$ under the additional assumption $x_+ u\in L^2$. The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity $f(u)$, when the ratio of the speeds of the solitons is small. The motivation is to accompany forthcoming works devoted to the collision of two solitons in the nonintegrable case. The arguments are refinements of previous works specialized to the case $u(t)\sim Q_{c_1}+Q_{c_2}$, for $0

with general C 2 nonlinearity f . Under an explicit condition on f and c > 0, there exists a solution in the energy space H 1 of (0.1) of the type u(t, x) = Q c (x − x 0 − ct), called soliton. Stability theory for Q c is well-known.
In [11], [14], we have proved that for f (u) = u p , p = 2, 3, 4, the family of solitons is asymptotically stable in some local sense in H 1 , i.e. if u(t) is close to Q c (for all t ≥ 0), then u(t, . + ρ(t)) locally converges in the energy space to some Q c+ as t → +∞, for some c + ∼ c. The main improvement in [14] is a direct proof, based on a localized Viriel identity on the solution u(t). As a consequence, we have obtained an integral estimate on u(t, . + ρ(t)) − Q c+ as t → +∞.
In [9] and [15], using the indirect approach of [11], we could extend the asymptotic stability result under general assumptions on f and Q c . However, without Viriel argument directly on the solution u(t), no integral estimate is available in that case.
The objective of this paper is twofold. The main objective is to prove that in the case f (u) = u p , p = 2, 3, 4, ρ(t) − c + t has limit as t → +∞ under the additional assumption x + u ∈ L 2 (R), which is consistent with a counterexample in [14]. This result persists for general nonlinearity if a Virial type estimate is assumed. The main motivation for this type of result is the determination of explicit shifts due to collision of two solitons in the nonintegrable case p = 4, see [16].
The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity f (u), when the ratio of the speeds of the solitons is small. The motivation is to accompany the two papers [16], [17], devoted to collisions of two solitons in the nonintegrable case. The arguments are refinements of [22], [18] specialized to the case u(t) ∼ Q c1 + Q c2 , for 0 < c 2 ≪ c 1 .

Introduction
We consider the generalized Korteweg-de Vries (gKdV) equations: for u(0) = u 0 ∈ H 1 (R), with a general C 2 nonlinearity f . We assume that for p = 2, 3 or 4, f (u) = u p + f 1 (u) where lim Denote F (s) = s 0 f (s ′ )ds ′ . Note that for (1.1), one can solve the Cauchy locally in time in H 1 , using the arguments of Kenig, Ponce and Vega [5] (using the norms given for f (u) = u 2 in H 1 ). Moreover, the following conservation laws holds for H 1 solutions: Recall that if Q c is a solution of is solution of (1.1). We call soliton such nontrivial traveling wave solution of (1.1). By well-known results on equation (1.3) (see [1], [15]), there exists c * (f ) > 0 such that c * (f ) = sup{c > 0 such that ∀c ′ ∈ (0, c), ∃ Q c ′ positive solution of (1.3)}.
Note that for f (u) = u p , c * (u p ) = +∞ and for all c > 0, Q c (x) = c

3) exists then
Q c is the unique (up to translation) positive solution of (1.3) and can be chosen even on R and decreasing on R + . From Weinstein [22], the soliton Q c is orbitally stable if Concerning asymptotic stability, we have proved in [15] the following general result Asymptotic stability ( [15], [14], [11]) Assume that f is C 2 and satisfies (1.2). Let 0 < c 0 < c * (f ). There exists α 0 > 0 such that if u(t) is a global (t ≥ 0) H 1 solution of (1.1) satisfying

5)
then the following hold.
Recall that the main improvement in [14] with respect to [11] is a direct proof, based on a localized Viriel etimate on the solution u(t), see Claim B.2 in the present paper for a similar result. As a consequence, we have obtained the following estimate: there exists K > 0 such that (1.7) A Viriel identity was proved in the case f (u) = u p for p = 2, 3, 4 see Proposition 6 in [11], and used under a localized form in [14] (see also the case p = 5 in [10]). In contrast, the proof of asymptotic stability in [9], [15] is for general f (u), but it is indirect, following the original approach of [11]. Thus, in this case, it is not known whether (1.7) holds. See further comments on this result in [15].

Refined asymptotics for power nonlinearities
Our main objective in this paper is to refine the convergence result for the power case, i.e. f (u) = u p with p = 2, 3 and 4 concerning the behavior of ρ(t) as t → +∞. In fact, the only requirement is that (1.7) holds, in particular, a virial type estimate around Q c is sufficient. A typical result in this direction is the following. where c(t) and ρ(t) are defined as in (1.6).
Recall that Pego and Weinstein [21] and Mizumachi [20] also obtained results of asymptotic stability in weighted spaces, where convergence of ρ(t) − t is proved. The results [21] and [20] depend on a spectral assumption which is proved only for p = 2, 3. Moreover, in [21], the initial data has to belong to an exponential weigthed space. This condition has been relaxed in [20], where the assumption is x>0 x 11 u 2 < +∞.
We point out two main motivations for this kind of results: • First, in [14], we gave the following counterexample: For any α > 0, there exists an H 1 solution u(t) of the KdV equation i.e. (1.1) with f (u) = u 2 , such that sup t∈R u(t) − Q(x − ρ(t)) H 1 ≤ α, and for some κ > 0, ρ(t), The initial data used in this construction contains a series of small solitons, which converges in H 1 but not in L 2 (x 2 dx). This proves that the convergence of ρ(t) − t as t → +∞ is not true in general and requires some additional decay on the solution. In this respect, the assumption x 2 u 2 < +∞ in Theorem 1 seems rather weak and improve the results in [21] and [20]. When looking for an L 2 condition independent of p for (1.1), we think that x 2 u 2 is optimal. Indeed, it seems that the relevant quantity is the L 1 norm, see [16], where it proved that during the collision of two solitons, the shifts on the trajectories are related to L 1 norms.
Thus, the natural question left open by Theorem 1 is whether assuming x>0 |u| < +∞ is sufficient to obtain convergence of ρ(t) − c + t. This might require a much more refined analysis.
• A second motivation for estimating ρ(t) − c + t as t → ∞ appears in the context of two soliton collisions. In a paper [16] concerning the collision of two solitons of different sizes for the gKdV equation (1.1) with f (u) = u 4 , we were able to compute the shift on the trajectories of the solitons resulting from their collision. The explicit shift is obtained for a fixed time T , long after the collision. The proof of Theorem 1 allows us to quantify the variation of this shift due to long time i.e. for t > T .
We also point out that the proof of Theorem 1 relies on several refinements of the proof of the asymptotic stability in [11], [14] and is independent from the methods in [21], [20]. Let and (1.7) do not give any conclusion We proceed as follows. First, we improve the monotonicity arguments on u(t) used in [11], [18] and [14]. The improvement is to prove monotonicity results on η(t), which are much more precise (see Claim B.3). This argument allows us to prove that (1.7) implies +∞ 0 |c(t) − c + |dt < +∞. Second, the control of +∞ 0 (ρ ′ (t) − c(t))dt is obtained through the equation of η(t), by noting that at the first order ρ ′ (t) − c(t) is the derivative of some bounded function of t. Note that we do not prove +∞ 0 |ρ ′ (t) − c(t)|dt < +∞.

Large time behavior in the two soliton case
A second objective of this paper is to provide asymptotic analysis in large time related to two soliton solutions of (1.1).
From [8] (see also [18]), there exist solutions u(t, x) of (1.1) which are asymptotic Nsoliton solutions at t → −∞ in the following sense: let N ≥ 1, c 1 > . . . > c N > 0, and (1.10) The behavior displayed by these solutions is stable in some sense. Considering for example the case of two solitons, there exist a large class of solutions such that as t ∼ −∞, where c 1 > c 2 and η(t) is a dispersion term small in the energy space H 1 with respect to Q c 1 , Q c 2 (see [18]). From the Physics point of view, the two solitons Q c 1 and Q c 2 have collide at some time t 0 . In the special case c 2 ≪ c 1 (or equivalently, Q c 2 H 1 ≪ Q c 1 H 1 ) and η(t) H 1 ≪ Q c 2 H 1 , for t close to −∞, we have introduced in [16] explicit computations allowing to understand the collision at the main orders, using a new nonlinear "basis" to write and compute an approximate solution v(t, x) up to any order of size.
Recall that the problem of collision of two solitons is a classical question in nonlinear wave propagation. In the so-called integrable cases (i.e. f (u) = u 2 and f (u) = u 3 ) it is wellknown that there exist explicit multi-soliton solutions, describing the elastic collision of several solitons (see Hirota [3], Lax [6] and the review paper Miura [19]). Note that in experiments, or numerically for more accurate nonintegrable models (see Craig et al. [2], Li and Sattinger [7] and other references in [16]), this remarkable property is mainly preserved, i.e. the collision of two solitons is almost elastic, however, a (very small) residual part is observed after the collision. Equation (1.1) being not integrable (unless f (u) = u 2 and f (u) = u 3 ), explicit Nsoliton solutions are not available in this case. The results obtained in [16] and [17], using the present paper, are the first rigorous results concerning inelastic (but almost elastic) collision in a nonintegrable situation. We refer to the introduction and the references in [16] for an overview on these questions.
In [16] and [17], the approximate solution is adapted to treat a large but fixed time interval around the collision region, but not the large time asymptotics (see for example Proposition 3.1 in [16]). In Proposition 2 below, we give stability and asymptotic stability results required to control the asymptotics in large time of the solutions constructed in [16] and [17]. In particular, we need a sharp stability result in the case where one soliton is small with respect of the other. We claim the following.
Proposition 2 Assume that f is C 2 and satisfies (1.2) for p = 2, 3 or 4. Let 0 < c 2 < c 1 < c * be such that (1.4) holds for c 1 , c 2 . Let 0 < c 1 < c * (f ) be such that (1.4) holds. There exist c 0 (c 1 ) and K 0 (c 1 ) > 0, continuous in c 1 such that for any 0 < c 2 < c 0 (c 1 ) and for any ω > 0, (1.12) (1.14) 2. Asymptotic stability. There exist c + 1 , c + 2 > 0 such that lim (1. 16) Remark. The time T c 1 ,c 2 corresponds to a time long after the collision of the two solitons (see [16]). In (1.12), since X 0 > 1 2 T c 1 ,c 2 , the two solitons are decoupled for any t ≥ t 1 . Proposition 2 follows directly from known arguments in Weinstein [22], and in [18], [15]. The only new point is the fact that one soliton is small with respect to the other. However, these statements are essential in [16] and [17]. In those works, the point is to show that even in a nonintegrable situation, the two soliton structure is preserved by collision. The method in [16], [17] concerns the collision problem in [−T c 1 ,c 2 , T c 1 ,c 2 ]. To obtain global in time results, it is essential to prove the global in time stability of the two soliton structure after the collision, i.e. for t > T c 1 ,c 2 , which is provided by Proposition 2. Proposition 2 is directly applied in [17]. A slightly more precise stability result is required in [16], see Proposition 4 in Section 2. Now, we claim an extension of Theorem 1 to the case of two solitons, with a qualitative control on lim t→+∞ ρ(t) − c + t.
Theorem 3 Under the assumptions of Proposition 2, assume further that f (u) = u p , for p = 2, 3 or 4 and x>0 x 2 u 2 (0, x)dx < +∞. Then, there exist x + 1 and x + 2 such that In the case p = 4, if in addition, for some κ > 0, α < κc  The main motivation of Theorem 3 is the following: in [16], in the same context as before, we were able to compute the main order of the shift on the trajectories of the solitons due to the collision at time t = T c 1 ,c 2 . Theorem 3 proves that the shifts do change at the main order in large time (for example, at the main order, the shift of Q 2 is a nonzero constant independent of c 2 , so that it is preserved by (1.19)). See proof of Theorem 1.2 in [16] for details.
The plan of the paper is as follows. In Section 2, we prove the stability part of Proposition 2. We focus on the case f (u) = u p for simplicity, the proof in the general case being exactly the same (see [15] and [18]). Moreover, by a scaling argument we consider only the case c 1 = 1, c 2 = c, where c is small enough.
In Section 3, we prove Theorem 3. First, we use the methods of localized Viriel estimates as in [14] to obtain the equivalent of (1.7) for two solitons. Next, we prove (1.17) and (1.19). The proof of Theorem 1 follows directly from the arguments of Section 3, thus it will be omitted.

Stability for large time of 2-soliton like solutions
Recall that we restrict ourselves to the case f (u) = u p (p = 2, 3, 4) and c 1 = 1, c 2 = c small enough. Let , which corresponds to the natural norm to study the stability of Q c .
Let u(t) be an H 1 solution of (1.1) such that for some t 1 ∈ R and X 0 ≥ T c /2, Then there exist Proposition 2 follows immediately from Proposition 4 with α = c ω (ω > 0) and a scaling argument.
Remark. Note that the proof of Proposition 4 does not need any new arguments with respect to [18]. We only need to check that the argument of [18] still applies to the situation where one soliton is small with respect to the other. Since Q c L 2 = c q Q L 2 (see Claim A.2), the assumption (2.1) does not seem optimal by a factor √ c. This is due to the fact that the appropriate norm for the stability of Q c is . H 1 c . Proof of Proposition 4. By time translation invariance, we may assume that t 1 = 0. Let X 0 ≥ T c /2 be such that Let D 0 > 2 to be chosen later, r = 1 400 and Observe that t * > 0 is well-defined since D 0 > 2, (2.4) and the continuity of t → u(t) in H 1 . The objective is to prove t * = +∞. For the sake of contradiction, we assume that t * is finite. First, we decompose the solution on [0, t * ] using modulation theory around the sum of two solitons (see proof of Claim 2.1 in Appendix A.1).

Claim 2.1 (Decomposition of the solution)
For α > 0, c > 0 small enough, independent of t * , there exist C 1 functions ρ 1 (t), ρ 2 (t), c 1 (t), c 2 (t), defined on [0, t * ], such that the function η(t) defined by (2.8) We define For m(t) = 1 2 (ρ 1 (t) + ρ 2 (t)), we set Note that I(t) corresponds at the main order to the L 2 norm of the solution u(t) at the right of the slow soliton R 2 (t), and the functional g(t) corresponds locally to the norm adapted to each soliton. In particular, we have g(t) ≤ η(t) H 1 c . We expand u(t) = R 1 (t) + R 2 (t) + η(t) in the three quantities u 2 (t), I(t) and E(u(t)).

Lemma 2.1 (Expansion of energy type quantities)
(2.14) Lemma 2.1 is proved in Appendix A.2. In the rest of this section, we assume α and c small enough so that .
We next obtain a contradiction from the following lemma.
If t * < +∞, then Lemma 2.2 and the continuity in H 1 of u(t) contradict the definition of t * . Therefore, we only have to prove Lemma 2.2.
Proof of Lemma 2.2.
In the nonlinear term u p+1 ψ ′ (x−m(t)), we expand u(t) = R 1 (t) + R 2 (t) + η(t). We obtain for α and c small enough, since η p−1 L ∞ ≤ K η p−1 H 1 ≤ Kc and 0 < R p−1 2 ≤ Kc. Moveover, by calculations similar to the ones of Claim A.3 and u L ∞ ≤ K, we have Step 2. Estimates on the scaling parameters. Let Proof of Claim 2.3. Since there are only two solitons, the proof follows only from the L 2 norm and the energy conservation, i.e. (2.11), (2.13) and (2.14). (When there are more than three solitons, the use of quantities such as I(t) is also needed, see [18].) Let t ∈ [0, t * ]. From (2.11) taken at time 0 and t, and u 2 (t) = u 2 (0), we have Then, from (2.14), we obtain |c 2q+1 Multiplying (2.19) by c 2 (0) and combining with (2.20), from (2.15), we obtain By (2.15), we obtain Using this estimate in (2.20), we obtain similarly Therefore, for ∆ 1 (t), ∆ 2 (t) small enough (by (2.6)), we obtain Step 3. Main argument of the proof of stability. For t ∈ [0, t * ], as in [18] we set The functional F coincides in a neighborhood of R 1 (respectively, R 2 ) with the functional introduced by Weinstein in [22] to prove the stability of R 1 (resp., R 2 ). We claim the following result on the quadratic part (in η) of F(u(t)).

Claim 2.4 Let
There On the one hand, using (2.11)-(2.14), we obtain the following estimate On the other hand, by conservation of E(u(t)) and u 2 (t), and by the monotonicity of (2.25) For α and c small enough, g(t), ∆ 1 (t), ∆ 2 (t) and D 0 (α + exp(− 1 2 c −r )) are small, and from (2.8), we obtain the following.
Therefore, we obtain and then choose α > 0, c > 0 small enough, so that all the previous estimates hold. Then, for all t ∈ [0, t * ], we have

Refined asymptotics for the 2-soliton structure
We claim the following.
Proposition 5 (Asymptotic stability) There exist K > 0, α 0 > 0, c 0 > 0 such that for any 0 < c < c 0 , 0 < α < α 0 the following if true. Let u(t) be an H 1 solution of (1.1) such that for 1 2 so that Proposition 4 applies with ρ 1 (t), ρ 2 (t). Then In the case p = 4, if in addition, for some κ > 0, α < κc 1 3 and x> 11 12 | ln c| Remark. To obtain the convergence of the translation parameters, one has to add an extra assumption on the initial data such as (3.5). Indeed, in the energy space, one can construct an explicit example where convergence does not hold (see [14]). Condition (3.5) is enough for our purposes and could be relaxed, and adapted for the cases p = 2, 3.
In what follows, we concentrate on the case f (u) = u p for p = 2, 3 or 4. The proof of the asymptotic stability (part 1 of Proposition 5) in the case of a general nonlinearity f (u) follows from [18] and [15]. Note that estimate (3.3) is a direct consequence of (2.28).
In the proof of Proposition 5, we need another proof of the asymptotic stability for f (u) = u p , for p = 2, 3 or 4, which is derived from the direct arguments of [14]. The interest of this direct approach is to obtain an estimate on the convergence (see Lemma 3.1), which is fundamental in proving the convergence of the translation parameters. For a general nonlinearity, this kind of property is open.
Proof of Proposition 5.
1. The argument presented now is very similar to [14], proof of Theorem 1, Step 3. We keep the notation of the proof of Proposition 4, in particular, the decomposition of u(t) introduced in Claim 2.1 and the conclusion of Claim 2.5. Now, we prove that c 1 (t) and c 2 (t) converge as t → +∞, and that η(t) converges to 0 in H 1 (x > ct/10) as t → +∞.
We first control η(t) around the solitons.

Lemma 3.1 (Asymptotic stability locally in space)
Let Then, Proof of Lemma 3.1. The proof is based on a localized Viriel type estimate. Consider Φ : R → R be an even smooth function such that Claim 3.1 (Viriel estimate) There exist A ≥ 5, K > 0, α 0 > 0 and c 0 > 0 such that for 0 < α < α 0 , 0 < c < c 0 and for all t ∈ [0, +∞), See the proof in Appendix B.1. Note that this result is very similar to Lemma 2 in [14]. In [14], the identity was established in the case of one soliton. Here, to treat the two soliton situation, we have to use an additional term in K 1 (t), K 2 (t).
From now on, we fix A ≥ 5 such that Claim 3.1 holds. Then Therefore, integrating (3.7) on [0, +∞) and using T c = c − 1 Thus Lemma 3.1 is proved. Now, we control the scaling parameters. Estimate (B.4) and Lemma 3.1 imply that c 1 (t) and c 2 (t) have limits as t → +∞, which we denote respectively by c + 1 and c + 2 . By the stability result (2.28), Now, we extend the convergence of η to 0 in a large region in space, following the proof of Theorem 1 in [14]. We give a sketch the proof (see [14], proof of Theorem 1, Step 3, for more details).
2. Now, we prove the second part of Proposition 5. Assume that x>0 Note that To prove that ρ j (t) − c + j t has a limit as t → +∞, we will study separately the existence of limits as t → +∞ of the two integrals above.

2a. Preliminary : Monotonicity results on η(t).
We introduce monotonicity results on η(t) (and not on u(t) as before) that are refinement of Claim 2.2.
We claim the following monotonicity results (see the proof in Appendix B.2).
Remark. The improvement of Claim 3.2 with respect to the monotonicity on u(t) in Claim 2.2, or Lemma 3 in [14] is that the upper bound can be integrated twice in time. Claim 3.2 is one example of monotonicity result on localized energy type quantities on η(t). In Appendix B.2, we prove a slightly more general version of Claim 3.2, wherex = x − ρ 1 (t) + σ 2 (t − t 0 ) − x 0 , where 0 < σ ≤ 1 2 and x 0 ∈ R, and we claim other monotonicity results to be used in this paper. Let Note that there exists K > 0 such that ψ(x) ≥ 1 K e − |x| 4 on R. Thus, we have g j (t) ≤ Kg j (t). We have the following consequence of Claim 3.2.

(3.15)
Proof of Claim 3.3. Integrating the conclusion of Claim 3.2 between t 0 and t, we obtain , we obtain the first estimate of (3.15). The estimate on |c 2q+1 2 (t)−c 2q+1 2 (t 0 )| is obtained in the same way using M 2 (t) and E 2 (t).
We claim the following lemma.
For the proof see Appendix B.3. Note that the proof is based only on Lemma 3.1 and monotonicity arguments such as Claim 3.2. We follow the same steps as in the proof of (3.11), using quantities M j (t), E j (t) instead of I σ,y 0 on the same lines. The proof of the monotonicity is the same as the one of Claim 3.2.
Lemma 3.4 (Estimate on ρ ′ j (t) − c j (t)) Assume that (3.12) holds. For j = 1, 2, Lemma 3.4 is proved in Appendix B.4. Note that it makes use of the following functional , which is an L 1 -type quantity, already introduced in [13]. For p = 3, another argument can be used. From Claim B.1, (B.5), where g j (t) is defined in (B.2). By (3.8)-(3.9), we obtain in this case: Such an integrability property cannot be proved from (B.3) for p = 2, 4, since we do not know whether or not +∞ 0 g j (t)dt < +∞.

A.2 Proof of Lemma 2.1
First, we recall well-known identities related to Q c for f (u) = u p .
. For x ≤ m(t), by the definition of ψ(x) and m(t) < ρ 1 (t), we have Proof of (A.11). We have for c small enough. Thus by integration in x, we obtain (A.11). Proof of (A. 12). For x ≥ m(t) Thus, by integration in x and for c small, we obtain (A.12).
Proof of (A. 13). Thus, by integration in x and for c small, we obtain (A.13).

A.3 Proof of Claim 2.4
The proof is based on the following well-known fact: There exists It is similar to [18], Proof of Lemma 4. Set Note that H 0 (t) and H(t) are easily compared. Indeed, we have Let ǫ 0 > 0. By (2.6), for α and c small enough, we have Thus, it is sufficient to prove (2.21) for H 0 (t) for some λ 0 > 0 independent of c and α. First, we consider a function Φ ∈ C 2 (R), . We recall the following claim from [18], page 355 (and references therein): This result is a localized version of (A. 16), and is easily proved by direct calculations. By a scaling argument, i.e. changing x into x √ c and using the definition of Q c , we have : Now, we consider η as in the proof of Lemma 2.2, i.e. satisfying the orthogonality con- Let λ 2 = min( λ 1 4 , 1 2 ). Since 1 − Φ 1 − Φ 2 ≥ 0, c 2 (t) > c 2 and c 1 (t) ≥ 1 2 , we have by (A.17) and (A.18): for B large enough. Similarly, for B large enough, since e − 1 2 |x−ρ 1 (t)| ≤ Kψ(x − m(t)), we have Therefore, for B large enough, we obtain This completes the proof of Claim 2.4.

B Appendix
B.1 Proof of Claim 3.1 -Localized Viriel estimate By explicit calculations, η(t) satisfies Step 1. Control of the geometrical parameters.
Step 2. Viriel identity in η and conclusion of the proof. For w ∈ H 1 (R), define Recall that the two quantities H ∞ (w, w) and H * ∞ (w, w) were introduced in [11] for a Viriel type identity. Here, by analogy, we define, A (x − ρ j (t)) . By straightforward calculations, we have Therefore, by using the equation of η, we have, for j = k, where we have used . From this identity, we claim Claim B.2 There exist A ≥ 5, κ 0 > 0 such that, for α small enough, and for all t ≥ 0, √ c j (t+Tc) . (B.14) By (2.15) and A ≥ 5, we have c 1 (t)/A < 1/4 and c 2 (t)/A < √ c/4 and thus we obtain the conclusion of Claim 3.1 from Claim B.2.
Proof of Claim B.2. By [11] Proposition 6 and localization arguments as [13] proof of Proposition 6 (see also proof of Claim 2.4 in the present paper), there exists A 0 > 0, λ 0 > 0, such that if A > A 0 then The claim means that all other terms in the previous identity can be absorbed by this term for some A ≥ 5 for α, c small enough up to and error term of size e − 1 8 √ c j (t+Tc) .
for A large enough. Now, A is fixed to such value. Next, we have from η H 1 ≤ Kαc q and (B.3) Therefore, Thus, for α small enough, for α small enough. Thus since Θ j is exponentially small around ρ k (t). Therefore, we need only estimate terms of the form For the last term in (B.17), we have, arguing as in the proof of (B.9), for α small enough. For the first term in (B.17), we integrate by parts, so that Finally, we treat all these terms similarly as in (B.9), so that, for α small enough, For E 3 , note that by (B.7), For the first term, since R j L 2 ≤ Kc q and η L 2 ≤ Kαc q , we have R j Θ j η ≤ Kαc 2q , and so by (B.4), and arguing for the other term as for E 2 , for α small enough, we get For the first term in E 4 , we proceed as for E 3 . The last two terms in E 4 are controled by Ke − 1 8 √ c j (t+Tc) since they contain products of exponentially decaying functions centered around ρ j (t) and ρ k (t). Thus, Since |Ψ j | ≤ K, R jx L 2 ≤ Kc q+ 1 2 j and η L 2 ≤ Kαc q , we have R jx Ψ j η ≤ Kαc 2q+ 1 2 , and then we obtain by (B.5) for α small enough. Thus, the proof of Claim B.2 is complete.

B.2 Proof of Claim 3.2 -Monotonicity results on η(t)
In this appendix, we prove monotonicity results for quantities defined in η(t). Claim 3.2 is a direct consequence of Claim B.3 below for x 0 = 0. Recall that ψ(x) is defined by (2.9). .
Note that the monotonicity results on E j (t) requires the addition of some quantity related to M j (t) (here the constant 1 100 is somewhat arbitrary, any small fixed positive constant would work). See also Lemma 1 in [8] for similar calculations in u(t).
Proof of Claim B.3. We prove the part of Claim B.3 concerning M 1 (t), E 1 (t), M 2 (t), E 2 (t). The rest is proved similarly.

B.3 Proof of Lemma 3.2
We follow similar steps as in the proof of (2.5), using monotonicity arguments on η(t) instead of u(t).
(ii) Estimate ofg 1 (t). We claim We claim the following preliminary result on J j (t) defined in (3.16).
Claim B.4 (i) Equation of J j (t): Then,