May  2008, 20(2): 177-218. doi: 10.3934/dcds.2008.20.177

Refined asymptotics around solitons for gKdV equations

1. 

Université de Versailles Saint-Quentin-en-Yvelines, Mathématiques, 45, av. des Etats-Unis, 78035 Versailles cedex, France

2. 

Université de Cergy-Pontoise and IHES, Laboratoire de mathématiques, UMR CNRS 8088, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

Received  May 2007 Revised  October 2007 Published  November 2007

We consider the generalized Korteweg-de Vries equation

$\partial_t u + \partial_x (\partial_x^2 u + f(u))=0, (t,x)\in \mathbb{R}\times \mathbb{R}$(1)

with $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution of (1) in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton.
    In [11], [13], it was 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 $Q_{c}$, then $u(t,.+\rho(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+$~$c$ and some function $\rho(t)$ such that $\rho'(t)$~$c^+$. Then, in [9] and [14], these results were extended with shorter proofs under general assumptions on $f$.
    The first objective of this paper is to give more information about the function $\rho(t)$. In the case $f(u)=u^p$, $p=2,3,4$ and under the additional assumption $x_+ u\in L^2(\mathbb{R})$, we prove that the function $\rho(t)-c^+ t$ has a finite limit as $t\to +\infty$.
    Second, we prove stability and asymptotic stability results for two solitons for a general nonlinearity $f(u)$, in the case where the ratio of the speeds of the two solitons is small.

Citation: Yvan Martel, Frank Merle. Refined asymptotics around solitons for gKdV equations. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 177-218. doi: 10.3934/dcds.2008.20.177
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