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