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Article overview
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Asymptotic stability of solitons of the gKdV equations with general
nonlinearity | | Yvan Martel
; Frank Merle
; | | Date: |
8 Jun 2007 | | Abstract: | We consider the generalized Korteweg-de Vries equation
$$partial_t u + partial_x (partial_x^2 u + f(u))=0, quad (t,x)in
[0,T) imes mathbb{R}, (1) $$
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 (1) of the type
$u(t,x)=Q_c(x-x_0-ct)$, called soliton.
In this paper, under general assumptions on $f$ and $Q_c$, we prove that the
family of soliton solutions around $Q_c$ is asymptotically stable in some local
sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $tgeq 0$), then
$u(t)$ locally converges in the energy space to some $Q_{c_+}$ as $t o
+infty$. Note in particular that we do not assume the stability of $Q_{c}$.
This result is based on a rigidity property of equation (1) around $Q_{c}$ in
the energy space whose proof relies on the introduction of a dual problem.
These results extend the main results in previous works devoted to the pure
power case. | | Source: | arXiv, arxiv.0706.1174 | | Services: | Forum | Review | PDF | Favorites | |
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