Asymptotic stability of solitons of the gKdV equations with general nonlinearity

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)\times \mathbb{R}, (1) with general $C^3$ 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 $t\ge 0$), then $u(t)$ locally converges in the energy space to some $Q_{c_{+}}$ as $t\to +\infty$. 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.

Corrected typos. Added comments. Minor changes. To appear in Mathematische Annalen

References in corpus (2)

• Asymptotic stability of N-soliton states of NLS
• Stability of two soliton collision for nonintegrable gKdV equations

Cited by in corpus (2)

• Stability of two soliton collision for nonintegrable gKdV equations
• Why are solitons stable?