Asymptotic stability of solitons of the gKdV equations with general nonlinearity
| dc.creator | Martel, Yvan | |
| dc.creator | Merle, Frank | |
| dc.date | 2007-06-08 | |
| dc.date | 2007-10-18 | |
| dc.date.accessioned | 2026-07-07T08:36:44Z | |
| dc.date.available | 2026-07-07T08:36:44Z | |
| dc.description | 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 $t\geq 0$), 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. | |
| dc.description | Corrected typos. Added comments. Minor changes. To appear in Mathematische Annalen | |
| dc.identifier | https://arxiv.org/abs/0706.1174 | |
| dc.identifier | http://arxiv.org/abs/0706.1174 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/140168 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35Q53, 35B40 | |
| dc.title | Asymptotic stability of solitons of the gKdV equations with general nonlinearity | |
| dc.type | text |