{
  "id": "0706.1174",
  "version": "v2",
  "published": "2007-06-08T12:47:06.000Z",
  "updated": "2007-10-18T17:03:16.000Z",
  "title": "Asymptotic stability of solitons of the gKdV equations with general nonlinearity",
  "authors": [
    "Yvan Martel",
    "Frank Merle"
  ],
  "comment": "Corrected typos. Added comments. Minor changes. To appear in Mathematische Annalen",
  "categories": [
    "math.AP"
  ],
  "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 $t\\geq 0$), 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-10-18T17:03:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q53",
      "35B40"
    ],
    "keywords": [
      "gkdv equations",
      "general nonlinearity",
      "asymptotic stability",
      "energy space",
      "generalized korteweg-de vries equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.1174M"
    }
  }
}