{
  "id": "1212.5027",
  "version": "v2",
  "published": "2012-12-20T13:31:38.000Z",
  "updated": "2013-07-10T06:14:36.000Z",
  "title": "Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation",
  "authors": [
    "Fabrice Béthuel",
    "Philippe Gravejat",
    "Didier Smets"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We pursue our work on the dynamical stability of dark solitons for the one-dimensional Gross-Pitaevskii equation. In this paper, we prove their asymptotic stability under small perturbations in the energy space. In particular, our results do not require smallness in some weighted spaces or a priori spectral assumptions. Our strategy is reminiscent of the one used by Martel and Merle in various works regarding generalized Korteweg-de Vries equations. The important feature of our contribution is related to the fact that while Korteweg-de Vries equations possess unidirectional dispersion, Schr\\\"odinger equations do not.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-10T06:14:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic stability",
      "dark solitons",
      "gross-pitaevskii equation",
      "energy space",
      "generalized korteweg-de vries equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.5027B"
    }
  }
}