{
  "id": "1205.1345",
  "version": "v1",
  "published": "2012-05-07T11:03:49.000Z",
  "updated": "2012-05-07T11:03:49.000Z",
  "title": "Nonlinear Schrödinger equations near an infinite well potential",
  "authors": [
    "Thomas Bartsch",
    "Mona Parnet"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The paper deals with standing wave solutions of the dimensionless nonlinear Schr\\\"odinger equation \\label{eq:abs1} i\\Phi_t(x,t) = -\\Delta_x\\Phi +V_\\la(x)\\Phi + f(x,\\Phi), \\quad x\\in\\R^N,\\ t\\in\\R,\\tag{$NLS_\\la$} where the potential $V_\\la:\\R^N\\to\\R$ is close to an infinite well potential $V_\\infty:\\R^N\\to\\R$, i. e. $V_\\infty=\\infty$ on an exterior domain $\\R^N\\setminus\\Om$, $V_\\infty|_\\Om\\in L^\\infty(\\Om)$, and $V_\\la\\to V_\\infty$ as $\\la\\to\\infty$ in a sense to be made precise. The nonlinearity may be of Gross-Pitaevskii type. A solution of \\eqref{eq:abs1} with $\\la=\\infty$ vanishes on $\\R^N\\setminus\\Om$ and satisfies Dirichlet boundary conditions, hence it solves \\label{eq:abs2} i\\Phi_t(x,t) &= -\\Delta_x\\Phi +V_\\la(x)\\Phi + f(x,\\Phi), &&\\quad x\\in\\Om,\\ t\\in\\R \\Phi(x,t) &= 0 &&\\quad x\\in\\pa\\Om,\\ t\\in\\R. \\tag{$NLS_\\infty$}. We investigate when a solution $\\Phi_\\infty$ of the infinite well potential \\eqref{eq:abs2} gives rise to nearby solutions $\\Phi_\\la$ of the finite well potential \\eqref{eq:abs1} with $\\la\\gg1$ large. Considering \\eqref{eq:abs2} as a singular limit of \\eqref{eq:abs1} we prove a kind of singular continuation type results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-07T11:03:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J61",
      "35J91",
      "35Q55",
      "58E05"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "satisfies dirichlet boundary conditions",
      "singular continuation type results",
      "wave solutions",
      "exterior domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.1345B"
    }
  }
}