{
  "id": "1004.1888",
  "version": "v1",
  "published": "2010-04-12T07:49:51.000Z",
  "updated": "2010-04-12T07:49:51.000Z",
  "title": "Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations",
  "authors": [
    "Stephen Gustafson",
    "Tuoc Van Phan"
  ],
  "comment": "55 pages, 4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider nonlinear Schr\\\"{o}dinger equations, $i\\partial_t \\psi = H_0 \\psi + \\lambda |\\psi|^2\\psi$ in $\\mathbb{R}^3 \\times [0,\\infty)$, where $H_0 = -\\Delta + V$, $\\lambda=\\pm 1$, the potential $V$ is radial and spatially decaying, and the linear Hamiltonian $H_0$ has only two eigenvalues $e_0 < e_1 <0$, where $e_0$ is simple, and $e_1$ has multiplicity three. We show that there exist two branches of small \"nonlinear excited state\" standing-wave solutions, and in both the resonant ($e_0 < 2e_1$) and non-resonant ($e_0 > 2e_1$) cases, we construct certain finite-codimension regions of the phase space consisting of solutions converging to these excited states at time infinity (\"stable directions\").",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-12T07:49:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35Q55"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "degenerate excited states",
      "stable directions",
      "nonlinear excited state",
      "linear hamiltonian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1888G"
    }
  }
}