{
  "id": "0909.1180",
  "version": "v2",
  "published": "2009-09-07T09:13:33.000Z",
  "updated": "2011-05-12T12:00:25.000Z",
  "title": "A Critical Centre-Stable Manifold for Schroedinger's Equation in R^3",
  "authors": [
    "Marius Beceanu"
  ],
  "comment": "Substantially revised from previous version",
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the focusing cubic semilinear Schroedinger equation in R^3 i \\partial_t \\psi + \\Delta \\psi + | \\psi |^2 \\psi = 0. It admits an eight-dimensional manifold of special solutions called ground state solitons. We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant. Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a soliton and radiation. Conversely, in a general setting, any solution that stays close to the soliton manifold for all time is in N. The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time-dependent linearized equation. The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here --- of the focusing cubic NLS in R^3 --- by the work of Marzuola-Simpson and Costin-Huang-Schlag.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-12T12:00:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q51"
    ],
    "keywords": [
      "critical centre-stable manifold",
      "schroedingers equation",
      "focusing cubic semilinear schroedinger equation",
      "soliton manifold",
      "ground state solitons"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1180B"
    }
  }
}