{
  "id": "0902.1643",
  "version": "v2",
  "published": "2009-02-10T19:29:59.000Z",
  "updated": "2009-08-17T07:05:20.000Z",
  "title": "A Critical Centre-Stable Manifold for the Cubic Focusing Schroedinger Equation in Three Dimensions",
  "authors": [
    "Marius Beceanu"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \\partial_t \\psi + \\Delta \\psi + |\\psi|^2 \\psi = 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(\\Gamma + vx - t|v|^2 + \\alpha^2 t)} \\phi(x-2tv-D, \\alpha), where \\phi(x, \\alpha) is a positive ground state solution of the semilinear elliptic equation -\\Delta \\phi + \\alpha^2\\phi = \\phi^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a solution starting on N remains on N for all positive time and for some finite negative time and N can be identified as the centre-stable manifold for this equation. The proof is based on the method of modulation, introduced by Soffer and Weinstein and adapted by Schlag to the L^2-supercritical case. Novel elements include a different linearization and new Strichartz-type estimates for the linear Schroedinger equation. The main result depends on a spectral assumption concerning the absence of embedded eigenvalues. We also establish several new estimates for solutions of the time-dependent and time-independent linear Schroedinger equation, which hold under sharper or more general conditions than previously known. Several of these estimates are based on a new approach that makes use of Wiener's Theorem in the context of function spaces.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-17T07:05:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q51"
    ],
    "keywords": [
      "cubic focusing schroedinger equation",
      "critical centre-stable manifold",
      "dimensions",
      "time-independent linear schroedinger equation",
      "real analytic manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.1643B"
    }
  }
}