{
  "id": "0801.3523",
  "version": "v2",
  "published": "2008-01-23T08:19:33.000Z",
  "updated": "2008-03-05T09:13:04.000Z",
  "title": "Nonlinear Schrödinger equation on real hyperbolic spaces",
  "authors": [
    "Jean-Philippe Anker",
    "Vittoria Pierfelice"
  ],
  "comment": "Version 1 : 18 January 2008. Version 2 : 29 February 2008",
  "journal": "Ann. Inst. Henri Poincar\\'e (C) Analyse Non Lin\\'eaire 26, 5 (2009) 1853-1869",
  "doi": "10.1016/j.anihpc.2009.01.009",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We consider the Schr\\\"odinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we get strong well-posedness results for NLS. Specifically, for small intial data, we prove $L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity $F$. On the other hand, if $F$ is gauge invariant, $L^2$ charge is conserved and hence, as in the Euclidean case, it is possible to extend local $L^2$ solutions to global ones. The corresponding argument in $H^1$ requires the conservation of energy, which holds under the stronger condition that $F$ is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles & Staffilani, for small radial $L^2$ data and for large radial $H^1$ data. The second important application of our global Strichartz estimates is \"scattering\" for NLS both in $L^2$ and in $H^1$, with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical power $\\gamma=1+\\frac4n$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1<\\gamma\\le1+\\frac4n$. If we restrict to defocusing nonlinearities $F$, we can extend the $H^1$ scattering results of Banica, Carles & Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearity : the geometry of hyperbolic spaces makes every power-like nonlinearity short range.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-03-05T09:13:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real hyperbolic spaces",
      "nonlinear schrödinger equation",
      "gauge invariance assumption",
      "euclidean case",
      "nonlinearity"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009AnIHP..26.1853A"
    }
  }
}