{
  "id": "math/0406058",
  "version": "v3",
  "published": "2004-06-03T11:40:03.000Z",
  "updated": "2007-11-29T10:08:56.000Z",
  "title": "The nonlinear Schrödinger equation on the hyperbolic space",
  "authors": [
    "Valeria Banica"
  ],
  "comment": "32 pages, final preprint version",
  "journal": "Comm. P.D.E. 32 (2007), no. 10, 1643-1677",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article we study some aspects of dispersive and concentration phenomena for the Schr\\\"odinger equation posed on hyperbolic space $\\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties : we prove that the dispersion inequality is valid, in a stronger form than the one on $\\mathbb{R}^n$. However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-11-29T10:08:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear schrödinger equation",
      "hyperbolic space",
      "concentration phenomena",
      "euclidean case",
      "blow-up turns"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6058B"
    }
  }
}