{
  "id": "1311.2275",
  "version": "v3",
  "published": "2013-11-10T14:14:46.000Z",
  "updated": "2014-10-08T21:50:42.000Z",
  "title": "Modified scattering for the cubic Schrödinger equation on product spaces and applications",
  "authors": [
    "Zaher Hani",
    "Benoit Pausader",
    "Nikolay Tzvetkov",
    "Nicola Visciglia"
  ],
  "comment": "47 pages. Minor corrections and several typos fixed",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the cubic nonlinear Schr\\\"odinger equation posed on the spatial domain $\\mathbb{R}\\times \\mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\\leq d\\leq 4)$. The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\\geq 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\\mathbb{R}\\times \\mathbb{T}^d$ (for any $d\\geq 2$) with infinitely growing high Sobolev norms $H^s$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-19T20:39:41.000Z",
      "comment": "47 pages. Added explicit construction and lower bounds on the infinite growth of Sobolev norms in Corollary 1.3. References added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-08T21:50:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "cubic schrödinger equation",
      "product spaces",
      "modified scattering",
      "applications",
      "construct modified wave operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.2275H"
    }
  }
}