{
  "id": "1307.1304",
  "version": "v2",
  "published": "2013-07-04T12:36:26.000Z",
  "updated": "2013-07-31T08:00:34.000Z",
  "title": "Global Smooth Solution of Nonlinear Schr$\\ddot{o}$dinger Equation",
  "authors": [
    "Yongqian Han"
  ],
  "comment": "13 pages. arXiv admin note: substantial text overlap with arXiv:1307.1012. This paper has been withdrawn by the author due to a crucial error in Section 2",
  "categories": [
    "math.AP"
  ],
  "abstract": "The spatially periodic initial problem and Cauchy problem for nonlinear Schr$\\ddot{o}$dinger equations are considered. The existence and uniqueness of global solution with infinite smooth initial data $u_0$, i.e. $u_0,\\;|u_0|^{2p}u_0\\in H^{\\infty}$, are established. Moreover these two problem with initial data $u_0\\in H^m$ are globally well-posed provided the Fourier frequency of $u_0$ is contained in a bounded compact set. The equations studied here cover $L^2$ and $\\dot{H}^1$ critical and supercritical, defocusing and focusing nonlinear Schr$\\ddot{o}$dinger equations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-31T08:00:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q40",
      "37K10"
    ],
    "keywords": [
      "dinger equation",
      "global smooth solution",
      "infinite smooth initial data",
      "spatially periodic initial problem",
      "focusing nonlinear schr"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.1304H"
    }
  }
}