{
  "id": "1909.07071",
  "version": "v1",
  "published": "2019-09-16T08:59:52.000Z",
  "updated": "2019-09-16T08:59:52.000Z",
  "title": "On the orbital stability of a family of traveling waves for the cubic Schr{ö}dinger equation on the Heisenberg group",
  "authors": [
    "Louise Gassot"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the focusing energy-critical Schr{\\\"o}dinger equation on the Heisenberg group in the radial case\\[i\\partial_t u-\\Delta_{\\mathbb{H}^1} u=|u|^2u,\\quad\\Delta_{\\mathbb{H}^1}=\\frac{1}{4}(\\partial_x^2+\\partial_y^2)+(x^2+y^2)\\partial_s^2,\\quad(t,x,y,s)\\in \\mathbb{R}\\times\\mathbb{H}^1,\\]which is a model for non-dispersive evolution equations. For this equation, existence of smooth global solutions and uniqueness of weak solutions in the energy space are open problems. We are interested in a family of ground state traveling waves parametrized by their speed $\\beta \\in (-1,1)$. We show that the traveling waves of speed close to $1$ present some orbital stability in the following sense. If the initial data is radial and close enough to one traveling wave, then there exists a global weak solution which stays close to the orbit of this traveling wave for all times. A similar result is proven for the limiting system associated to this equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-16T08:59:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heisenberg group",
      "orbital stability",
      "cubic schr",
      "dinger equation",
      "smooth global solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}