{
  "id": "2011.00919",
  "version": "v1",
  "published": "2020-11-02T12:01:09.000Z",
  "updated": "2020-11-02T12:01:09.000Z",
  "title": "Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data",
  "authors": [
    "Qiaoyuan Cheng",
    "Yiling Yang",
    "Engui Fan"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AP",
    "nlin.SI"
  ],
  "abstract": "We apply $\\bar{\\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schr\\\"{o}dinger equation $$ q_t+iq_{xx}-ia (\\vert q \\vert^2q)_x -2b^2\\vert q \\vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \\in H^{2,2}(\\mathbb{R})$. In the asymptotic expression, the leading order term $\\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\\mathcal{O}(t^{-3/4})$ from a $\\overline\\partial$ equation",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-02T12:01:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted sobolev initial data",
      "long-time asymptotic behavior",
      "mixed schrödinger equation",
      "steepest descent method",
      "essentially minimal regularity assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}