{
  "id": "2101.05942",
  "version": "v1",
  "published": "2021-01-15T02:28:35.000Z",
  "updated": "2021-01-15T02:28:35.000Z",
  "title": "Soliton resolution for the Hirota equation with weighted Sobolev initial data",
  "authors": [
    "Jin-Jie Yang",
    "Shou-Fu Tian",
    "Zhi-Qiang Li"
  ],
  "comment": "43 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "In this work, the $\\overline{\\partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(\\mathbb{R})=\\{f\\in L^{2}(\\mathbb{R}): f',xf\\in L^{2}(\\mathbb{R})\\}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=\\left\\{(x,t)\\in \\mathbb{R}\\times\\mathbb{R}: x=x_{0}+vt ~\\text{with}~ x_{0}\\in[x_{1},x_{2}]\\right\\}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $\\mathcal {O}(t^{-1/2})$ in the continuous spectrum, $\\mathcal {N}(\\mathcal {I})$ soliton solutions in the discrete spectrum and error order $\\mathcal {O}(t^{-3/4})$ from the $\\overline{\\partial}$ equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-15T02:28:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted sobolev initial data",
      "hirota equation",
      "soliton resolution",
      "initial value belong",
      "steepest descent method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}