{
  "id": "2101.02489",
  "version": "v1",
  "published": "2021-01-07T11:11:00.000Z",
  "updated": "2021-01-07T11:11:00.000Z",
  "title": "Long-time asymptotic behavior of the modified Camassa-Holm equation",
  "authors": [
    "Yiling Yang",
    "Engui Fan"
  ],
  "comment": "50 pages. arXiv admin note: substantial text overlap with arXiv:2012.15496",
  "categories": [
    "math.AP",
    "nlin.SI"
  ],
  "abstract": "In this paper, we apply $\\overline\\partial$ steepest descent method to study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation \\begin{align} &m_{t}+\\left(m\\left(u^{2}-u_{x}^{2}\\right)\\right)_{x}+\\kappa u_{x}=0, \\quad m=u-u_{x x}, &u(x, 0)=u_{0}(x), \\end{align} where $\\kappa$ is a positive constant. It is shown that the solution of the Cauchy problem can be characterized via a the solution of Riemann-Hilbert problem. In a fixed space-time solitonic region $x/t\\in(-\\infty,-1/4-\\delta_0]\\cup[2+\\delta_0,+\\infty)$ with a positive small constant $\\delta_0$, we further compute the long time asymptotic expansion of the solution $u(x,t)$, which implies soliton resolution conjecture and can be characterized with an $N(\\Lambda)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region; the residual error order $\\mathcal{O}(|t|^{-1+2\\rho})$ from a $\\overline\\partial$ equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-07T11:11:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "long-time asymptotic behavior",
      "modified camassa-holm equation",
      "long time asymptotic behavior",
      "long time asymptotic expansion",
      "implies soliton resolution conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}