{
  "id": "2204.07287",
  "version": "v1",
  "published": "2022-04-15T01:50:38.000Z",
  "updated": "2022-04-15T01:50:38.000Z",
  "title": "Long time asymptotic behavior for the nonlocal mKdV equation in space-time solitonic regions-II",
  "authors": [
    "Xuan Zhou",
    "Engui Fan"
  ],
  "comment": "59 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions \\begin{align*} &q_t(x,t)-6\\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\\ \\ \\lim_{x\\to \\pm\\infty} q_{0}(x)=q_{\\pm}, \\end{align*} where $|q_{\\pm}|=1$ and $q_{+}=\\delta q_{-}$, $\\sigma\\delta=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<\\xi<6$ with $\\xi=\\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $\\xi<-6$ and $\\xi>6$. Based on the Riemann-Hilbert problem of the the Cauchy problem, further using the $\\bar{\\partial}$ steepest descent method, we derive different long time asymptotic expansions of the solution $q(x,t)$ in above two different space-time solitonic regions. In the region $\\xi<-6$, phase function $\\theta(z)$ has four stationary phase points on the $\\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\\mathcal{N}(\\Lambda)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function ${\\rm Im}\\nu(\\zeta_i)$. In the region $\\xi>6$, phase function $\\theta(z)$ has four stationary phase points on $i\\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\\mathcal{N}(\\Lambda)$-soliton with diverse residual error order $\\mathcal{O}(t^{-1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-15T01:50:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "long time asymptotic behavior",
      "space-time solitonic regions-ii",
      "real nonlocal mkdv equation",
      "stationary phase points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 59,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}