{
  "id": "2108.06284",
  "version": "v1",
  "published": "2021-08-13T15:14:40.000Z",
  "updated": "2021-08-13T15:14:40.000Z",
  "title": "Long time asymptotics for the defocusing mKdV equation with finite density initial data in different solitonic regions",
  "authors": [
    "Taiyang Xu",
    "Zechuan Zhang",
    "Engui Fan"
  ],
  "comment": "74 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We investigate the long time asymptotics for the Cauchy problem of the defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data in different solitonic regions \\begin{align*} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \\quad (x,t)\\in\\mathbb{R}\\times \\mathbb{R}^{+}, &q(x,0)=q_{0}(x), \\quad \\lim_{x\\rightarrow\\pm\\infty}q_{0}(x)=\\pm 1, \\end{align*} where $q_0\\mp 1\\in H^{4,4}(\\mathbb{R})$. Based on the spectral analysis of the Lax pair, we express the solution of the mKdV equation in terms of an Riemann-Hilbert problem. In our previous article, we have obtained long time asymptotics and soliton solutions for the mKdV equation in the solitonic region $\\xi\\in(-6,-2)$ with $\\xi=\\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for the solitonic region $\\xi\\in(-\\infty,-6)\\cup(-2,+\\infty)$. For $\\xi<-6$, there exist four stationary phase points on jump contour, and the asymptotic approximations can be characterized with an $N$-soliton on discrete spectrums and a leading order term $O(t^{-1/2})$ on continuous spectrum up to a residual error order $O(t^{-3/4})$. For $\\xi>-2$, the leading term of asymptotic expansion is described by the soliton solution and the error order $\\mathcal{O}(t^{-1})$ comes from a $\\bar{\\partial}$-problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-13T15:14:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite density initial data",
      "long time asymptotics",
      "solitonic region",
      "defocusing mkdv equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}