{
  "id": "2204.01299",
  "version": "v1",
  "published": "2022-04-04T08:14:13.000Z",
  "updated": "2022-04-04T08:14:13.000Z",
  "title": "On the Cauchy problem of defocusing mKdV equation: long-time asymptotics under step-like initial data",
  "authors": [
    "Taiyang Xu",
    "Engui Fan"
  ],
  "comment": "51 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 step-like initial data, i.e., $q_{0}(x)=q(x,t=0)=C_{L}$, as $x<0$ and $q_{0}(x)=C_{R}$ as $x>0$, where $C_{L}>C_{R}>0$ are arbitrary positive real numbers. We firstly develop the direct scattering theory to establish the Riemann-Hilbert (RH) problem associated with step-like initial data. Then by introducing the related $g$ function in different space-time regions and using the steepest descent analysis, we deform the original matrix valued RH problem to explicitly solving models. Finally we obtain the different long-time asymptotic behavior of the solution of the Cauchy problem for defocusing mKdV equation in four different space-time regions $\\mathcal{R}_{\\xi,I}, \\mathcal{R}_{\\xi,II},\\mathcal{R}_{\\xi,III}$ and $\\mathcal{R}_{\\xi,IV} $ in the half-plane, where $\\mathcal{R}_{\\xi,I}$ and $ \\mathcal{R}_{\\xi,IV}$ are far left field regions; and $\\mathcal{R}_{\\xi,II}$ and $ \\mathcal{R}_{\\xi,III}$ are rarefaction wave regions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T08:14:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "step-like initial data",
      "defocusing mkdv equation",
      "cauchy problem",
      "long-time asymptotic",
      "space-time regions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}