{
  "id": "2410.17405",
  "version": "v1",
  "published": "2024-10-22T20:26:23.000Z",
  "updated": "2024-10-22T20:26:23.000Z",
  "title": "The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves",
  "authors": [
    "Elliot Blackstone",
    "Louise Gassot",
    "Patrick Gérard",
    "Peter D. Miller"
  ],
  "comment": "63 pages, 18 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\\mathrm{ZD}(t,x;\\epsilon)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;\\epsilon)$ of the Benjamin-Ono equation with dispersion parameter $\\epsilon>0$ and initial data $u_0$ satisfies $u(t,x;\\epsilon)-u^\\mathrm{ZD}(t,x;\\epsilon)\\to 0$ in the locally uniform sense as $\\epsilon\\to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases this convergence implies strong $L^2(\\mathbb{R})$ convergence. The asymptotic profile $u^\\mathrm{ZD}(t,x;\\epsilon)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-22T20:26:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35C20",
      "35Q51",
      "41A60"
    ],
    "keywords": [
      "benjamin-ono equation",
      "dispersive shock waves",
      "zero-dispersion limit",
      "generation",
      "generic rational initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}