{
  "id": "1601.02154",
  "version": "v1",
  "published": "2016-01-09T20:11:14.000Z",
  "updated": "2016-01-09T20:11:14.000Z",
  "title": "The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations",
  "authors": [
    "H. A. Erbay",
    "S. Erbay",
    "A. Erkip"
  ],
  "comment": "24 pages, to appear in Discrete and Continuous Dynamical Systems",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\\epsilon$ and $\\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-09T20:11:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q53",
      "35Q74",
      "74J30",
      "35C20"
    ],
    "keywords": [
      "boussinesq equation",
      "camassa-holm equation",
      "long-wave limit",
      "equations model bidirectional wave propagation",
      "nonlocal wave equations model bidirectional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102154E"
    }
  }
}