{
  "id": "2003.03114",
  "version": "v1",
  "published": "2020-03-06T10:08:14.000Z",
  "updated": "2020-03-06T10:08:14.000Z",
  "title": "Well-posedness and convergence of a variational discretization of the Camassa-Holm equation",
  "authors": [
    "Sondre Tesdal Galtung",
    "Xavier Raynaud"
  ],
  "comment": "62 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We show how the two-component Camassa-Holm (2CH) system, for which the Camassa-Holm equation is a special case, can be derived from variational principles by introducing a potential energy. The variation is done in Lagrangian coordinates, but the Euler-Lagrange equations have a pure Eulerian formulation. After discretizing the kinetic and potential energies, we use the same variational principles to derive a semi-discrete system of equations as an approximation of the 2CH system. In the discrete case, the Euler$-$Lagrange equations are only available in Lagrangian variables. In this derivation there naturally appears a discrete Sturm-Liouville operator whose coefficients depend on the initial data. We show the existence of a decaying Green's function for this operator. Then, we prove that the semi-discrete system admits globally defined unique solutions, and preserves discrete versions of the energy and momentum for the 2CH system. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions which converges to the solution of the 2CH system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-06T10:08:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q51",
      "35A15",
      "35A35",
      "37K58",
      "39A60",
      "65M80",
      "37K60",
      "34B24"
    ],
    "keywords": [
      "camassa-holm equation",
      "variational discretization",
      "2ch system",
      "globally defined unique solutions",
      "admits globally defined unique"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}