{
  "id": "1203.3228",
  "version": "v1",
  "published": "2012-03-14T22:15:32.000Z",
  "updated": "2012-03-14T22:15:32.000Z",
  "title": "On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type",
  "authors": [
    "Mats Ehrnström",
    "Mark D. Groves",
    "Erik Wahlén"
  ],
  "comment": "37 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "nlin.PS"
  ],
  "abstract": "We consider a class of pseudodifferential evolution equations of the form $$u_t + (n(u) + Lu)_x = 0,$$ in which $L$ is a linear smoothing operator and $n$ is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimisation principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-03-14T22:15:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q53",
      "35A15",
      "76B15"
    ],
    "keywords": [
      "solitary-wave solutions",
      "whitham type",
      "whitham equation",
      "pseudodifferential evolution equations",
      "concentration-compactness methods"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/25/10/2903",
      "journal": "Nonlinearity",
      "year": 2012,
      "month": "Oct",
      "volume": 25,
      "number": 10,
      "pages": 2903
    },
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012Nonli..25.2903E"
    }
  }
}