{
  "id": "math-ph/0411060",
  "version": "v2",
  "published": "2004-11-18T15:24:38.000Z",
  "updated": "2005-03-10T10:07:55.000Z",
  "title": "Wave equation with concentrated nonlinearities",
  "authors": [
    "Diego Noja",
    "Andrea Posilicano"
  ],
  "comment": "Revised version. To appear in Journal of Physics A: Mathematical and General, special issue on Singular Interactions in Quantum Mechanics: Solvable Models",
  "doi": "10.1088/0305-4470/38/22/022",
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of $\\CO^n$ and a discrete set $Y\\subset\\RE^3$ with $n$ elements, we define a nonlinear operator $\\Delta_{V,Y}$ on $L^2(\\RE^3)$ which coincides with the free Laplacian when restricted to regular functions vanishing at $Y$, and which reduces to the usual Laplacian with point interactions placed at $Y$ when $V$ is linear and is represented by an Hermitean matrix. We then consider the nonlinear wave equation $\\ddot \\phi=\\Delta_{V,Y}\\phi$ and study the corresponding Cauchy problem, giving an existence and uniqueness result in the case $V$ is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problem: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part $\\dot\\zeta+V(\\zeta)$. Main properties of the solution are given and, when $Y$ is a singleton, the mechanism and details of blow-up are studied.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-03-10T10:07:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concentrated nonlinearities",
      "nonlinear wave equation",
      "free wave equation",
      "inhomogeneous ordinary differential equation",
      "free laplacian"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2005,
      "month": "Jun",
      "volume": 38,
      "number": 22,
      "pages": 5011
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005JPhA...38.5011N"
    }
  }
}