{
  "id": "1002.0870",
  "version": "v1",
  "published": "2010-02-04T00:22:31.000Z",
  "updated": "2010-02-04T00:22:31.000Z",
  "title": "Geometry of Darboux-Manakov-Zakharov systems and its application",
  "authors": [
    "Peter J. Vassiliou"
  ],
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations \\label{GDMZabstract} \\frac{\\partial^2 u}{\\partial x_i\\partial x_j}=f_{ij}\\Big(x_k,u,\\frac{\\partial u}{\\partial x_l}\\Big), 1\\leq i<j\\leq n, k,l\\in\\{1,...,n\\} for a real-valued function $u(x_1,...,x_n)$ are studied with particular reference to the linear systems in this equation class. System (\\ref{GDMZabstract}) will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive geometric tools for explicitly constructing involutive systems of the form (\\ref{GDMZabstract}), essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multi-dimensional $n$-wave resonant interaction system and its modified version as well as constructing new examples of semi-Hamiltonian systems of hydrodynamic type. The general theory is illustrated by a study of these applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-04T00:22:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58A17",
      "58A30",
      "58D19",
      "58J45",
      "35N05",
      "53B05"
    ],
    "keywords": [
      "application",
      "wave resonant interaction system",
      "semilinear partial differential equations",
      "complicated nonlinear integrability conditions",
      "intrinsic geometric properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.0870V"
    }
  }
}