{
  "id": "1905.11976",
  "version": "v1",
  "published": "2019-05-28T17:50:43.000Z",
  "updated": "2019-05-28T17:50:43.000Z",
  "title": "A connection between the classical r-matrix formalism and covariant Hamiltonian field theory",
  "authors": [
    "Vincent Caudrelier",
    "Matteo Stoppato"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We bring together aspects of covariant Hamiltonian field theory and of classical integrable field theories in $1+1$ dimensions. Specifically, our main result is to obtain for the first time the classical $r$-matrix structure within a covariant Poisson bracket for the Lax connection, or Lax one form. This exhibits a certain covariant nature of the classical $r$-matrix with respect to the underlying spacetime variables. The main result is established by means of several prototypical examples of integrable field theories, all equipped with a Zakharov-Shabat type Lax pair. Full details are presented for: $a)$ the sine-Gordon model which provides a relativistic example associated to a classical $r$-matrix of trigonometric type; $b)$ the nonlinear Schr\\\"odinger equation and the (complex) modified Korteweg-de Vries equation which provide two non-relativistic examples associated to the same classical $r$-matrix of rational type, characteristic of the AKNS hierarchy. The appearance of the $r$-matrix in a covariant Poisson bracket is a signature of the integrability of the field theory in a way that puts the independent variables on equal footing. This is in sharp contrast with the single-time Hamiltonian evolution context usually associated to the $r$-matrix formalism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-28T17:50:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "covariant hamiltonian field theory",
      "classical r-matrix formalism",
      "covariant poisson bracket",
      "connection",
      "integrable field theories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}