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Russian Mathematics

, Volume 62, Issue 12, pp 50–58 | Cite as

Darboux System as Three-Dimensional Analog of Liouville Equation

  • R. Ch. KulaevEmail author
  • A. K. Pogrebkov
  • A. B. Shabat
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Abstract

We discuss the problems of the connections of the modern theory of integrability and the corresponding overdetermined linear systems with works of geometers of the late nineteenth century. One of these questions is the generalization of the theory of Darboux–Laplace transforms for second-order equations with two independent variables to the case of three-dimensional linear hyperbolic equations of the third order. In this paper we construct examples of such transformations. We consider applications to the problem of orthogonal curvilinear coordinate systems in ℝ3.

Keywords

Darboux system integrable systems Goursat problem third-order hyperbolic equation 

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Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  • R. Ch. Kulaev
    • 1
    • 2
    Email author
  • A. K. Pogrebkov
    • 3
  • A. B. Shabat
    • 4
  1. 1.North-Ossetian State University named after K. L. KhetagurovVladikavkazRussia
  2. 2.Southern Mathematical Institute Vladikavkaz Scientific Center of the Russian Academy of SciencesVladikavkazRussia
  3. 3.V. A. Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  4. 4.L. D. Landau Institute of Theoretial Physics of Russian Academy of SciencesChernogolovka, Moscow Obl.Russia

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