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


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.


Darboux system integrable systems Goursat problem third-order hyperbolic equation 


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  1. 1.
    Tsarev, S. P. “The Geometry of Hamilton Systems of Hydrodynamic Type. The Generalized Hodograph Method”, Math. USSR, Izv. 37, No. 2, 397–419 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Rogers, C., Schief, W. K. Bäcklund and Darboux Transformations: Geometry and Modern Application in Soliton Theory (Cambridge Univ. Press, Cambridge, 2002).CrossRefzbMATHGoogle Scholar
  3. 3.
    Dubrovin, B. A., Novikov, S. P. “Hydrodynamics ofWeaklyDeformedSoliton Lattices”, Differential geometry and Hamiltonian theory. Russ. Math. Surv. 44, No. 6, 35–124 (1989).Google Scholar
  4. 4.
    Zakharov, V. E. “Description of the n-Orthogonal Curvilinear Coordinate Systems and Hamiltonian Integrable Systems of Hydrodynamic Type. Part 1. Integration of the Lame Equations”, DukeMath. J. 94, No. 1, 103–139 (1998).CrossRefzbMATHGoogle Scholar
  5. 5.
    Eisenhart L.P. A treatise on the differential geometry of curves and surfaces (Kessinger Publ., LLC, 2010).Google Scholar
  6. 6.
    Krichever, I. M. “Algebraic-Geometric n-Orthogonal Curvilinear Coordinate Systems and Solutions of Associativity Equations”, Funkts. analiz i ego prilozh. 31, No. 1, 32–50 (1997) [inu Russian].MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhegalov, V. I., Mironov, A. I. Differential Equations With Higher Partial Derivatives (Kazan, 2001) [in Russian].Google Scholar
  8. 8.
    Dubrovin, B. A., Matveev, V. B. and Novikov, S.P. “Nonlinear Equations of Korteweg-de Vries Type, Finite- Band Linear Operators and Abelian Varieties”, Usp.Mat. Nauk 31, No. 1, 55–136 (1976) [in Russian].Google Scholar
  9. 9.
    Drach, U. “Sur l’Integration par Quadratures de l’Equation Differentielle y = [ϕ(x) + h]y”, Compt. Rend. Acad. Sci. 168, 337–340 (1919).zbMATHGoogle Scholar
  10. 10.
    Pogrebkov, A. K. “Symmetries of the Hirota Difference Equation”, SIGMA 13 (2017), 053, 14 pages; Scholar

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© 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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