Darboux System: Liouville Reduction and an Explicit Solution

  • R. Ch. Kulaev
  • A. K. Pogrebkov
  • A. B. Shabat


For a Darboux system in ℝ3, we introduce a class of solutions for which an auxiliary second-order linear problem satisfies the factorization condition. We show that this reduction provides the (local) solvability of the Darboux system, and present an explicit solution to this problem for two types of dependent variables. We also construct explicit formulas for the Lamé coefficients and solutions to the associated linear problem. The previously known reduction to a weakly nonlinear system is shown to be a particular case of the approach proposed.


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • R. Ch. Kulaev
    • 1
    • 2
  • A. K. Pogrebkov
    • 3
    • 4
  • A. B. Shabat
    • 5
    • 6
  1. 1.North-Ossetian State University named after K. L. KhetagurovVladikavkazRussia
  2. 2.Southern Mathematical Institute — the Affiliate of Vladikavkaz Scientific Centre of Russian Academy of SciencesVladikavkazRussia
  3. 3.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  4. 4.National Research University “Higher School of Economics,”MoscowRussia
  5. 5.L.D. Landau Institute for Theoretical Physics of Russian Academy of SciencesChernogolovkaRussia
  6. 6.Karachay-Cherkess State University named after U. D. AliyevKarachaevskRussia

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