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Darboux System: Liouville Reduction and an Explicit Solution

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Abstract

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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Authors and Affiliations

  1. North-Ossetian State University named after K. L. Khetagurov, ul. Vatutina 44–46, Vladikavkaz, 362025, Russia

    R. Ch. Kulaev

  2. Southern Mathematical Institute — the Affiliate of Vladikavkaz Scientific Centre of Russian Academy of Sciences, ul. Vatutina 53, Vladikavkaz, 362027, Russia

    R. Ch. Kulaev

  3. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. K. Pogrebkov

  4. National Research University “Higher School of Economics,”, ul. Myasnitskaya 20, Moscow, 101000, Russia

    A. K. Pogrebkov

  5. L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, pr. Akademika Semenova 1a, Chernogolovka, Moscow oblast, 142432, Russia

    A. B. Shabat

  6. Karachay-Cherkess State University named after U. D. Aliyev, ul. Lenina 29, Karachaevsk, 369202, Russia

    A. B. Shabat

Authors
  1. R. Ch. Kulaev
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  2. A. K. Pogrebkov
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  3. A. B. Shabat
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Correspondence to R. Ch. Kulaev.

Additional information

Original Russian Text © R.Ch. Kulaev, A.K. Pogrebkov, A.B. Shabat, 2018, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Vol. 302, pp. 268–286.

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Kulaev, R.C., Pogrebkov, A.K. & Shabat, A.B. Darboux System: Liouville Reduction and an Explicit Solution. Proc. Steklov Inst. Math. 302, 250–269 (2018). https://doi.org/10.1134/S0081543818060123

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  • DOI: https://doi.org/10.1134/S0081543818060123

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