This article is cited in 1 paper

Darboux system as three-dimensional analog of Liouville equation

R. Ch. Kulaev^ab, A. K. Pogrebkov^c, A. B. Shabat<sup>d

a</sup> North-Ossetian State University named after K.L. Khetagurov, 44–46 Vatutin str., Vladikavkaz, 362025 Russia
^b Southern Mathematical Institute, Vladikavkaz Scientific Center of the Russian Academy of Sciences, 22 Markus str., Vladikavkaz, 362027 Russia
^c V.A. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin str., Moscow, 119991 Russia
^d L.D. Landau Institute of Theoretial Physics of Russian Academy of Sciences, 1a Academician Semenov Ave., Chernogolovka, Moscow Obl., 142432 Russia

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 $\mathbb{R}^3$.

Keywords: Darboux system, integrable systems, Goursat problem, third-order hyperbolic equation.

UDC: 517.958

Received: 21.11.2017

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2018, 62:12, 50–58

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