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Darboux system as three-dimensional analog of Liouville equation. (English. Russian original) Zbl 1412.35211
Russ. Math. 62, No. 12, 50-58 (2018); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2018, No. 12, 60-69 (2018).
Summary: 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 \(\mathbb{R}^3\).

MSC:
35Q05 Euler-Poisson-Darboux equations
35L25 Higher-order hyperbolic equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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