Geodesic
Darboux system as three-dimensional analog of Liouville equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2018), pp. 60-69

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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, third-order hyperbolic equation.
Mots-clés : Goursat problem 
@article{IVM_2018_12_a3,
     author = {R. Ch. Kulaev and A. K. Pogrebkov and A. B. Shabat},
     title = {Darboux system as three-dimensional analog of {Liouville} equation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {60--69},
     publisher = {mathdoc},
     number = {12},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2018_12_a3/}
}
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R. Ch. Kulaev; A. K. Pogrebkov; A. B. Shabat. Darboux system as three-dimensional analog of Liouville equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2018), pp. 60-69. http://geodesic.mathdoc.fr/item/IVM_2018_12_a3/