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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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##### References:
 [1] Tsarev, S. P., The Geometry of Hamilton Systems of Hydrodynamic Type. The Generalized Hodograph Method, Math. USSR, Izv., 37, 397-419, (1991) · Zbl 0796.76014 [2] Rogers, C., Schief, W. K. Bäcklund and Darboux Transformations: Geometry and Modern Application in Soliton Theory (Cambridge Univ. Press, Cambridge, 2002). · Zbl 1019.53002 [3] Dubrovin, B. A.; Novikov, S. P., Hydrodynamics ofWeaklyDeformedSoliton Lattices, Differential geometry and Hamiltonian theory. Russ. Math. Surv., 44, 35-124, (1989) · Zbl 0712.58032 [4] Zakharov, V. E., Description of the n-Orthogonal Curvilinear Coordinate Systems and Hamiltonian Integrable Systems of Hydrodynamic Type. Part 1. Integration of the Lame Equations, DukeMath. J., 94, 103-139, (1998) · Zbl 0963.37068 [5] Eisenhart L.P. A treatise on the differential geometry of curves and surfaces (Kessinger Publ., LLC, 2010). · JFM 40.0626.02 [6] Krichever, I. M., Algebraic-Geometric n-Orthogonal Curvilinear Coordinate Systems and Solutions of Associativity Equations, Funkts. analiz i ego prilozh., 31, 32-50, (1997) · Zbl 1004.37052 [7] Zhegalov, V. I., Mironov, A. I. Differential Equations With Higher Partial Derivatives (Kazan, 2001) [in Russian]. [8] Dubrovin, B. A.; Matveev, V. B.; Novikov, S. P., Nonlinear Equations of Korteweg-de Vries Type, Finite- Band Linear Operators and Abelian Varieties, (1976) · Zbl 0326.35011 [9] Drach, U., Sur l’Integration par Quadratures de l’Equation Differentielle y = [ϕ(x) + h]y, Compt. Rend. Acad. Sci., 168, 337-340, (1919) · JFM 47.0412.01 [10] Pogrebkov, A. K., Symmetries of the Hirota Difference Equation, (2017) · Zbl 1372.35267
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