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 Izv. Vyssh. Uchebn. Zaved. Mat., 2020, Number 4, Pages 43–53 (Mi ivm9560)

Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$

R. Ch. Kulaevabc, A. B. Shabatde

a North Ossetia State University 46 Vatutina str., Vladikavkaz, 362025 Russia
b Southern Mathematical Institute VSC RAS, 22 Markusa str., Vladikavkaz, 362027 Russia
c Institute of Mathematics, Ufa Federal Research Centre RAS, 112 Chernyshevsky str., Ufa, 450008 Russia
d Landau Institute for Theoretical Physics RAS, 1-A Akademika Semenova Ave., Chernogolovka, 142432 Russia
e Adygea State University, 208 Pervomayskaya str., Maykop, 385000 Russia

Abstract: We construct a reduction of the three-dimensional Darboux system for the Christoffel symbols, which describes conjugate curvilinear coordinate systems. The reduction is determined by one additional algebraic condition on the Christoffel symbols. It is shown that the corresponding class of solutions of the Darboux system is parametrized by six functions of one variable (two for each of three independent variables). Explicit formulas for Darboux system solutions are given. For the case when Christoffel symbols are constants, the linear system associated with the Darboux system is studied. In this formulation, this system is reduced to the three-dimensional Goursat problem for a third-order equation with data on the coordinate planes. It is shown that the solution to the Goursat problem allows the separation of variables and is determined by its values on the coordinate lines.

Keywords: three-dimensional Darboux system, integrable systems, three-dimensional Goursat problem, systems of hydrodynamic type equations, Hamiltonian system.

 Funding Agency Grant Number Russian Science Foundation 15-11-20007

DOI: https://doi.org/10.26907/0021-3446-2020-4-43-53

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UDC: 517.95
Revised: 17.07.2019
Accepted: 25.09.2019

Citation: R. Ch. Kulaev, A. B. Shabat, “Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 4, 43–53

Citation in format AMSBIB
\Bibitem{KulSha20} \by R.~Ch.~Kulaev, A.~B.~Shabat \paper Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$ \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2020 \issue 4 \pages 43--53 \mathnet{http://mi.mathnet.ru/ivm9560} \crossref{https://doi.org/10.26907/0021-3446-2020-4-43-53}