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

*(English. Russian original)*Zbl 1445.37048
Russ. Math. 64, No. 4, 35-43 (2020); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2020, No. 4, 43-53 (2020).

Summary: 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 parameterized by six functions of one variable (two for each of three independent variables). We give explicit formulas for solutions of the Darboux system. Under the additional assumption that the Christoffel symbols are constants, a linear system associated with the Darboux one is studied. In the case, the system is reduced to the three-dimensional Goursat problem for a third-order equation with data located on the coordinate planes. It is shown that the solution to the Goursat problem enables us to separate the variables, and it is determined by its values on the coordinate lines.

##### MSC:

37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |

37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |

##### Keywords:

three-dimensional Darboux system; integrable systems; three-dimensional Goursat problem; systems of hydrodynamic type equations; Hamiltonian system
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\textit{R. Ch. Kulaev} and \textit{A. B. Shabat}, Russ. Math. 64, No. 4, 35--43 (2020; Zbl 1445.37048); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2020, No. 4, 43--53 (2020)

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##### References:

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