Система Дарбу и разделение переменных в задаче Гурса для уравнения третьего порядка в R3

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 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.