Geometry of a quasilinear system of two partial differential equations of the 1st and 2nd order in two functions of two independent variables

Abstract

Introduction. In 1904–1905, the paper [5] of French mathematician E. Cartan was published. In Moscow State University (MGU), the ideas of Cartan were applied by Prof. S. P. Finikov [7], and also by his students G. F. Laptev, A. M. Vasil’ev, and many others. In the 40s–80s of the 20th century, in Moscow State University and in other colleges inside Russia and outside Russia, representatives of geometers of these scientific schools worked. In Moscow State University, there was the scientific workshop on theory of differential-geometric structures chaired by A. M. Vasil’ev. The work [17] is one of the first works devoted to the geometry of three partial differential equations (in two independent variables) that reduce to a system of three exterior differential equations of the second degree in a five-dimensional manifold. This work considers a wide range of problems related to differential-geometric structures should be noted. The description of intermediate integrals, the invariant linearizability criteria of the system considered, the methods for finding conservation laws and their use for reducing systems to simpler ones, and also the illustration of all this via examining the classical systems of continuous medium mechanics. A considerable part of these results was generalized by Kh. O. Kil’p [8–10] to quasilinear systems of n equations with n unknowns and two independent variables. The corresponding structures are no longer g-structures, as in the cases n = 2, 3. However, it was revealed that the method for their studying was not more complicated than that for g-structures. Kh. O. Kil’p also studied systems of three first-order partial differential equations without additional conditions [8] and a number of special problems referring to particular classes of quasilinear systems [9], and others. In the works of E. M. Kan, the case of a quasilinear system of four equations with pairwise coinciding characteristics was considered in detail, and in the works [13, 14] of L. N. Orlova, the case of a system of two equations, one of which contains first-order partial derivatives and the other firstand secondorder partial derivatives, was considered.