ABOUT ANALYTICAL METHOD FOR SOLVING THE CAUCHY PROBLEM OF TWO QUASILINEAR HYPERBOLIC EQUATIONS SYSTEM

Abstract

The applicability of the V. Lychagin "manual" integration method is analyzed with respect to systems of two quasilinear hyperbolic differential equations of the first order with two independent variables t, x and two unknown functions u = u (t, x) and v = v (t, x). The systems under consideration are a special case of Jacobi systems, for which V. Lychagin proposed an analytical method for solving the initial-boundary value problem. Each of the equations of the system is associated with a differential 2-form on four-dimensional space. This pair of forms uniquely determines the field of linear operators, which, for hyperbolic equations, generates an almost product structure. This means that the tangent space of four-dimensional space in each point is a direct sum of two-dimensional own-subspaces of the given operator and, thus, two 2-dimensional distributions are defined. If at least one of these distributions is completely integrable, then it is possible to construct a vector field along which shifts keep the solution of the original system of equations. Thus, the solution of the initial-boundary value problem for the system under consideration can be obtained analytically by shifting the initial curve along the trajectories of the given vector field. As an example, the Buckley-Leverett system of equations describing the process of nonlinear one-dimensional two-phase filtration in a porous medium is considered. To construct the solution of the Cauchy problem, a curve of the initial data is chosen; the solution of the Buckley-Leverett system is obtained by shifting this curve along the trajectories of the vector field (this vector field is defined up to multiplication by a function). The cross-sections of the components of this graph for different instants of time are brought in the figure. The graph shows that at some point of time the solution stops being unambiguous. At this point, the solution breaks and a shock wave appears.