Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems

Abstract

. The objective of this paper is to construct geometrically Riemann k -wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two approaches to the construction of Riemann k -waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the k -wave problem and discuss in detail the conditions for the existence of k -wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the k -wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.