Conditional symmetries and Riemann invariants for hyperbolic systems of PDEs

Abstract

This paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that this overdetermined system admits rank-k solutions expressible in terms of Riemann invariants. Finally, examples of applications of the proposed approach to the fluid dynamics equations in (k + 1) dimensions are discussed in detail. Several new soliton-like solutions (among them kinks, bumps and multiple wave solutions) have been obtained. These solutions remain bounded even when the Riemann invariants admit a gradient catastrophe. Some physical interpretation of these results is discussed.