Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

TL;DR

This paper introduces a generalized method for finding rank-k solutions of hyperbolic PDE systems using Riemann invariants, expanding the classical approach with specific side conditions and demonstrated through hydrodynamic examples.

Contribution

It develops a direct method with side conditions for rank-k solutions, generalizing Riemann invariant techniques for multi-dimensional hyperbolic systems.

Findings

Derived necessary and sufficient conditions for solutions in terms of Riemann invariants.

Obtained new classes of closed-form solutions for hydrodynamic systems.

Validated the approach with multiple illustrative examples.

Abstract

In this paper we employ a "direct method" in order to obtain rank-k solutions of any hyperbolic system of first order quasilinear differential equations in many dimensions. We discuss in detail the necessary and sufficient conditions for existence of these type of solutions written in terms of Riemann invariants. The most important characteristic of this approach is the introduction of specific first order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method of solving multi-dimensional systems of PDEs. We have demonstrated the usefulness of our approach through several examples of hydrodynamic type systems; new classes of solutions have been obtained in a closed form.