Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

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.

30 pages

References in corpus (4)

• On integrability of (2+1)-dimensional quasilinear systems
• Integrable hydrodynamic chains
• The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type
• Hydrodynamic reductions of the heavenly equation

Cited by in corpus (6)

• Extended symmetry analysis of isothermal no-slip drift flux model
• Multimode solutions of first-order elliptic quasilinear systems obtained from Riemann invariants
• Elliptic solutions of isentropic ideal compressible fluid flow in (3+1) dimensions
• Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model
• Conditionally invariant solutions of the rotating shallow water wave equations
• Conditional symmetries and Riemann invariants for inhomogeneous hydrodynamic-type systems