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

TL;DR

This paper develops a geometric framework for constructing Riemann k-wave solutions to first-order quasilinear hyperbolic systems, combining symmetry reduction and characteristics methods, with applications to hydrodynamic systems.

Contribution

It introduces a unified geometric approach for Riemann k-wave solutions, integrating two existing methods and providing explicit formulas for solutions.

Findings

Derived conditions for existence of k-wave solutions.

Established a Frobenius theorem-based formula for solutions.

Applied the theory to hydrodynamic systems including Brownian motion.

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.