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

A. M. Grundland, J. de Lucas

TL;DR

The paper develops a geometric framework to construct $k$-wave solutions for general first-order quasilinear hyperbolic systems by combining symmetry reduction with the generalized method of characteristics. It establishes existence criteria via a modified Frobenius approach, introduces hodograph-type representations with potential wave functions, and provides closed-form constructions of multi-wave solutions, including conditions for inhomogeneous-to-homogeneous reductions. The authors illustrate the theory with hydrodynamic-type examples, notably the Brownian motion equation, demonstrating simple and double Riemann waves and their nonlinear superpositions. This work offers a systematic, geometry-driven pathway to generate and analyze nonlinear Riemann wave superpositions in multidimensional hyperbolic PDEs with potential applications in fluid dynamics and field theories.

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.

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

TL;DR

The paper develops a geometric framework to construct -wave solutions for general first-order quasilinear hyperbolic systems by combining symmetry reduction with the generalized method of characteristics. It establishes existence criteria via a modified Frobenius approach, introduces hodograph-type representations with potential wave functions, and provides closed-form constructions of multi-wave solutions, including conditions for inhomogeneous-to-homogeneous reductions. The authors illustrate the theory with hydrodynamic-type examples, notably the Brownian motion equation, demonstrating simple and double Riemann waves and their nonlinear superpositions. This work offers a systematic, geometry-driven pathway to generate and analyze nonlinear Riemann wave superpositions in multidimensional hyperbolic PDEs with potential applications in fluid dynamics and field theories.

Abstract

The objective of this paper is to construct geometrically Riemann -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 -waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the -wave problem and discuss in detail the conditions for the existence of -wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the -wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.
Paper Structure (9 sections, 4 theorems, 205 equations)

This paper contains 9 sections, 4 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Riemann invariants
  3. A general inhomogeneous first-order system
  4. Reduction to homogeneous first-order systems
  5. On $k$-wave solutions
  6. Conditions for the existence of $k$-wave solutions
  7. A geometric approach to hydrodynamic equations
  8. Examples
  9. Acknowledgements

Key Result

Proposition 6.1

(Simple Riemann wave solutions) Suppose that there exists a map $\varphi:\mathbb{R}^{p+q}\rightarrow \mathbb{R}$ such that the set of implicitly defined relations between the variables $u,x$, and $s$, namely can be solved so that $s$ and $u$ can be given as a graph over an open subset $D\subset E$, i.e. one has $s=s(x)$ and $u=f(x)$ for $x$ in an open subset $A\subset \mathbb{R}^p$. Assume that $

Theorems & Definitions (7)

  • Proposition 6.1
  • proof
  • Theorem 6.2
  • proof
  • Proposition 7.1
  • Theorem 8.1
  • proof