Conservation laws and exact solutions of a nonlinear acoustics equation by classical symmetry reduction

TL;DR

This paper analyzes symmetries and conservation laws of a nonlinear acoustics equation, deriving solutions like shock waves relevant for biomedical ultrasound applications.

Contribution

It provides a complete symmetry classification and conservation laws for a generalized Westervelt equation, including potential symmetries and explicit traveling wave solutions.

Findings

Classification of point symmetries for the equation

Derivation of local conservation laws related to sound wave mass

Traveling wave solutions leading to shock waves

Abstract

Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely used in nonlinear acoustics and it is especially important in biomedical applications such as ultra-sound imaging in human tissue. Modern methods are applied to uncover point symmetries and conservation laws that can lead to useful developments concerning solutions and their properties. A complete classification of point symmetries is shown for the arbitrary function. Local low-order conservation laws related to net mass of sound waves are obtained by the multiplier method. Two potential systems are derived yielding potential symmetries and nonlocal conservation laws. For the physical case interesting for this equation, travelling wave solutions are studied leading to shock waves.