Symmetry analysis and hidden variational structure of Westervelt's equation in nonlinear acoustics

TL;DR

This paper explores the symmetry properties and hidden variational structures of Westervelt's equation in nonlinear acoustics, revealing new conservation laws, exact solutions, and connections to linear wave equations.

Contribution

It introduces novel conserved integrals, hidden symmetries, and variational structures, advancing the understanding of Westervelt's equation's mathematical properties.

Findings

New conserved integrals identified

Mapping to linear wave equation achieved

Hidden variational structures uncovered

Abstract

Westervelt's equation is a nonlinear wave equation that is widely used to model the propagation of sound waves in a compressible medium, with one important application being ultra-sound in human tissue. Two fundamental aspects of this equation -- symmetries and conservation laws -- are studied in the present work by modern methods. Numerous results are obtained: new conserved integrals; potential systems yielding hidden symmetries and nonlocal conservation laws; mapping of Westervelt's equation in the undamped case into a linear wave equation; exact solutions arising from the mapping; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and conservation laws.