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

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 the general dissipative version of Westervelt’s 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; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and higher-order conservation laws.