Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries
Willy Hereman, Rehana Naz
TL;DR
The paper analyzes conservation laws for nonlinear PDEs arising in elasticity and acoustics across Cartesian, cylindrical, and spherical geometries. It employs two complementary techniques—the scaling-homogeneity method and the multiplier method—to derive conserved densities and fluxes for shear-wave models in cylindrical coordinates and for Khokhlov-Zabolotskaya-Kuznetsov (KZK) and Westervelt-type equations in multiple coordinate systems, revealing infinite families of conservation laws in certain geometries. A generalized implicit constitutive framework with $\epsilon = F(\sigma)$ is developed, yielding densities independent of $F$ while fluxes depend on integrals of $F$, and the results connect to KZK and Westervelt-type models in higher dimensions. The work provides computational tools (e.g., ConservationLawsMD.m, GeM) and advances understanding of conservation laws in nonlinear elasticity and acoustics with potential implications for stability analyses and numerical methods.
Abstract
Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave propagation in a circular cylinder and a cylindrical annulus. Next, using the multiplier method, conservation laws are derived for a parameterized system of constitutive equations in cylindrical coordinates involving a general expression for the Cauchy stress. Conservation laws for the Khokhlov-Zabolotskaya-Kuznetsov equation and Westervelt-type equations in various coordinate systems are also presented.