Noether symmetries and conservation laws of a class of time-dependent multidimensional nonlinear wave equations

TL;DR

This paper derives conservation laws for a class of damped nonlinear wave equations using Noether's theorem, revealing symmetry structures and associated conserved quantities.

Contribution

It identifies the symmetry algebra of these equations and finds additional conservation laws for specific nonlinear forms, extending previous understanding.

Findings

Symmetry algebra includes Euclidean algebra $ ext{E}(n)$ for general damping and nonlinear terms.

Additional conservation laws emerge when symmetry algebra enlarges to a subalgebra of $ ext{Conf}(1,n)$.

Conservation of linear and angular momentum is established for the general case.

Abstract

Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational symmetries span a Euclidean algebra $\euclid(n)$ of space translations and rotations. They produce conservation of linear and angular momentums. For some specific forms of these two terms symmetry algebra is enlarged to a subalgebra of the conformal algebra $\conf(1,n)$ and in this case more interesting conservation laws are found.