Conservation laws and non-invariant solutions of anisotropic wave equations with a source

Abstract

Linear and nonlinear waves in anisotropic media are used in various fields, e.g. in biomechanics, biomedical acoustics, etc. The present paper is devoted to discussion of nonlinear anisotropic wave equations with a source from point of view of their conservation laws and exact solutions associated with conservation laws. Nonlinearly self-adjoint wave equations with special source terms are singled out.The conservation laws associated with symmetries of the nonlinearly self-adjoint wave equations are computed and used for constructing exact solutions. The obtained solutions are different from group invariants solutions, in particular, from steady state and traveling wave solutions.The paper is designed for the application oriented readers. Its main goal is to introduce readers, interested in solutions of mathematical models having real world applications, to the recent method of conservation laws for constructing exact solutions of partial differential equations using conservation laws.