Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

TL;DR

This paper performs a comprehensive group classification of variable coefficient (1+1)-dimensional telegraph equations, identifying invariant models, transformations, exact solutions, and conservation laws using Lie symmetry methods.

Contribution

It provides a complete classification of symmetries and solutions for a broad class of variable coefficient telegraph equations, including new invariant models and conservation laws.

Findings

New nonlinear invariant models with non-trivial invariance algebras

Exact solutions constructed via Lie and conditional transformations

Classification of local conservation laws with order 0

Abstract

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Furthermore, the possible additional equivalence transformations between equations from the class under consideration are investigated. Exact solutions of special forms of these equations are also constructed via classical Lie method and generalized conditional transformations. Local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations.