Group-theoretical analysis of variable coefficient nonlinear telegraph equations

TL;DR

This paper conducts a comprehensive symmetry and conservation law classification of variable coefficient nonlinear telegraph equations, identifying new invariant models and deriving exact solutions through Lie and nonclassical symmetry methods.

Contribution

It introduces a systematic approach using gauges and furcate split to classify symmetries and conservation laws for these equations, revealing new invariant models.

Findings

New nonlinear invariant models with non-trivial invariance algebra

Exact solutions obtained via classical Lie reduction

Complete classification of nonclassical symmetries and conservation laws

Abstract

Given a class of differential equations with arbitrary element, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member the structure of its Lie symmetry group, conditional symmetry and conservation law under some proper equivalence transformations groups. In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable. The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical Lie reduction. The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.