Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries

TL;DR

This paper develops a method combining approximate symmetries and singularity analysis to find exact invariant solutions of a nonlinear wave equation with small dissipation.

Contribution

It introduces a systematic approach using Lie symmetry algebra and subalgebra classification to derive new invariant solutions for the nonlinear wave equation.

Findings

Identified conjugacy classes of subalgebras for the equation

Found new invariant solutions via symmetry reduction

Extended symmetry analysis to integro-differential form

Abstract

In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra of this equation and identify the conjugacy classes of the one-dimensional subalgebras of this Lie algebra. We show that the subalgebra classification of the integro-differential form of the nonlinear wave equation is much larger than the one obtained from the original wave equation. A systematic use of the symmetry reduction method allows us to find new invariant solutions of this wave equation.