On the paper "Symmetry analysis of wave equation on sphere" by H. Azad and M. T. Mustafa

TL;DR

This paper extends the symmetry analysis of the wave equation on a sphere to nonlinear Klein-Gordon equations, utilizing scalar curvature and group classification on (pseudo) Riemannian manifolds.

Contribution

It introduces a comprehensive symmetry classification for nonlinear Klein-Gordon equations on the 2D sphere, expanding previous work on the wave equation.

Findings

Complete group classification of nonlinear Klein-Gordon equations on S^2

Extension of symmetry analysis methods to nonlinear equations

Identification of symmetries specific to the sphere geometry

Abstract

Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl., 333 (2007) 1180--1888] to nonlinear Klein-Gordon equations on the two-dimensional sphere.