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

Abstract

Keywords: Lie point symmetry Noether symmetry Conservation laws Wave equations on the sphere Scalar curvature Using the scalar curvature of the product manifold S 2 × 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, M.T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl. 333 (2007) 1180-1188] to nonlinear Klein-Gordon equations on the two-dimensional sphere.

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

“…Theorem 1 applied to the situations studied in [43,5,23,39] immediately gives the results on group classification, obtained in these works, for semilinear Poisson and wave equations. In this way the projection on the space of independent variables of the symmetry group of the critical Klein-Gordon…”

“…For some applications of symmetries and conservation laws see [10,13,15,27,22]. We observe that in [26,27] Ibragimov has established the fact that the projection of the Lie symmetry group on the space of independent variables is a sub-group of the conformal group of the considered manifold.…”

“…In [22] one of us, using the results of the present paper, generalized [5] for the semilinear wave equation on the sphere S 2 :…”

“…(1). Examples of symmetry analysis involving some particular hyperbolic cases of (1) can be found in [5,22,24].…”

“…We merely mention the Poisson equations in R n , in particular those involving critical exponents, taking in (1) f (u) = u n+2 n−2 and the Euclidean metric; the Klein-Gordon equation, taking in (1) the metric ds 2 = −dt 2 + dx 2 + dy 2 + dz 2 and f (u) = u; the semilinear wave equations in R 1+n , with f (u) = 0 and the metric ds 2 = −dt 2 + δ ij dx i dx j in (1); and the Klein-Gordon equation on the S 2 sphere. These particular equations have been studied in [27,35,6,7,24,43,5,22,25,39]. The interested reader may also consult the book [27] of Ibragimov, where various aspects of symmetry analysis of differential equations on manifolds are presented.…”