Symmetry analysis of wave equation on sphere

Abstract

The symmetry classification problem for wave equation on sphere is considered. Symmetry algebra is found and a classification of its subalgebras, up to conjugacy, is obtained. Similarity reductions are performed for each class, and some examples of exact invariant solutions are given.

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…”

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

“…For the symmetry So ∞ , with g b = 0, the conservation law is Div(S) = 0, where S = (S 1 , S 2 , S 3 ) and S 1 = bu x − b x u, S 2 = e −2x bu y − e −2x b y u, 5. For the symmetry So ∞ , with g b + b = 0, the conservation law is Div(S) = 0, where S is given in (112).…”

“…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.…”

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