Symmetry Reduction for Nonlinear wave Equations in Riemannian and Pseudo-Riemannian Spaces

Abstract

This chapter discusses symmetry reduction for nonlinear wave equations in Riemannian and pseudo -Riemannian spaces. Nonlinear equations of the form H(Δu, u) = 0 are discussed, where H is an arbitrary sufficiently smooth function of three variables, u =u(x 1 …, x n ) scalar function and Δand (▽.) 2 are the Laplace and gradient square operators on some n-dimensional Riemannian or pseudo -Riemannian manifold M. It is shown that this equation can be reduced to a differential equation in k (1 ≤ k ≤ n) variables by making the Ansatz that one should depend only on some new variables ξa (x 1 ,….,x n ) (a=1,…,k). The chapter discusses “degenerate symmetry variables” involving arbitrary functions of null variables in pseudo -Riemannian spaces.