Symmetry reduction for nonlinear relativistically invariant equations

Abstract

Symmetry reduction is studied for the relativistically invariant scalar partial differential equation H(⧠u,(∇u)2,u)=0 in (n+1)-dimensional Minkowski space M(n,1). The introduction of k symmetry variables ξ1, ... ,ξk as invariants of a subgroup G of the Poincaré group P(n,1), having generic orbits of codimension k≤n in M(n,1), reduces the equation to a PDE in k variables. All codimension-1 symmetry variables in M(n,1) (n arbitrary), reducing the equation studied to an ODE are found, as well as all codimension-2 and -3 variables for the low-dimensional cases n=2,3. The type of equation studied includes many cases of physical interest, in particular nonlinear Klein–Gordon equations (such as the sine–Gordon equation) and Hamilton–Jacobi equations.