Symmetry group of the nonlinear Klein-Gordon equation

Abstract

Maximal symmetry groups for nonlinear Klein-Gordon equations of soliton physics have been obtained using Lie's method (Rudra 1984) of extended groups. For general nonlinearity, including the sine-Gordon and the double sine-Gordon equation, the Poincare group is the maximal symmetry group. It has been shown that the symmetry group is larger than the Poincare group only for power law type nonlinearity V0 Psi n and exponential type nonlinearity V0exp(-n Psi ). For all exponential and power types with n not=0,1,3 the symmetry group is the Weyl group containing the extra scaling transformation over and above the Poincare transformations. When n=0 the symmetry group is the semi-direct product of the Abelian group of Psi translation and the Weyl group. When n=1 the symmetry group is the direct product of the Poincare group and the Abelian group of Psi scaling. When n=3 the conformal group is the maximal symmetry group.