Conservation Laws for the Nonlinear Klein-Gordon Equation in (1+1)-, (2+1), and (3+1)-dimensions

Abstract

We study soliton solutions to the Klein-Gordon equation via Lie symmetries and the travelling-wave ansatz. It is shown, by taking a linear combination of the spatial and temporal Lie point symmetries, that soliton solutions naturally exist, and the resulting field lies in the complex plane. We normalize the field over a finite spatial interval, and thereafter, specify one of the integration constants in terms of the other. Solutions to a specific type of nonlinear Klein-Gordon equation are studied via the sine-cosine method, and a real soliton wave is obtained. Lastly, the multiplier method is used to construct conservation laws for this particular nonlinear Klein-Gordon equation in (3 + 1)-dimensions.