Some Remarks on Similarity and Soliton Solutions of Nonlinear Klein-Gordon Equation

Abstract

The three-dimensional nonlinear Klein-Gordon [, Higgs field and Yang-Milles] (3D–KG [, H and YM]) equation is first reduced to the 2D nonlinear Schrödinger (2D–NLS) and 2D–KG [, H and YM] equations, and secondly to the 1D–NLS and 1D–KG [, H and YM] equations by similarity transformations. It is shown that similar type soliton solutions of the 3D–KG, H and YM equations, which have singularity on a plane in (x, y, z, t) space, are obtained by substituting the soliton solutions of the 1D–NLS or 1D–KG (or H) equation into the similarity transformations. The soliton solutions of the YM equation are also investigated.