Vortex dynamics for nonlinear Klein–Gordon equation

We consider the vortex motion laws governed by the nonlinear Klein–Gordon equation. Three limits are determined when the light speed c→∞ and ϵ→0. It is shown that the vortex motion law governed by the nonlinear Klein–Gordon equation is an intermediate state that connects both the Kirchhoffʼs law for the Schrödinger equation and the vortex motion law for the wave equation.