Static analysis for coupled nonlinear Klein-Gordon equations with asymmetric parameter settings

Abstract

Klein-Gordon equations describe the dynamics of waves/particles in sub-atomic scales. For a system of nonlinear Klein-Gordon equations, a systematic analysis of the time evolution for their spatially uniform solutions has been performed \cite{21takei}. In the study, the parameters (mass, wave propagation speed, and the force parameters) are chosen to be symmetric between the two single equations. Symmetric parameter settings are equivalent to assume the interacting two same particles. In this paper, for a system of nonlinear Klein-Gordon equations with asymmetric parameter settings, the time evolution for their spatially uniform solutions are studied. This is equivalent to assume the interacting two different particles. As a result, based on the high precision numerical scheme \cite{22takei}, the existence of divergent and bounded solutions that depend on parameter settings is revealed. The competition, coherence, and decoherence of different waves are shown to appear depending on the choice of asymmetrically-implemented parameter values.