Interacting sine-Gordon solitons and classical particles: A dynamic equivalence

Abstract

Motivated by the work of Fogel et al., who showed that the dynamics of the classical sine-Gordon soliton in the presence of external perturbations is essentially the dynamics of a Newtonian particle, we have looked for such particlelike behavior in those cases where sine-Gordon solitons interact with each other rather than with an applied field. We find that, although a direct soliton-particle equivalence is not possible in these cases, there is an indirect representation in terms of the poles of the Hamiltonian density when the latter is assumed to be a function of a complex space variable. These poles have well-defined motions in the complex space plane, and we show that there is a strong correlation between the dynamics of the poles, considered as classical particles, and those of the interacting solitons they represent. Thus, all the qualitative features of the two-soliton (and/or antisoliton) interactions, as well as analytic expressions for the forces between them, are predicted by analyzing the motions of the poles of the Hamiltonian density in the complex space plane. If this "particle" picture is used to identify the individual silitons in a two-soliton profile, then it turns out that the solitons move at a constant speed, but with a changing rest mass. Hence, solitons do not maintain their "free" mass identity when interacting with each other. A few analogies with quantum behavior are also discussed.