Breather-breather interactions in sine-Gordon systems using collective coordinate approach

Abstract

We investigate the interaction between breathers in sine-Gordon systems using a collective coordinate approach. Focusing on the in-phase or out-of-phase oscillation mode of two identical breathers, we derive a simple ordinary differential equation for the collective coordinate: the separation between the centers of mass of the two breathers R. Analytic solutions of this equation can reproduce quantitatively the results of a direct numerical simulation of the sine-Gordon equation over the whole parameter range. The interaction between the two breathers is attractive (repulsive) for the in-phase (out-of-phase) oscillation mode with an asymptotic exponential dependence on R. We find that the interaction within the innermost kink-antikink (kink-kink) pair for the in-phase (out-of-phase) case plays the most significant role in determining the sign and the strength of the effective breather-breather interaction. We also find an internal oscillation mode of the in-phase oscillating breather pair and obtain an analytic expression for the angular frequency of the mode.