The coarse-grain description of interacting sine-Gordon solitons with varying widths
Ivan Christov - Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, United States (email)
Abstract: We study the dynamics of the sine-Gordon equation's kink soliton solutions under the coarse-grain description via two "collective variables": the position of the "center" of a soliton and its characteristic width ("size"). Integral expressions for the interaction potential and the quasi-particles' cross-masses are derived. However, these cannot be evaluated in closed form when the solitons have varying widths, so we develop a perturbation approach with the velocity of the faster soliton as the small parameter. This enables us to derive a system of four coupled second-order ODEs, one for each collective variable. The resulting initial-value problem is very stiff and numerical instabilities make it difficult to solve accurately, so a semi-empirical iterative approach to its solution is proposed. Then, we demonstrate that, even though it appears the solitons pass through each other, the quasi-particles actually "exchange" their pseudomasses during a collision.
Keywords: Solitons, Variational approximation, Quasi-particles, sine-Gordon equation, Nonlinear-wave quantization
Received: July 2008; Revised: June 2009; Published: September 2009.