The coarse-grain description of interacting sine-Gordon solitons with varying widths

Pages: 171 - 180, Issue Special, September 2009

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Ivan Christov - Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, United States (email)
C. I. Christov - Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, 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
Mathematics Subject Classification:  Primary: 35Q51, 35Q53; Secondary: 49S05, 81V25

Received: July 2008;      Revised: June 2009;      Published: September 2009.