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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q51, 35Q53; Secondary: 49S05, 81V25.

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