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We have developed a computer simulation program to study the dynamical behavior of soliton solutions of the sine-Gordon equation in the presence of external perturbations. Our work extends numerical and formal mathematical analysis on the sine-Gordon system in four directions. First, we demonstrate that lossless soliton propagation on a lattice is complicated by a lattice pinning effect and the generation of "harmonic excitations" as "radiation. " We define regimes according to the coefficient ₀^2 of the nonlinear potential term in which propagation can (₀^21) or cannot (₀^21) occur. Second, we study two examples of perturbation which are of particular importance in condensed matter: (i) a model impurity binds low-velocity solitons but merely space shifts those with high velocities, and (ii) spatial inhomogeneities in the coefficient of the nonlinear term ₀^2 cause the soliton to adjust its velocity and shape in the regions of imperfection. We find that the results of Fogel et al. , who treat these types of perturbation in a linear perturbation theory, are accurate to better than 25% as long as the small parameter does not exceed 0. 1. Third we demonstrate that their conclusion that solitons can be treated as classical particles obeying Newton's laws is in excellent agreement with the simulation results. Finally we indicate several applications of our simulation results for the propagation of a quantum of flux along a Josephson-junction transmission line.
Currie et al. (1977) studied this question.