Numerical simulation of sine-Gordon soliton dynamics in the presence of perturbations

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 ${\omega }_0^2$ of the nonlinear potential term in which propagation can (${\omega }_0^2\lesssim 1$) or cannot (${\omega }_0^2\gtrsim 1$) 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 ${\omega }_0^2$ 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 $\varphi$ 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.