Dynamics of sine-Gordon solitons in the presence of perturbations

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Abstract
We have examined the dynamical behavior of soliton solutions of the sine-Gordon equation in the presence of weak external perturbations. Three examples, of particular importance in condensed matter, are described in detail: (i) a model impurity is found to bind low-velocity solitons but merely phase shift those with high velocities, (ii) an external static driving "force" together with damping causes the soliton to accelerate to a terminal velocity, and (iii) spatial inhomogeneities in the coefficient of the nonlinear term cause the soliton to adjust its velocity and shape in the regions of imperfection. In all cases we find that solitons maintain their integrity to a high degree. These calculations are based on a linear perturbation theory which emphasizes the use of a translation mode, and we are led to conclude that, in many respects, sine-Gordon solitons behave as classical particles whose dynamics are governed by Newton's law.