Theory of boundary effects on sine-Gordon solitons

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Abstract
We examine the properties of solutions to the sine‐Gordon equation in the presence of various boundary conditions. Reflection from the boundary of a semi‐infinite system with a fixed or free endpoint is found to be explainable in terms of the standard soliton‐soliton and soliton‐antisoliton solutions, respectively, for the infinite system. We also consider both nonlinear standing‐wave and solitonic solutions to the sine‐Gordon equation on a system of finite length L. Through the use of the application by Costabile et al. of the separation of variables ansatz due to Lamb, analytic expressions in terms of Jacobi elliptic functions are found for these two types of solutions, which assume the forms appropriate to the infinite and semi‐infinite system, respectively, as L→∞. Concentrating on the solitonic solution, we examine its symmetry properties, relations among its characteristic parameters, and give plots of its waveform in several cases. The effect of the boundaries on this solution is such that the oscillation of the soliton center closely resembles that of a particle in a symmetric potential, whose shape is determined straightforwardly from the time dependence of the soliton position and of the potential energy of the system.