Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

Abstract

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter $\e$ tends to zero. Assuming natural initial data having the profile of a moving $-2\pi$ kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all $\e>0$ sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small $\e$. This sequence of exact solutions is analogous to that of the well-known $N$-soliton (or higher-order soliton) solutions of the focusing nonlinear Schr\"odinger equation. Plots of exact solutions for small $\e$ reveal certain features that emerge in the semiclassical limit. For example, in the limit $\epsilon\to 0$ one observes the appearance of nonlinear caustics. In the appendices we give a self contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic $L^1$-Sobolev initial data, and Appendix B establishes the well-posedness for $L^p$-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).