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

TL;DR

This paper derives exact semiclassical solutions for the sine-Gordon equation with specific initial data, revealing nonlinear caustics and advancing understanding of integrable systems in the small dispersion limit.

Contribution

It analytically computes scattering data and exact solutions for the sine-Gordon equation in the semiclassical limit, including a comprehensive inverse scattering framework.

Findings

Emergence of nonlinear caustics as  o 0

Explicit exact solutions for small 

Validation of inverse scattering for initial data in Sobolev spaces

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).