# Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation - Nonlinear Sciences > Exactly Solvable and Integrable Systems

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation - Nonlinear Sciences > Exactly Solvable and Integrable Systems - Download this document for free, or read online. Document in PDF available to download.

Abstract: The use of the sine-Gordon equation as a model of magnetic flux propagationin Josephson junctions motivates studying the initial-value problem for thisequation 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 ofthis completely integrable Cauchy problem for all $\e>0$ sufficiently small,and further we invert the scattering transform to calculate the solution for asequence of arbitrarily small $\e$. This sequence of exact solutions isanalogous to that of the well-known $N$-soliton (or higher-order soliton)solutions of the focusing nonlinear Schr\-odinger equation. Plots of exactsolutions for small $\e$ reveal certain features that emerge in thesemiclassical limit. For example, in the limit $\epsilon\to 0$ one observes theappearance of nonlinear caustics. In the appendices we give a self containedaccount of the Cauchy problem from the perspectives of both inverse scatteringand classical analysis (Picard iteration). Specifically, Appendix A contains acomplete formulation of the inverse-scattering method for generic $L^1$-Sobolevinitial data, and Appendix B establishes the well-posedness for $L^p$-Sobolevinitial data (which in particular completely justifies the inverse-scatteringanalysis in Appendix A).

Author: ** Robert Buckingham Peter D. Miller**

Source: https://arxiv.org/