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 ε tends to zero. Assuming natural initial data having the profile of a moving − 2 π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε > 0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε . This sequence of exact solutions is analogous to that of the well-known N -soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε ↓ 0 one observes the appearance of nonlinear caustics, i.e. curves in space–time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases. 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).