## Exact Solutions of Semiclassical Non-characteristic Cauchy Problems for the Sine-Gordon Equation

**Robert Buckingham and Peter D. Miller
Department of Mathematics, University of Michigan, Ann Arbor
**

### 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 *-2π* 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. Plots of exact solutions for
small *ε* reveal certain features that emerge in the semiclassical limit. For example, 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*-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).

^{p}*A density plot of the cosine of one of the semiclassical solutions of the sine-Gordon equation.*