2007/05/21 by Robert Buckingham Peter D. Miller, Miller, Robert Buckingham Peter D.
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Pattern Formation and Solitons (nlin.PS) #nlin.PS #nlin.SI
paper · pdf · doi:10.48550/arxiv.0705.3159
49 pages, 10 figures
arxiv created 2007/05/21 · arxiv updated 2009/12/01
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 -2π 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 Schrödinger equation. Plots of exact solutions for small \e reveal certain features that emerge in the semiclassical limit. For example, in the limit ε→ 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 L1-Sobolev initial data, and Appendix B establishes the well-posedness for Lp-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).