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# memo_has_moved_text();The sine-Gordon equation in the semiclassical limit: Dynamics of fluxon condensates

Robert J. Buckingham, Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, Ohio 45221 and Peter D. Miller, Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, Michigan 48109

Publication: Memoirs of the American Mathematical Society
Publication Year 2013: Volume 225, Number 1059
ISBNs: 978-0-8218-8545-1 (print); 978-1-4704-1059-9 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00672-1
Published electronically: February 20, 2013
Keywords:Sine-Gordon, semiclassical asymptotics, Riemann-Hilbert problems
MSC (2010): Primary 35Q51, 35B40, 35Q15

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Chapters

• Chapter 1. Introduction
• Chapter 2. Formulation of the Inverse Problem for Fluxon Condensates
• Chapter 3. Elementary Transformations of $\mathbf {J}(w)$
• Chapter 4. Construction of $g(w)$
• Chapter 5. Use of $g(w)$
• Appendix A. Proofs of Propositions Concerning Initial Data
• Appendix B. Details of the Outer Parametrix in Cases $\mathsf {L}$ and $\mathsf {R}$

### Abstract

We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. We show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham’s formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.