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The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
Robert J. Buckingham, University of Cincinnati, OH, and Peter D. Miller, University of Michigan, Ann Arbor, MI
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Memoirs of the American Mathematical Society
2013; 136 pp; softcover
Volume: 225
ISBN-10: 0-8218-8545-6
ISBN-13: 978-0-8218-8545-1
List Price: US$74 Individual Members: US$44.40
Institutional Members: US\$59.20
Order Code: MEMO/225/1059

The authors 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. They 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.

• Introduction
• Formulation of the inverse problem for fluxon condensates
• Elementary transformations of $$\mathbf{J}(w)$$
• Construction of $$g(w)$$
• Use of $$g(w)$$
• Appendix A. Proofs of propositions concerning initial data
• Appendix B. Details of the outer parametrix in cases $$\mathsf{L}$$ and $$\mathsf{R}$$
• Bibliography