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This paper has been published in Journal d'Analyse Mathématique, volume 118, pages 397-492. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0807653. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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The Sine-Gordon Equation in the Semiclassical Limit: Critical Behavior Near a Separatrix

Robert J. Buckingham and Peter D. Miller
Department of Mathematical Sciences, University of Cincinnati
Department of Mathematics, University of Michigan, Ann Arbor


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. Subject to suitable conditions of a general nature, we analyze the fluxon condensate solution approximating the given initial data for small time near points where the initial data crosses the separatrix of the phase portrait of the simple pendulum. We show that the solution is locally constructed as a universal curvilinear grid of superluminal kinks and grazing collisions thereof, with the grid curves being determined from rational solutions of the Painlevé-II system.