The sine-Gordon equation in the Semiclassical limit: Critical behavior near a separatrix

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.