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Presented By: Department of Mathematics

Student AIM Seminar

Semi-classical sine-Gordon equation, universality at phase transition and the gradient catastrophe


In this talk I am going to discuss the universal behaviours of of the semi-classical limit of the sine-Gordon equation. We consider a class of solutions with pure impulse initial data below threshold that decay at $|x|\to \infty$. We are particularty interested in a neighbourhood of the gradient catastrophy that contains both modulated plane waves and spikes. We aim to describe the solutions using special funtions. Besides the gradient catastrophe point (we think of it as a more degenerate point than other generic locations of phase transition), we are also intested in describing when the first time the phase transition appear for genral location and the behaviours of the solutions nearby. These phase transitions have universality in the sense that the space-time locations and solutions behave the same way in the asymptotic limit for different initial data chosen from the class we consider. The location only depends on very simple parameters. We use the Deift-Zhou steepest descent method for the analysis and the result is inspired by Tovbis-Bertola. Speaker(s): Luby Lu (University of Michigan)

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