Seminar Event Detail


Integrable Systems and Random Matrix Theory

Date:  Monday, January 22, 2018
Location:  1866 East Hall (4:00 PM to 5:00 PM)

Title:  Semi-Classical Sine-Gordan Equation, Universality at the Gradient Catastrophe

Abstract:   We consider a class of solutions with pure impulse initial data below critical value such that within small time only librational-type waves are generated and the solutions should decay when |x| goes to infinity. In a neighbourhood of a certain gradient catastrophe point that contains both modulated plane waves and localized structures or ''spikes'', the asymptotic behaviour of the solutions can be universally described by analyzing a Riemann-Hilbert problem related to Painleve I equation Tritronquee solutions. It is a well-known fact that the solutions to Painleve equations have poles. In fact we show the locations of the poles are directly linked to where the ''spikes'' happen. In suitable scaling limit, we are able describe the first correction of the solution (compared to before breaking happens) using Painleve I Tritronquee solution away from the ''spikes''. I will explain how I want to then modify the Riemann-Hilbert problem to describe the ''spike'' shape. Notice that this is universal in the sense that the local asymptotics is not sensitive to the initial condition as long as it falls into a large class; it is only the space-time location of the transition that depends on the initial data. As a sidenote, the unique symmetry of the catastrophe location will also result in a symmetry of the corresponding Riemann-Hilbert problem, which allows us to describe the solution in more details. We lose this property when the initial condition is no longer an even function. Our technique is the Deift-Zhou steepest descent method related to an approach of universality for the focusing nonlinear Schrodinger equation.

Files:


Speaker:  Bingying Lu
Institution:  University of Michigan

Event Organizer:   Thomas Bothner and Guilherme Silva    bothner@umich.edu, silvag@umich.edu

 

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