Critical asymptotics for the Korteweg-de Vries equation in the small dispersion limit

Tom Claeys

Tuesday 26th January, 2010 15:00-16:00 Mathematics Building, room 204

Abstract

We consider the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit. Up to a certain time, the solution to this problem can be approximated by the solution to the Hopf equation. After the time of gradient catastrophe for the Hopf equation, there is an interval where this is no longer the case. On this interval, the KdV solution oscillates heavily and can be approximated by elliptic theta-functions. We discuss three different transitions from the Hopf asymptotics to the elliptic asymptotics. One transition is described by a higher order Painlevé I equation, another one by the Hastings-McLeod solution to the Painlevé II equation, and the last one is a solitonic transition which is of different nature. The talk will be based on joint work with T. Grava.

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