Painlevé asymptotics for the KdV equation in the small dispersion limit

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 a 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 two different transitions from the Hopf asymptotics to the elliptic asymptotics. One transition is described by a higher order Painlev\'e I equation, another one by the Hastings-McLeod solution to the Painlev\'e II equation.

This is joint work with T. Grava.