Asymptotics for the Korteweg-de Vries equation and perturbations using Riemann-Hilbert methods
Small dispersion asymptotics for solutions to the Korteweg-de Vries equation can be obtained using Riemann-Hilbert problems. In critical regimes, this leads to asymptotic expansions in terms of Painlev\'e transcendents. I will give an overview of this procedure, and I will discuss some obstacles that occur when considering non-integrable perturbations of the Korteweg-de Vries equation.
This is based on joint work with Tamara Grava, and on joint work in progress with Tamara Grava and Ken McLaughlin.