Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg - de Vries equation in the small-dispersion limit

In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\'e II equation. We prove our results using the Riemann-Hilbert approach