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

TL;DR

This paper derives a universal asymptotic expansion involving Painleve II solutions for the Korteweg-de Vries equation's oscillatory behavior near the leading edge in the small dispersion limit, using Riemann-Hilbert techniques.

Contribution

It provides the first detailed second-order asymptotic expansion near the oscillatory zone's leading edge for KdV solutions in the small dispersion regime.

Findings

Universal asymptotic expansion involving Painleve II

Second-order corrections to the leading edge behavior

Validation via Riemann-Hilbert analysis

Abstract

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.