Painleve II Asymptotics near the Leading Edge of the Oscillatory Zone for the Korteweg-de Vries Equation in the Small-Dispersion Limit

摘要

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 Painleve II equation. We prove our results Using the Riemann-Hilbert approach.