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.
Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg - de Vries equation in the small-dispersion limit
Grava, Tamara
2010-01-01
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.File in questo prodotto:
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