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arXiv · 0812.4142

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

T. Claeys·T. Grava
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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 Painlevé II equation. We prove our results using the Riemann-Hilbert approach.

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T. Claeys, T. Grava. 2008-12-22. Painleve II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small dispersion limit. https://arxiv.org/abs/0812.4142

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