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

Claeys, Tom;Grava, T
(2010) Communications on Pure and Applied Mathematics — Vol. 63, n° 2, p. 203-232 (2010)

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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 Painleve II equation. We prove our results Using the Riemann-Hilbert approach. (C) 2009 Wiley Periodicals, Inc.
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Claeys, T., & Grava, T. (2010). Painleve II Asymptotics near the Leading Edge of the Oscillatory Zone for the Korteweg-de Vries Equation in the Small-Dispersion Limit. Communications on Pure and Applied Mathematics, 63(2), 203-232. https://doi.org/10.1002/cpa.20277 (Original work published 2010)