Numerical study of a multiscale expansion of the Korteweg de Vries equation

Tamara Grava and Christian Klein

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Submission date: 19. Jun. 2007
Pages: 22
published in: Proceedings of the Royal Society of London / A, 464 (2008) 2091, p. 733-757 
DOI number (of the published article): 10.1098/rspa.2007.0249
Bibtex
with the following different title: Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painleve-II equation
Keywords and phrases: double scaling limits, Painleve equation, dispersive equation
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Abstract:
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order , , is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as in the interior of the Whitham oscillatory zone, it is known to be only of order near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlevé-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order .