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MiS Preprint
57/2007

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

Tamara Grava and Christian Klein

Abstract

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\epsilon^2$, $\epsilon\ll 1$, 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 $\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\epsilon^{1/3}$ 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 $\epsilon^{2/3}$.

Received:
Jun 19, 2007
Published:
Jun 19, 2007
Keywords:
double scaling limits, Painleve equation, dispersive equation

Related publications

inJournal
2008 Repository Open Access
Tamara Grava and Christian Klein

Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painleve-II equation

In: Proceedings of the Royal Society of London / A, 464 (2008) 2091, pp. 733-757