@article {2008,
title = {Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlev{\'e}-II equation},
journal = {Proc. R. Soc. A 464 (2008) 733-757},
number = {arXiv.org;0708.0638v3},
year = {2008},
abstract = {The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\e^2$, $\\\\e\\\\ll 1$, 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 Painlev\\\\\\\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\\\\epsilon^{2/3}$.},
doi = {10.1098/rspa.2007.0249},
url = {http://hdl.handle.net/1963/2592},
author = {Tamara Grava and Christian Klein}
}