%0 Journal Article
%J Proc. R. Soc. A 464 (2008) 733-757
%D 2008
%T Numerical study of a multiscale expansion of the Korteweg-de Vries equation and PainlevĂ©-II equation
%A Tamara Grava
%A Christian Klein
%X 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 $\\\\epsilon^{2/3}$.
%B Proc. R. Soc. A 464 (2008) 733-757
%G en_US
%U http://hdl.handle.net/1963/2592
%1 1530
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-02-25T15:08:44Z\\nNo. of bitstreams: 1\\n0708.0638v3.pdf: 453744 bytes, checksum: 05291095860df236125f0d9f8c676fbb (MD5)
%R 10.1098/rspa.2007.0249