

Preprint 57/2007
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
.