Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations

Tamara Grava and Christian Klein

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Submission date: 14. Nov. 2005
Pages: 41
published in: Communications on pure and applied mathematics, 60 (2007) 11, p. 1623-1664 
DOI number (of the published article): 10.1002/cpa.20183
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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 of wave-length of order . 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. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of between and . The numerical results are compatible with a difference of order within the `interior' of the Whitham oscillatory zone, of order at the left boundary outside the Whitham zone and of order at the right boundary outside the Whitham zone.