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

 dc.contributor.author Grava, Tamara en_US dc.contributor.author Klein, Christian en_US dc.date.accessioned 2005 en_US dc.date.accessioned 2011-09-07T20:28:34Z dc.date.available 2005 en_US dc.date.available 2011-09-07T20:28:34Z dc.date.issued 2005 en_US dc.identifier.citation Comm. Pure Appl. Math. 60 (2007) 1623-1664 en_US dc.identifier.uri http://preprints.sissa.it/xmlui/handle/1963/1788 en_US dc.description.abstract The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\epsilon^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\epsilon$. 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 $\epsilon$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $\epsilon$ within the `interior' of the Whitham oscillatory zone, of order $\epsilon^{1/3}$ at the left boundary outside the Whitham zone and of order $\epsilon^{1/2}$ at the right boundary outside the Whitham zone. en_US dc.format.extent 905542 bytes en_US dc.format.mimetype application/pdf en_US dc.language.iso en_US en_US dc.relation.ispartofseries SISSA;91/2005/FM en_US dc.relation.ispartofseries arXiv.org;math-ph/0511011 en_US dc.relation.uri 10.1002/cpa.20183 en_US dc.title Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations en_US dc.type Preprint en_US dc.contributor.department Mathematical Physics en_US dc.contributor.area Mathematics en_US
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