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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
2/2007

Numerical study of a multiscale expansion of KdV and Camassa-Holm equation

Tamara Grava and Christian Klein

Abstract

We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation.

Received:
Jan 9, 2007
Published:
Jan 9, 2007
MSC Codes:
54C40, 14E20, 46E25, 20C20
Keywords:
differential geometry, algebraic geometry

Related publications

inBook
2008 Repository Open Access
Tamara Grava and Christian Klein

Numerical study of a multiscale expansion of the Korteweg-de-Vries and Camassa-Holm equation

In: Integrable systems and random matrices : in honor of Percy Deift ; Conference on Integrable Systems, Random Matrices, and Applications in Honor of Percy Deift's 60th birthday, May 22 - 26, 2006, Courant Institute of Mathematical Sciences, New Yor Universi / Jinho Baik (ed.)
Providence, RI : American Mathematical Society, 2008. - pp. 81-98
(Contemporary mathematics ; 458)