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

revised version: January 2007
Tamara Grava, and Christian Klein
(Please use for correspondence this email).

Submission date: 09. Jan. 2007
Pages: 19
published 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 University / J. Baik (ed.)
Providence, RI : American Mathematical Society, 2008. - P. 81 - 98
(Contemporary mathematics ; 458) 
with the following different title: Numerical study of a multiscale expansion of the Korteweg-de-Vries and Camassa-Holm equation
MSC-Numbers: 54C40, 14E20, 46E25, 20C20
Keywords and phrases: differential geometry, algebraic geometry
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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.