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

T. Grava·C. Klein
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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

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T. Grava, C. Klein. 2007-02-12. Numerical study of a multiscale expansion of KdV and Camassa-Holm equation. https://arxiv.org/abs/math-ph/0702038

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