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

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

arXiv comment17 pages, 13 figures

References in corpus (3)

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• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• On Hamiltonian perturbations of hyperbolic systems of conservation laws

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