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

17 pages, 13 figures

References in corpus (1)

• On Hamiltonian perturbations of hyperbolic systems of conservation laws

Cited by in corpus (6)

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• Regularization of Hele-Shaw flows, multiscaling expansions and the Painleve I equation
• Numerical study of the long wavelength limit of the Toda lattice
• Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations
• On universality of critical behaviour in Hamiltonian PDEs
• Painleve II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small dispersion limit