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

TL;DR

This paper numerically investigates the behavior of solutions to the KdV and Camassa-Holm equations near singularity formation, comparing them with asymptotic solutions derived from Painlevé equations and Whitham theory.

Contribution

It introduces a numerical approach to validate asymptotic descriptions of KdV and Camassa-Holm solutions near breakup, connecting them with Painlevé I hierarchy solutions.

Findings

Painlevé I hierarchy solutions accurately describe KdV near breakup

Quantitative comparison with Hopf and Whitham equations

Qualitative insights into Camassa-Holm behavior

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 Painlev\'e 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