Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations

TL;DR

This paper numerically investigates the formation of dispersive shocks in generalized Korteweg-de Vries and Kawahara equations, revealing local behaviors near gradient catastrophe points and providing higher-order correction approximations.

Contribution

It introduces a numerical analysis of dispersive shock formation in Hamiltonian regularizations, linking local solutions to Painlevé-type equations and deriving higher-order dispersive corrections.

Findings

Local behavior near gradient catastrophe described by Painlevé-type solutions

Dispersive solutions approximate transport solutions with order ε^2 accuracy

Higher-order ε^4 corrections are derived and validated numerically

Abstract

This article is concerned with a conjecture by one of the authors on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasilinear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behaviour of their solution near the point of gradient catastrophe for the transport equation is described locally by a special solution of a Painlev\'e-type equation. This local description holds also for solutions to equations where blow up can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\epsilon^2$ where $\epsilon^2$ is the small dispersion parameter. Corrections up to order $\epsilon^4 $ are obtained and tested numerically.