Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations

TL;DR

This paper conducts a comprehensive numerical analysis of solutions to generalized Korteweg-de Vries equations, focusing on soliton stability, blow-up mechanisms, and the influence of small dispersion, revealing new insights into singularity formation.

Contribution

It numerically identifies blow-up mechanisms and scaling laws in generalized KdV equations, extending theoretical results to broader initial data and small dispersion limits.

Findings

Solitons are unstable and tend to blow up or radiate away.

In the critical case, the Martel-Merle-Raphaël blow-up mechanism is confirmed numerically.

Blow-up time scales exponentially with the small dispersion parameter.

Abstract

We present a detailed numerical study of solutions to general Korteweg-de Vries equations with critical and supercritical nonlinearity. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the $L_{2}$ critical case, the blow-up mechanism by Martel, Merle and Rapha\"el can be numerically identified. In the limit of small dispersion, it is shown that a dispersive shock always appears before an eventual blow-up. In the latter case, always the first soliton to appear will blow up. It is shown that the same type of blow-up as for the perturbations of the soliton can be observed which indicates that the theory by Martel, Merle and Rapha\"el is also applicable to initial data with a mass much larger than the soliton mass. We study the scaling of the blow-up time $t^{*}$ in dependence of the small dispersion parameter $\epsilon$ and find an exponential dependence $t^{*}(\epsilon)$ and that there is a minimal blow-up time $t^{*}_{0}$ greater than the critical time of the corresponding Hopf solution for $\epsilon\to0$. To study the cases with blow-up in detail, we apply the first dynamic rescaling for generalized Korteweg-de Vries equations. This allows to identify the type of the singularity.