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

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 Raphaël 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 Raphaël 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 $ε$ and find an exponential dependence $t^{*}(ε)$ and that there is a minimal blow-up time $t^{*}_{0}$ greater than the critical time of the corresponding Hopf solution for $ε\to0$. 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.