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 $t^{\ast }$ in dependence of the small dispersion parameter $\epsilon$ and find an exponential dependence $t^{\ast }(\epsilon )$ and that there is a minimal blow-up time $t_0^{\ast }$ greater than the critical time of the corresponding Hopf solution for $\epsilon \to 0$. 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.

Major changes, small dispersion limit added

References in corpus (2)

• Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations
• Blow up for the critical gKdV equation III: exotic regimes

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