  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.
