  In this paper we consider the slightly $L^2$-supercritical gKdV equations
$\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity
$5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and
stability of a blow-up dynamic with self-similar blow-up rate in the energy
space $H^1$ and give a specific description of the formation of the singularity
near the blow-up time.
