  In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical
gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time
$T<+\infty$. We expect to construct a natural extension of $u(t)$ after the
blow-up time. To do this, we consider the solution $u_{\gamma}(t)$ to the
saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-\gamma
u|u|^{q-1})_x=0$ with the same initial data, where $\gamma>0$ and $q>5$. A
standard argument shows that $u_{\gamma}(t)$ is always global in time and for
all $t<T$, $u_{\gamma}(t)$ converges to $u(t)$ in $H^1$ as
$\gamma\rightarrow0$. We prove in this paper that for all $t\geq T$,
$u_{\gamma}(t)$ converges to some $v(t)$ as $\gamma\rightarrow0$, in a certain
sense. This limiting function $v(t)$ is a weak solution to the unperturbed
$L^2$-critical gKdV, hence can be viewed as a natural extension of $u(t)$ after
the blow-up time.
