  We fully revisit the near soliton dynamics for the mass critical (gKdV)
equation.
  In Part I, for a class of initial data close to the soliton, we prove that
only three scenario can occur:
  (BLOW UP) the solution blows up in finite time $T$ in a universal regime with
speed $1/(T-t)$;
  (SOLITON) the solution is global and converges to a soliton in large time;
  (EXIT) the solution leaves any small neighborhood of the modulated family of
solitons in the scale invariant $L^2$ norm.
  Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this
class that any nonpositive energy initial data (except solitons) yields finite
time blow up, thus obtaining the classification of the solitary wave at zero
energy.
  In Part II, we classify minimal mass blow up by proving existence and
uniqueness (up to invariances of the equation) of a minimal mass blow up
solution $S(t)$. We also completely describe the blow up behavior of $S(t)$.
  Second, we prove that $S(t)$ is the universal attractor in the (EXIT) case,
i.e. any solution as above in the (EXIT) case is close to $S$ (up to
invariances) in $L^2$ at the exit time. In particular, assuming scattering for
$S(t)$ (in large positive time), we obtain that any solution in the (EXIT)
scenario also scatters, thus achieving the description of the near soliton
dynamics.
