  We study the nonlinear-damping continuation of singular solutions of the
critical and supercritical NLS. Our simulations suggest that for generic
initial conditions that lead to collapse in the undamped NLS, the solution of
the weakly-damped NLS $$
i\psi_t(t,\X)+\Delta\psi+|\psi|^{p-1}\psi+i\delta|\psi|^{q-1}\psi=0,\qquad0<\delta
\ll 1, $$ is highly asymmetric with respect to the singularity time, and the
post-collapse defocusing velocity of the singular core goes to infinity as the
damping coefficient $\delta$ goes to zero. In the special case of the
minimal-power blowup solutions of the critical NLS, the continuation is a
minimal-power solution with a higher (but finite) defocusing velocity, whose
magnitude increases monotonically with the nonlinear damping exponent $q$.
