  We consider a general class of focusing $L^2$-critical nonlinear
Schr\"odinger equations with lower order perturbations, for which the
pseudo-conformal symmetry and the conservation law of energy are absent. In
dimensions one and two, we construct Bourgain-Wang type solutions concentrating
at $K$ distinct singularities, $1\leq K<\infty$, and prove that they are unique
if the asymptotic behavior is within the order $(T-t)^{4+}$, for $t$ close to
the blow-up time $T$. These results apply to the canonical nonlinear
Schr\"odinger equations and, through the pseudo-conformal transform, in
particular yield the existence and conditional uniqueness of non-pure
multi-solitons. Furthermore, through a Doss-Sussman type transform, these
results also apply to stochastic nonlinear Schr\"odinger equations, where the
driving noise is taken in the sense of controlled rough path.
