Multi-bubble Bourgain-Wang solutions to nonlinear Schr\"odinger equation

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.