Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equation

Abstract

We consider a general class of focusing $L^2$-critical nonlinear Schrödinger 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, $1\le K<\infty$, 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 Schrödinger 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 Schrödinger equations, where the driving noise is taken in the sense of controlled rough path.