Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equations

Abstract

We study a general class of focusing L 2 L^2 -critical nonlinear Schrödinger equations with lower order perturbations, in the possible absence of the pseudo-conformal symmetry and the conservation law of energy. In dimensions one and two, we construct multi-bubble Bourgain-Wang type blow-up solutions, which behave like a sum of pseudo-conformal blow-up solutions that concentrate at K K distinct singularities, 1 ≤ K > ∞ 1\leq K>{\infty } , and a regular profile. Moreover, we obtain the uniqueness in the energy class where the convergence rate is within the order ( T − t ) 4 + (T-t)^{4+} , for t t close to the blow-up time T T . These results in particular apply to the canonical nonlinear Schrödinger equations and, through the pseudo-conformal transform, yield the existence and conditional uniqueness of non-pure multi-solitons, which behave asymptotically as a sum of multi-solitons and a dispersive part. Thus, the results provide new examples of the mass quantization conjecture and the soliton resolution conjecture. Furthermore, through a Doss-Sussman type transform, we obtain multi-bubble Bourgain-Wang solutions for stochastic nonlinear Schrödinger equations, where the driving noise is taken in the sense of controlled rough path.