Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equations

Abstract:

We study a general class of focusing $L^2$-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 $K$ distinct singularities, $1\leq K<{\infty }$, and a regular profile. Moreover, we obtain the uniqueness in the energy class where the convergence rate is within the order $(T-t)^{4+}$, for $t$ close to the blow-up time $T$. 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.