Recent progress on multi-bubble blow-ups and multi-solitons to (stochastic) focusing nonlinear Schr\"odinger equations

TL;DR

This paper reviews recent advances in understanding the long-time behavior of focusing nonlinear Schrödinger equations, including stochastic and deterministic models, highlighting new constructions of blow-up solutions and solitons, and their qualitative properties.

Contribution

It introduces new constructions of multi-bubble blow-up solutions and multi-solitons, and analyzes their uniqueness and qualitative properties in both stochastic and deterministic settings.

Findings

Construction and uniqueness of multi-bubble blow-up solutions

Existence of non-pure multi-solitons

Qualitative properties of stochastic blow-up solutions

Abstract

We review the recent progress on the long-time behavior for a general class of focusing $L^2$-critical nonlinear Schr\"odinger equations (NLS) with lower order perturbations. Two canonical models are the stochastic NLS driven by linear multiplicative noise and the classical deterministic NLS. We show the construction and uniqueness of the corresponding blow-up solutions and solitons, including the multi-bubble Bourgain-Wang type blow-up solutions and non-pure multi-solitons, which provide new examples for the mass quantization conjecture and the soliton resolution conjecture. The refined uniqueness of pure multi-bubble blow-ups and pure multi-solitons to NLS under very low asymptotical rate is also reviewed. Finally, as a new result, we prove the qualitative properties of stochastic blow-up solutions, including the concentration of mass, universality of critical mass blow-up profiles, as well as the vanishing of the virial at the blow-up time.