Rigidity of smooth finite-time blow-up for equivariant self-dual Chern-Simons-Schr\"odinger equation

TL;DR

This paper classifies the long-term behavior of smooth, finite-time blow-up solutions to the equivariant self-dual Chern-Simons-Schr"odinger equation, showing they are essentially pseudoconformal blow-up solutions with a universal structure.

Contribution

It provides the first full classification of the dynamics, including scale and phase, for smooth solutions of a non-integrable nonlinear Schr"odinger equation with solitons.

Findings

Any finite-time blow-up solution decomposes into a contracting soliton and radiation.

Solutions either blow up in the pseudoconformal regime, scatter to zero, or scatter to a soliton.

The asymptotic profile has a universal singular structure.

Abstract

We consider the long time dynamics for the self-dual Chern-Simons-Schr\"odinger equation (CSS) within equivariant symmetry. (CSS) is a self-dual $L^{2}$-critical equation having pseudoconformal invariance and solitons. In this paper, we show that any $m$-equivariant, $m\geq1$, $H^{3}$ finite-time blow-up solution to (CSS) is a pseudoconformal blow-up solution. More precisely, such a solution decomposes into the sum of one modulated soliton that contracts at the pseudoconformal rate $\lambda(t)\sim T-t$, and a radiation. Applying the pseudoconformal transform in reverse, we also obtain a refined soliton resolution theorem for $m$-equivariant, $m\geq1$, $H^{3,3}$ solutions: any such solutions blow up in the pseudoconformal regime, scatter (to $0$), or scatter to a modulated soliton with some fixed scale and phase. To our knowledge, this is the first result on the full classification of the dynamics of arbitrary smooth and spatially decaying solutions, including the dynamics of scale and phase, in the class of nonlinear Schr\"odinger equations which are not known to be completely integrable. Our analysis not only builds upon the previous works, especially the soliton resolution theorem by the author, Kwon, and Oh, but also refines all steps of the arguments typically employed in the forward construction of blow-up dynamics. The key feature of the proof is that we can identify the singular and regular parts of any $H^{3}$ finite-time blow-up solutions, such that the evolution of the singular part is governed by a universal modulation dynamics while the regular part is kept $H^{3}$-bounded even up to the blow-up time. As a byproduct, we also observe that the asymptotic profile has a universal singular structure.