Vortex Collapse for the L2-Critical Nonlinear Schr\"odinger Equation

TL;DR

This paper proves the existence of vortex collapse at a specific rate for the L2-critical nonlinear Schrödinger equation with spin m=1, extending previous work and highlighting potential unbounded mass concentration during collapse.

Contribution

It demonstrates vortex collapse with a log-log rate for m=1, expanding understanding of singularity formation in the L2-critical nonlinear Schrödinger equation.

Findings

Existence of vortex collapse at log-log rate for m=1

Extension of previous collapse results from m=0 to m=1

Discussion of challenges for m >= 2 or broken symmetry

Abstract

The focusing cubic nonlinear Schr\"odinger equation in two dimensions admits vortex solitons, standing wave solutions with spatial structure, Qm(r,theta) = e^{i m theta} Rm(r). In the case of spin m = 1, we prove there exists a class of data that collapse with the vortex soliton profile at the log-log rate. This extends the work of Merle and Rapha\"el, (the case m = 0,) and suggests that the L2 mass that may be concentrated at a point during generic collapse may be unbounded. Difficulties with m >= 2 or when breaking the spin symmetry are discussed.