A determination of the blowup solutions to the focusing NLS with mass equal to the mass of the soliton

TL;DR

This paper classifies all blowup solutions to the focusing mass-critical nonlinear Schrödinger equation with mass equal to the soliton, showing they are either solitons or their pseudoconformal transformations.

Contribution

It establishes a rigidity result that completely characterizes blowup solutions with critical mass in dimensions 2 to 15.

Findings

Only solitons and their pseudoconformal transformations blow up at critical mass.

Rigidity holds for dimensions 2 through 15.

Classifies blowup solutions in the mass-critical focusing NLS.

Abstract

In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{\"o}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons.