A pseudoconformal compactification of the nonlinear Schr\"odinger equation and applications

TL;DR

This paper introduces a pseudoconformal compactification of the nonlinear Schrödinger equation via lens transformation, providing a new perspective that links free and potential-including NLS, and reinterprets existing results within this framework.

Contribution

It presents a novel interpretation of the lens transformation as a pseudoconformal compactification, connecting different forms of NLS and offering new insights into their analysis.

Findings

Reinterpreted existing NLS results in the compactified setting

Established a link between free and harmonic potential NLS

Provided a new geometric perspective on NLS transformations

Abstract

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schr\"odinger equation (NLS) without potential with a nonlinear Schr\"odinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offering a new perspective to view this theory.