Singularity formation for the 1-D cubic NLS and the Schr\"odinger map on $\mathbb S^2$

TL;DR

This paper investigates singularity formation in the 1-D cubic NLS and Schr"odinger map on the sphere, revealing geometric quantities that become singular and connecting these to fluid mechanics phenomena.

Contribution

It introduces a geometric framework to define solutions beyond singularities for the 1-D cubic NLS with delta-like initial data and links these to fluid mechanics phenomena.

Findings

Identification of geometric quantities that are singular at initial time.

Use of geometric framework to extend solutions beyond singularities.

Connection between singularities in NLS and fluid mechanics phenomena.

Abstract

In this note we consider the 1-D cubic Schr\"odinger equation with data given as small perturbations of a Dirac-$\delta$ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schr\"odinger map to define the solution beyond the singularity time. Then, we find some natural and well defined geometric quantities that are not regular at time zero. Finally, we make a link between these results and some known phenomena in fluid mechanics that inspired this note.