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

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