Singularity formation for the 1-D cubic NLS and the Schrödinger map on <inline-formula><tex-math id="M1">\begin{document}$\mathbb S^2$\end{document}</tex-math></inline-formula>

Abstract

In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-δ 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 Schrödinger 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.

Locations

• Communications on Pure &amp Applied Analysis - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF