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.2000 Mathematics Subject Classification. Primary: 76B47, 35Q55; Secondary: 35BXX.

Singularities and Global Solutions in the Schrodinger-Hartree Equation

“…Besides the explicit solutions of (1)-( 2) such as straight line, circle and helix, one important class is a oneparameter family of self-similar solution curves, that develop a corner in finite time [6]. The dynamics of (1) for one-corner curves have been well-studied both theoretically and numerically and motivated the study of curves with multiple corners [7,8]. In particular, the evolution of regular planar curves in the Euclidean case capture some qualitative feature of real fluids; for instance, the axis-switching phenomenon observed in the case of non-circular jets [9,10,Fig.…”