# American Institute of Mathematical Sciences

July  2018, 17(4): 1317-1329. doi: 10.3934/cpaa.2018064

## Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$

 1 Sorbonne Université, Laboratoire Jacques-Louis Lions (UMR 7598), BC 187, 4 place Jussieu, 75005 Paris, France 2 Departamento de Matemáticas, Universidad del Pais Vasco, BCAM Alameda Mazarredo 14, 48009 Bilbao, Spain

* Corresponding author: Luis Vega

Received  February 2017 Revised  June 2017 Published  April 2018

Fund Project: The first author is partially supported by ANR project "SchEq" ANR-12-JS01-0005-01. The second author is partially supported by MINECO projects MTM2014-53850-P and SEV-2013-0323. and by the Basque Government projects IT-641-13 and BERC 2014-2017. Both authors are partially supported by ERC Advanced Grant 2014 669689 - HADE

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.

Citation: Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064
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