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

  • * Corresponding author: Luis Vega

    * Corresponding author: Luis Vega
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.
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  • 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.

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

    Citation:

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