Standing Ring Blowup Solutions for Cubic NLS

Details Standing Ring Blowup Solutions for Cubic NLS Standing Ring Blowup Solutions for Cubic NLS

2010-02-05

arxiv.org/abs/1002.1267v2

Mathematics

Analysis of PDEs

54 pages. Introduction Updated

Scientific paper

We prove there exist solutions to the focusing cubic nonlinear Schr\"odinger equation in three dimensions that blowup on a circle, in the sense of L^2 concentration on a ring, bounded H^1 norm outside any surrounding toroid, and growth of the global H^1 norm with the log-log rate. Analogous behaviour occurs in any higher dimension. That is, there exists data for which the corresponding evolution by cubic NLS explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

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