May

03

Monday, May 10, 2010, 3:00 p.m., in BA 6183, 40 St. George St. PhD Candidate: Ian Zwiers PhD Advisor: Jim Colliander Thesis Title: Standing Ring Blowup Solutions for the Cubic Nonlinear Schrodinger Equation Thesis: http://www.math.toronto.edu/~izwiers Thesis Abstract: The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2 norm concentration on a ring, bounded H1norm outside any surrounding toroid, and growth of the global H1 norm with the log-log rate. Analogous behaviour occurs in any dimension N \ge 3. That is, there exists data on N dimensions for which the corresponding evolution by the cubic nonlinear Schrodinger equation 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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