## Departmental PhD Thesis Exam – Ian Zwiers

Monday, May 10, 2010, 3:00 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Ian Zwiers

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.