Standing Ring Blowup Solutions for the Cubic Nonlinear Schrodinger Equation

• Main Author: Zwiers, Ian
• Other Authors: Colliander, James
• Language: en_ca
• Published: 2012
• Subjects: Nonlinear Schrodinger Equation; Blowup and Singularity Formation; 0405
• Online Access: http://hdl.handle.net/1807/33836

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 H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data 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.