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Conference on Non-linear Phenomena in Mathematical Physics: Dedicated to Cathleen Synge Morawetz on her 85th birthday
September 18-20, 2008
Fields Institute
Toronto, Ontario, Canada

Organizers
Jim Colliander (University of Toronto), Susan Friedlander (University of Southern California) Irene M. Gamba (U. Texas at Austin) - Committee Chair, Fern Hunt (NIST) Barbara L. Keyfitz (Fields Institute, Canada), Walter Strauss (Brown University)

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Scattering For the Focusing 2D Quintic Nonlinear Schrödinger Equation
by
Cristi Darley Guevara
Arizona State University
Coauthors: Fernando Carreon (Arizona State University), Svetlana Roudenko(Arizona State University)

Abstract

Recent developments for the energy critical nonlinear Schrödinger equation (NLS) in 3d, and nonlinear wave equation (NLW) by Carlos Kenig and Frank Merle have attracted attention from Harmonic Analysis and PDE audience. Their approach is based on concentration-compactness method which dates back to works of P.-L. Lions and the localized virial argument. It gives a sharp threshold for the scattering and finite time blow up of solutions at least in the case of radial data, and in many problems can be extended to nonradial data as well. These methods have been recently applied to the focusing cubic NLS in 3d by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko as well as to the mass critical (both focusing and defocusing) NLS in 2 and higher dimensions by Killip-Tao-Visan, Killip-Visan-Zhang. Using the above techniques, we characterize the behavior of H1 solutions to the focusing quintic NLS in R2, namely,
i ∂t u+\triangle u+|u|4u=0,
(x, t) ∈ R2×R.
We obtain scattering for globally existing solutions (under an a priori mass-energy threshold) and mention how this extends to a general mass supercritical and energy subcritical NLS with H1 data.

Date received: July 12, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxk-07.