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International Mathematics Research Notices Advance Access originally published online on December 25, 2008
International Mathematics Research Notices (2009) 2009:414-432, doi:10.1093/imrn/rnn135 published on February 3, 2009
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org.

Global Solutions for 3D Quadratic Schrödinger Equations

Pierre Germain, Nader Masmoudi and Jalal Shatah

Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA

Correspondence: Correspondence to be sent to: pgermain{at}courant.nyu.edu

This is the firstof the three papers where we present a new method based on the concept of space-time resonance to prove global existence of small solutions to nonlinear dispersive equations. The idea is that time resonances (dynamical systems resonances) correspond to interactions between plane waves; but since for dispersive equations we deal with localized solutions, it is crucial to take also into account the traveling speeds of the different wave packets. Here we show how this idea, and the analytical method that this naturally suggests, leads to a simple proof of global existence and scattering for quadratic nonlinear Schrödinger equations in three dimensions.


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