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The Scientific World Journal
Volume 2014, Article ID 214821, 11 pages
http://dx.doi.org/10.1155/2014/214821
Review Article

Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions

1Department of Mathematics, College of Science, Yanbian University, No.977 Gongyuan Road, Yanji, Jilin 133002, China
2Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Received 20 January 2014; Accepted 2 March 2014; Published 31 March 2014

Academic Editors: D. Baleanu and C. M. Khalique

Copyright © 2014 Chunhua Li and Nakao Hayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in -dimensional space, where . The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown.