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Journal of Fourier Analysis and Applications

, Volume 25, Issue 4, pp 1447–1486 | Cite as

NLS in the Modulation Space \(M_{2,q}({\mathbb {R}})\)

  • N. PattakosEmail author
Article

Abstract

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space \(M_{2,q}^{s}({\mathbb {R}})\), \(1\le q\le 2\) and \(s\ge 0.\) In addition, for either \(s\ge 0\) and \(1\le q\le \frac{3}{2}\) or \(\frac{3}{2}<q\le 2\) and \(s>\frac{2}{3}-\frac{1}{q}\) we show that the Cauchy problem is unconditionally wellposed in \(M_{2,q}^{s}({{\mathbb {R}}}).\) It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.

Keywords

Normal form method Modulation spaces Nonlinear Schrödinger Local wellposedness Unconditional uniqueness 

Mathematics Subject Classification

35A01 35A02 35D30 35J10 

Notes

Acknowledgements

The author gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. He would also like to thank Peer Kunstmann from KIT for his helpful comments and fruitful discussions. Finally, he would like to thank the referees of the paper for their constructive criticism.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

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