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

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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.

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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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  1. Department of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    N. Pattakos

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  1. N. Pattakos
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Pattakos, N. NLS in the Modulation Space \(M_{2,q}({\mathbb {R}})\). J Fourier Anal Appl 25, 1447–1486 (2019). https://doi.org/10.1007/s00041-018-09655-9

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  • DOI: https://doi.org/10.1007/s00041-018-09655-9

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