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Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

  • Leonid Chaichenets
  • Dirk Hundertmark
  • Peer KunstmannEmail author
  • Nikolaos PattakosEmail author
Article
  • 26 Downloads

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 \(M_{p,q}^{s}({\mathbb {R}})\) where \(1\le q\le 2\), \(2\le p<\frac{10q'}{q'+6}\) and \(s\ge 0\). Moreover, for either \(1\le q\le \frac{3}{2}, s\ge 0\) and \(2\le p\le 3\) or \(\frac{3}{2}<q\le \frac{18}{11}, s>\frac{2}{3}-\frac{1}{q}\) and \(2\le p\le 3\) or \(\frac{18}{11}<q\le 2, s>\frac{2}{3}-\frac{1}{q}\) and \(2\le p<\frac{10q'}{q'+6}\) we show that the Cauchy problem is unconditionally wellposed in \(M_{p,q}^{s}({\mathbb R}).\) This improves Pattakos (J Fourier Anal Appl, 2018.  https://doi.org/10.1007/s00041-018-09655-9), where the case \(p=2\) was considered and the differentiation-by-parts technique was introduced to a problem with continuous Fourier variable. Here, the same technique is used, but more delicate estimates are necessary for \(p\ne 2\).

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Notes

Acknowledgements

The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Dirk Hundertmark also thanks Alfried Krupp von Bohlen und Halbach Foundation for their financial support.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsInstitute for Analysis, Karlsruhe Institute of Technology (KIT)KarlsruheGermany

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