Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

  • Leonid Chaichenets
  • Dirk Hundertmark
  • Peer KunstmannEmail author
  • Nikolaos PattakosEmail author


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., 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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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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