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Periodic and Quasi-Periodic Solutions for Reversible Unbounded Perturbations of Linear Schrödinger Equations

  • Zhaowei LouEmail author
  • Jianguo Si
Article
  • 26 Downloads

Abstract

In this paper, we consider a new class of derivative nonlinear Schrödinger equations with reversible nonlinearities of the form
$$\begin{aligned} \mathrm {i}u_t+u_{xx}+|u_x|^{4}u=0,\quad (t, x)\in {\mathbb {R}}\times {\mathbb {T}}. \end{aligned}$$
We obtain real analytic, linearly stable periodic solutions and quasi-periodic ones with two basic frequencies via infinite dimensional Kolmogorov–Arnold–Moser (KAM) theory for reversible systems. By investigating the gauge invariance and the compact form of vector fields, in our KAM iterative procedure, we remove the usual Diophantine restrictions on tangential frequencies and only use the Melnikov non-resonance conditions. In the proof, we also use Birkhoff normal form techniques due to the lack of external parameters in the equation above.

Keywords

Periodic solution Quasi-periodic solution DNLS KAM theory Reversible vector field Birkhoff normal form 

Mathematics Subject Classification

37K55 35Q41 35B15 

Notes

Acknowledgements

The first author was supported by Research Foundation of Nanjing University of Aeronautics and Astronautics “YAH18085” (No. 56YAH18085). Both authors were supported by the National Natural Science Foundation of China (Grant No. 11571201).

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Copyright information

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

Authors and Affiliations

  1. 1.College of ScienceNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.School of MathematicsShandong UniversityJinanPeople’s Republic of China

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