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Stability of KAM tori for nonlinear Schrödinger equation


About this Title

Hongzi Cong, Jianjun Liu and Xiaoping Yuan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 239, Number 1134
ISBNs: 978-1-4704-1657-7 (print); 978-1-4704-2751-1 (online)
DOI: http://dx.doi.org/10.1090/memo/1134
Published electronically: July 27, 2015
Keywords:KAM tori, Normal form, Stability, $p$-tame property

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Table of Contents


Chapters

  • Preface
  • Chapter 1. Introduction and main results
  • Chapter 2. Some notations and the abstract results
  • Chapter 3. Properties of the Hamiltonian with $p$-tame property
  • Chapter 4. Proof of Theorem 2.9 and Theorem 2.10
  • Chapter 5. Proof of Theorem 2.11
  • Chapter 6. Proof of Theorem 1.1
  • Chapter 7. Appendix: technical lemmas

Abstract


We prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation

subject to Dirichlet boundary conditions , where is a real Fourier multiplier. More precisely, we show that, for a typical Fourier multiplier , any solution with the initial datum in the -neighborhood of a KAM torus still stays in the -neighborhood of the KAM torus for a polynomial long time such as for any given with , where is a constant depending on and as .

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