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Stability of KAM tori for nonlinear Schrödinger equation
About this Title
Hongzi Cong, School of Mathematical Sciences, Dalian University of Technology, Dalian Liaoning 116024, China, Jianjun Liu, School of Mathematical Sciences, Sichuan University, Chengdu Sichuan 610065, China and Xiaoping Yuan, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
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: https://doi.org/10.1090/memo/1134
Published electronically: July 27, 2015
Keywords: KAM tori,
Normal form,
Stability,
$p$-tame property,
KAM technique
MSC: Primary 37K55, 37J40; Secondary 35B35, 35Q55
Table of Contents
Chapters
- Preface
- 1. Introduction and main results
- 2. Some notations and the abstract results
- 3. Properties of the Hamiltonian with $p$-tame property
- 4. Proof of Theorem and Theorem
- 5. Proof of Theorem
- 6. Proof of Theorem
- 7. Appendix: technical lemmas
Abstract
We prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation \[ \sqrt {-1}\, u_{t}=u_{xx}-M_{\xi }u+\varepsilon |u|^2u,\] subject to Dirichlet boundary conditions $u(t,0)=u(t,\pi )=0$, where $M_{\xi }$ is a real Fourier multiplier. More precisely, we show that, for a typical Fourier multiplier $M_{\xi }$, any solution with the initial datum in the $\delta$-neighborhood of a KAM torus still stays in the $2\delta$-neighborhood of the KAM torus for a polynomial long time such as $|t|\leq \delta ^{-\mathcal {M}}$ for any given $\mathcal M$ with $0\leq \mathcal {M}\leq C(\varepsilon )$, where $C(\varepsilon )$ is a constant depending on $\varepsilon$ and $C(\varepsilon )\rightarrow \infty$ as $\varepsilon \rightarrow 0$.- Dario Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), no. 2, 253–285. MR 1962462, DOI 10.1007/s00220-002-0774-4
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