# Stability of KAM Tori for Nonlinear Schrödinger Equation

Share this page
*Hongzi Cong; Jianjun Liu; Xiaoping Yuan*

The authors 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, they 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 \(\varepsilon\rightarrow0\).

#### Table of Contents

# Table of Contents

## Stability of KAM Tori for Nonlinear Schrodinger Equation

- Cover Cover11 free
- Title page i2 free
- Preface vii8 free
- Chapter 1. Introduction and main results 110 free
- Chapter 2. Some notations and the abstract results 312 free
- Chapter 3. Properties of the Hamiltonian with 𝑝-tame property 1726
- Chapter 4. Proof of Theorem 2.9 and Theorem 2.10 3342
- Chapter 5. Proof of Theorem 2.11 4756
- Chapter 6. Proof of Theorem 1.1 6978
- Chapter 7. Appendix: technical lemmas 7584
- Bibliography 8392
- Index 8594 free
- Back Cover Back Cover1100