eBook ISBN:  9781470427511 
Product Code:  MEMO/239/1134.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $47.40 
eBook ISBN:  9781470427511 
Product Code:  MEMO/239/1134.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $47.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 239; 2015; 85 ppMSC: Primary 37; Secondary 35
The authors prove the long time stability of KAM tori (thus quasiperiodic solutions) for nonlinear Schrödinger equation \[\sqrt{1}\, u_{t}=u_{xx}M_{\xi}u+\varepsilonu^2u,\] 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

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


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The authors prove the long time stability of KAM tori (thus quasiperiodic solutions) for nonlinear Schrödinger equation \[\sqrt{1}\, u_{t}=u_{xx}M_{\xi}u+\varepsilonu^2u,\] 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 \(\varepsilon\rightarrow0\).

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