| eBook ISBN: | 978-1-4704-2751-1 |
| Product Code: | MEMO/239/1134.E |
| List Price: | $79.00 |
| MAA Member Price: | $71.10 |
| AMS Member Price: | $47.40 |
| eBook ISBN: | 978-1-4704-2751-1 |
| Product Code: | MEMO/239/1134.E |
| List Price: | $79.00 |
| MAA Member Price: | $71.10 |
| AMS Member Price: | $47.40 |
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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 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 \(\varepsilon\rightarrow0\).
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Table of Contents
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Chapters
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Preface
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1. Introduction and main results
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2. Some notations and the abstract results
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3. Properties of the Hamiltonian with $p$-tame property
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4. Proof of Theorem and Theorem
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5. Proof of Theorem
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6. Proof of Theorem
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7. Appendix: technical lemmas
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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 \(\varepsilon\rightarrow0\).
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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
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7. Appendix: technical lemmas
