Stability of KAM tori for nonlinear Schrödinger equation
著者
書誌事項
Stability of KAM tori for nonlinear Schrödinger equation
（Memoirs of the American Mathematical Society, no. 1134）
American Mathematical Society, [2016], c2015
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注記
"January 2016, volume 239, number 1134 (sixth of 6 numbers)."
Includes bibliographical references and index
内容説明・目次
内容説明
The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrodinger 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 $\varepsilon\rightarrow0$.
目次
Introduction and main results
Some notations and the abstract results
Properties of the Hamiltonian with $p$-tame property
Proof of Theorem 2.9 and Theorem 2.10
Proof of Theorem 2.11
Proof of Theorem 1.1
Appendix: technical lemmas
Bibliography
Index
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