Stability of KAM Tori for Nonlinear Schrödinger Equation
( Memoirs of the American Mathematical Society )
Publication series ：Memoirs of the American Mathematical Society
Author： Hongzi Cong;Jianjun Liu;Xiaoping Yuan
Publisher： American Mathematical Society
Publication year： 2016
E-ISBN: 9781470427511
P-ISBN(Hardback): 9781470416577
Subject： O413.1 quantum mechanics (wave mechanics, matrix mechanics)
Keyword： 暂无分类
Language： ENG
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Stability of KAM Tori for Nonlinear Schrödinger Equation
Description
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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