ABSTRACT

We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing ($\lambda =1$) or focusing ($\lambda =-1$) case, $\mathrm{i}{u}_t=-\mathrm{\Delta }u+\lambda |u{|}^{2}u,x\in {\mathbb{T}}^d,t\in \mathbb{R}.$ More precisely, in a Gevrey space $G_{\sigma }:=\left\{u:|\parallel u{\parallel }_{\sigma }^2=\sum _{a\in {\mathbb{Z}}^d}{e}^{2\sigma \sqrt{|a|}}|u_a{|}^2<\mathrm{\infty }\right\}$ for some positive constant σ, we show that solution with the initial datum in the $4ϵ$-neighborhood of the plane wave solution still stays in the $Cϵ$-neighborhood ( C>4 ) of the plane wave solution for a subexponential long time $|t|\le {ϵ}^{-\zeta |\ln ϵ{|}^{\varrho }},$ where $\zeta =min\{\frac{1}{4},\sigma -{\sigma }^{\prime }\},\sigma >{\sigma }^{\prime }>0$ and $0<\varrho <1/6$.

中文翻译：

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圆环上薛定谔方程平面波解的长期稳定性

摘要

我们证明了在散焦中非线性薛定谔方程（NLS）的平面波解的长期轨道稳定性（$\lambda =1$) 或聚焦 ($\lambda =-1$） 案子，$\mathrm{一世}{你}_{吨}=-\mathrm{\Delta }你+\lambda |你{|}^2你,X\in {\mathbb{吨}}^d,吨\in \mathbb{R}.$更准确地说，在 Gevrey 空间中$G_{\sigma }:=\left\{你：|\parallel 你{\parallel }_{\sigma }^2=\sum _{\mathrm{一个}\in {\mathbb{Z}}^d}{e}^{2\sigma \sqrt{|\mathrm{一个}|}}|{你}_{\mathrm{一个}}{|}^2<\mathrm{\infty }\right\}$对于一些正常数σ，我们用初始数据在$4\epsilon$-平面波解的邻域仍然停留在$C\epsilon$-次指数长时间内平面波解的邻域 ( C >4 )$|吨|\le {\epsilon }^{-\zeta |\ln \epsilon {|}^{\varrho }},$在哪里$\zeta =分钟\{\frac{1}{4},\sigma -{\sigma }^{\text{'}}\},\sigma >{\sigma }^{\text{'}}>0$和$0<\varrho <1/6$.