In this paper, we prove the existence of small amplitude solutions which are almost-periodic in time for a quasi-periodically forced nonlinear Schrödinger equation

$$\begin{aligned} \sqrt{-1}u_{t}-u_{xx}+M_{\xi }u+f(\tilde{\omega }t)|u|^2u=0 \end{aligned}$$

with periodic boundary conditions

$$\begin{aligned} u(t,x+2\pi )=u(t,x),~~t\in \mathbb {R}, \end{aligned}$$

where \(M_{\xi }\) is a real Fourier multiplier, \(f(\tilde{\theta })(\tilde{\theta }=\tilde{\omega }t)\) is real analytic and the forced frequency vector \(\tilde{\omega }\in \mathbb {R}^{b}\) is fixed and Diophantine.

中文翻译：

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准周期受迫非线性薛定谔方程的近周期解

在本文中，我们证明了准周期性受迫非线性薛定谔方程在时间上几乎是周期性的小幅度解的存在性

$$\begin{aligned} \sqrt{-1}u_{t}-u_{xx}+M_{\xi }u+f(\tilde{\omega }t)|u|^2u=0 \end{对齐}$$

具有周期性边界条件

$$\begin{aligned} u(t,x+2\pi )=u(t,x),~~t\in \mathbb {R}, \end{aligned}$$

其中\(M_{\xi }\)是实数傅立叶乘数，\(f(\tilde{\theta })(\tilde{\theta }=\tilde{\omega }t)\)是实解析的，并且强制频率向量\(\tilde{\omega }\in \mathbb {R}^{b}\)是固定的并且丢番图。