We establish the uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schr\"odinger equation with wave operator (NLSW), in which the strength of the nonlinearity is characterized by $\varepsilon^{2p}$ with $\varepsilon \in (0, 1]$ a dimensionless parameter and $p \in \mathbb{N}^+$. When $0 < \varepsilon \ll 1$, the long-time dynamics of the problem is equivalent to that of the NLSW with $O(1)$-nonlinearity and $O(\varepsilon)$-initial data. The NLSW is numerically solved by the EWI-FP method which combines an exponential wave integrator for temporal discretization with the Fourier pseudospectral method in space. We rigorously establish the uniform $H^1$-error bound of the EWI-FP method at $O(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$ up to the time at $O(1/\varepsilon^{\beta})$ with $0 \leq \beta \leq 2p$, the mesh size $h$, time step $\tau$ and $m \geq 2$ an integer depending on the regularity of the exact solution. Finally, numerical results are provided to confirm our error estimates of the EWI-FP method and show that the convergence rate is sharp.

翻译：我们建立了指数波积分傅里叶拟谱（EWI-FP）方法用于含波动算子的非线性薛定谔方程（NLSW）长时间动力学时的均匀误差界，其中非线性强度由$\varepsilon^{2p}$表征，$\varepsilon \in (0, 1]$为无量纲参数，$p \in \mathbb{N}^+$。当$0 < \varepsilon \ll 1$时，该问题的长时间动力学等价于具有$O(1)$非线性强度和$O(\varepsilon)$初始数据的NLSW。我们采用EWI-FP方法对该方程进行数值求解，该方法结合了时间离散的指数波积分与空间上的傅里叶拟谱方法。我们严格建立了EWI-FP方法在$O(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$量级上的均匀$H^1$误差界，直至$O(1/\varepsilon^{\beta})$时间的计算结果，其中$0 \leq \beta \leq 2p$，网格尺寸为$h$，时间步长为$\tau$，$m \geq 2$为依赖于精确解正则性的整数。最后，数值结果验证了EWI-FP方法的误差估计，并表明收敛阶是最优的。