We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schr\"odinger equation (NLSE) with $ L^\infty $-potential and/or locally Lipschitz nonlinearity under the assumption of $ H^2 $-solution of the NLSE. For the semi-discretization in time by the first-order Gautschi-type EWI, we prove an optimal $ L^2 $-error bound at $ O(\tau) $ with $ \tau>0 $ being the time step size, together with a uniform $ H^2 $-bound of the numerical solution. For the full-discretization scheme obtained by using the Fourier spectral method in space, we prove an optimal $ L^2 $-error bound at $ O(\tau + h^2) $ without any coupling condition between $ \tau $ and $ h $, where $ h>0 $ is the mesh size. In addition, for $ W^{1, 4} $-potential and a little stronger regularity of the nonlinearity, under the assumption of $ H^3 $-solution, we obtain an optimal $ H^1 $-error bound. Furthermore, when the potential is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method which has the same error bound as the Fourier spectral method while its computational cost is similar to the standard Fourier pseudospectral method. Our new error bounds greatly improve the existing results for the NLSE with low regularity potential and/or nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

翻译：针对具有 $ L^\infty $ 势和/或局部Lipschitz非线性的非线性薛定谔方程（NLSE），在假设解属于 $ H^2 $ 空间的前提下，我们建立了指数波积分器（EWI）的最优误差估计。对于采用一阶Gautschi型EWI的时间半离散格式，我们证明了数值解具有 $ O(\tau) $ 的最优 $ L^2 $ 误差界（其中 $ \tau>0 $ 为时间步长），同时数值解满足一致 $ H^2 $ 有界性。对于结合空间傅里叶谱方法得到的全离散格式，我们在无时间步长 $ \tau $ 与网格尺寸 $ h>0 $ 耦合条件的情况下，证明了 $ O(\tau + h^2) $ 的最优 $ L^2 $ 误差界。此外，针对 $ W^{1,4} $ 势与稍强正则性的非线性项，在 $ H^3 $ 解假设下，我们得到了最优 $ H^1 $ 误差界。进一步地，当势函数正则性较低而非线性项充分光滑时，我们提出了一种扩展傅里叶伪谱方法，该方法在保持与傅里叶谱方法相同误差界的同时，计算成本与标准傅里叶伪谱方法相当。我们的新误差界显著改进了现有关于低正则势和/或非线性NLSE的结果。大量数值实验验证了误差估计的正确性及其最优性。