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)的最优误差界。对于时间半离散化采用一阶Gautschi型EWI，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），且数值解满足一致的$H^2$有界性。对于空间采用Fourier谱方法得到的全离散格式，我们证明了在$\tau$与$h$无需任何耦合条件时具有$O(\tau + h^2)$的最优$L^2$误差界（其中$h>0$为网格尺寸）。此外，对于$W^{1,4}$势和具有更强正则性的非线性，在解属于$H^3$空间的假设下，我们得到了最优的$H^1$误差界。进一步地，当势能具有低正则性而非线性足够光滑时，我们提出了一种扩展Fourier拟谱方法，其误差界与Fourier谱方法相当，而计算复杂度与标准Fourier拟谱方法相近。我们的新误差界显著改进了现有关于低正则势和/或非线性的NLSE的研究结果。丰富的数值实验结果验证了误差估计的准确性并证明了其最优性。