We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form $ |\psi|^{2\sigma}\psi $ with $ \psi $ being the wave function and $ \sigma > 0 $ being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For ``good" potential and nonlinearity ($H^2$-potential and $\sigma \geq 1$), we establish an optimal second-order error bound in the $L^2$-norm. For low regularity potential and nonlinearity ($L^\infty$-potential and $\sigma > 0$), we obtain a first-order $L^2$-norm error bound accompanied with a uniform $H^2$-norm bound of the numerical solution. Moreover, adopting a new technique of \textit{regularity compensation oscillation} (RCO) to analyze error cancellation, for some non-resonant time steps, the optimal second-order $L^2$-norm error bound is proved under a weaker assumption on the nonlinearity: $\sigma \geq 1/2$. For all the cases, we also present corresponding fractional order error bounds in the $H^1$-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.

翻译：本文针对低正则势与形如$ |\psi|^{2\sigma}\psi $（其中$\psi$为波函数，$\sigma>0$为非线性指数）的典型幂律非线性非线性薛定谔方程（NLSE），提出并分析了一种新型对称Gautschi型指数波积分器（sEWI）。该积分器为显式格式，其稳定性条件中对时间步长的限制独立于网格尺寸。我们在势函数和非线性项的不同正则性假设下，严格建立了sEWI的误差估计。对于“良好”势函数与非线性项（$H^2$正则势与$\sigma \geq 1$），我们得到了$L^2$范数下的最优二阶误差界；而对于低正则势函数与非线性项（$L^\infty$正则势与$\sigma > 0$），我们推导出$L^2$范数下的一阶误差界，并同时给出了数值解的一致$H^2$范数界。此外，通过引入一种新的“正则补偿振荡”（RCO）技术来分析误差抵消效应，在部分非共振时间步长下，即便将非线性项假设减弱至$\sigma \geq 1/2$，仍可证明最优二阶$L^2$范数误差界。针对所有情形，我们还给出了对应的$H^1$范数分数阶误差界——该范数为能量意义上的自然范数。大量数值结果验证了我们的误差估计，并展示了sEWI在势函数与非线性项正则性要求显著降低、以及具备近质量与能量守恒的优良长时间行为等方面的优越性。