We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the long-time error bounds of different low-regularity integrators for the nonlinear Schr\"odinger equation (NLSE) with small initial data characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in $H^r$ is proven at $O(\varepsilon \tau)$ for solutions in $H^r$ with $r > 1/2$ up to the time $T_{\varepsilon } = T/\varepsilon $ with fixed $T > 0$. Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time $T_{\varepsilon } = T/\varepsilon ^2$. With the help of the regularity compensation oscillation (RCO) technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduce the long-time error by a factor of $\varepsilon^2$ compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.

翻译：我们针对三次非线性薛定谔方程(NLSE)提出一种新型非共振低正则积分器，其长时间误差估计在偏微分方程(PDE)意义下达到最优。主要思想在于在离散化过程中精确处理零模态。对于长时间误差估计，我们严格建立了不同低正则积分器在含小初始数据（由无量纲参数$\varepsilon \in (0, 1]$表征）的非线性薛定谔方程(NLSE)中的长时间误差界。首先针对二次NLSE的低正则积分器，其中积分被精确计算，我们证明在$H^r$空间（$r > 1/2$）中，解在固定时间$T>0$内达到$T_{\varepsilon } = T/\varepsilon$时间尺度时，具有$O(\varepsilon \tau)$阶的改进均匀一阶收敛性。随后，在长时间框架下将改进的均匀长时间误差界推广至对称二阶低正则积分器。针对三次NLSE，我们设计了精确处理零模态的新型非共振一阶和对称二阶低正则积分器，并严格完成了在$T_{\varepsilon } = T/\varepsilon ^2$时间尺度内的误差分析。借助正则补偿振荡(RCO)技术，我们为新非共振低正则格式建立了改进的均匀误差界，与经典的低正则积分器相比，此类格式将三次NLSE的长时间误差进一步降低$\varepsilon^2$倍。数值算例验证了误差估计结果，并在长时间模拟中与经典时间分裂方法进行了比较。