In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schr\"odinger equation on the $d$ dimensional torus $\mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y. Bruned and K. Schratz, arXiv:2005.01649]. It is explicit and efficient to implement. Here, we present an alternative derivation, and we give a rigorous error analysis. In particular, we prove second-order convergence in $H^\gamma(\mathbb T^d)$ for initial data in $H^{\gamma+2}(\mathbb T^d)$ for any $\gamma > d/2$. This improves the previous work in [Kn\"oller, A. Ostermann, and K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967-1986]. The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.

翻译：在本文中, 我们分析非线性立方体 Schr\ 的新的指数型集成器。 特别是, 我们证明, 美元( mathbb T ⁇ d) 美元( mathbb T ⁇ d$ 美元) 的二等联产方程式。 最近, 这个计划也从[ Y. Brunned 和 K. Schratz: 2005. 01649] 中装饰树木的大背景中衍生出来。 这改进了[ Kn\"oller, A. Ostermann, 和 K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967- 1986] 中以前的工作。 这个计划的设计基于一种能接近非线性频率互动的新方法。 这使我们得以与复杂的工作结构进行任意的理论实验。 使得Numeralal 能够与这个复杂的工作结构进行任意的理论实验。