非线性薛定谔方程的二阶低正则积分
A second order low-regularity integrator for the nonlinear Schrödinger equation
摘要：
本文分析了$d$维环面$\mathbb T^d$上非线性三次薛定谔方程的一个新的指数型积分器.该方案最近也在[Y. Bruned和K. Schratz，arXiv：2005.01649]。它是明确和有效的实施。在这里，我们提出了一种替代的推导，我们给出了严格的误差分析。特别是，我们证明了二阶收敛$H^γ(\mathbb T^d)$的初始数据在$H^{γ+2}(\mathbb T^d)$的任何$γ> d/2$。这改进了[Knöller，A. Ostermann和K. Schratz，SIAM J. Numer. Anal. 57（2019），1967-1986]。该方案的设计是基于一种新的方法来近似的非线性频率相互作用。这使我们能够处理任意维度的复杂共振结构。数值实验与理论结果相一致，补充了这一工作。
英文摘要：
In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schrödinger 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 $H^γ(\mathbb T^d)$ for initial data in $H^{γ+2}(\mathbb T^d)$ for any $γ> d/2$. This improves the previous work in [Knöller, 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.
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