A second-order low-regularity integrator for the nonlinear Schrödinger equation

In this paper, we analyze a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the d dimensional torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^{d}$\end{document}. The scheme has also been derived recently in a wider context of decorated trees (Bruned et al. in Forum Math. Pi 10:1–76, 2022). It is explicit and efficient to implement. Here, we present an alternative derivation and give a rigorous error analysis. In particular, we prove the second-order convergence in Hγ(Td)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\gamma }(\mathbb{T}^{d})$\end{document} for initial data in Hγ+2(Td)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\gamma +2}(\mathbb{T}^{d})$\end{document} for any γ>d/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma > d/2$\end{document}. This improves the previous work (Knöller et al. in SIAM J. Numer. Anal. 57:1967–1986, 2019).