Modified scattering for a dispersion-managed nonlinear Schrödinger equation

We prove sharp L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrödinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged version of the dispersion-managed NLS in the strong dispersion management regime. The proof adapts techniques from Hayashi and Naumkin (Am. J. Math. 120(2):369–389, 1998) and Kato and Pusateri (Differ Integral Equ 24(9–10):923–940, 2011), which established small-data modified scattering for the standard 1d cubic NLS.