Low-Regularity Integrator for the Davey–Stewartson II System

We consider the Davey–Stewartson system in the hyperbolic–elliptic case (DS-II) in two dimensional case. It is a mass-critical equation, and was proved recently by Nachman et al. (Invent Math 220(2):395–451, 2020) the global well-posedness and scattering in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}. In this paper, we give the numerical study on this model and construct a first order low-regularity integrator for the DS-II in the periodic case. It only requires the boundedness of one additional derivative of the solution to get the first order convergence. The Fast Fourier Transform is exploited to speed up the numerical implementation. By rigorous error analysis, we prove that the numerical scheme provides first order convergence in Hγ(T2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\gamma }({\mathbb {T}}^{2})$$\end{document} for rough initial data in Hγ+1(T2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\gamma +1}({\mathbb {T}}^{2})$$\end{document} with γ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma > 1$$\end{document}. The optimality of the convergence is conformed by numerical experience.