Quasi-Periodic Solutions of Derivative Beam Equation on Flat Tori

In this paper, we consider a class of nonlinear beam equations on flat tori TLd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}^d_{\mathfrak {L}}$$\end{document}, utt+Δ2u=ϵ(-Δ)αf(ωt,x,(-Δ)βu),0<α+β≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{tt}+\Delta ^2u=\epsilon (-\Delta )^{\alpha } f(\omega t,x,(-\Delta )^{\beta }u), \quad 0<\alpha +\beta \le 1 \end{aligned}$$\end{document}and prove that the equation admits many quasi-periodic solutions with the non-resonant frequencies ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}. The main proof is based on an abstract Nash-Moser type implicit function theorem developed in Berti and Bolle (Nonlinearity 25(9):2579–2613, 2012; J Eur Math Soc 15(1):229–286, 2013).