Invariant tori of full dimension for higher-dimensional beam equations with almost-periodic forcing

Jie Rui,Yi Wang

doi:10.1186/s13661-020-01374-9

Abstract

In this paper, we focus on the class of almost-periodically forced higher-dimensional beam equations utt+(−Δ+μ)2u+ψ(ωt)u=0,μ>0,t∈R,x∈Rd,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}+(-\\Delta +\\mu )^{2}u+\\psi (\\omega t)u=0,\\quad \\mu >0, t \\in \\mathbb{R}, x\\in \\mathbb{R}^{d}, $$\\end{document} subject to periodic boundary conditions, where psi (omega t) is real analytic and almost-periodic in t. We show the existence of almost-periodic solutions for this equation under some suitable hypotheses. In the proof, we improve the KAM iteration to deal with the infinite-dimensional frequency omega =(omega _{1},omega _{2},ldots).

Highlights

1 Introduction Recently, many researchers focus on some physical models appeared in dynamics of the suspension bridge, nonrelativistic quantum mechanics, supersymmetric field theoriesk and inflation cosmology [1,2,3,4,5,6,7,8,9,10,11]
We focus on the reducibility of the linear d-dimensional beam equation with almost-periodic forcing utt + (– + μ)2u + ψ(ωt)u = 0, μ > 0, t ∈ R, x ∈ Rd, (1.1)
There is almost-periodic forcing in higher dimensions, because the reducibility is complex and doubtful

Summary

Introduction

Many researchers focus on some physical models appeared in dynamics of the suspension bridge, nonrelativistic quantum mechanics, supersymmetric field theoriesk and inflation cosmology [1,2,3,4,5,6,7,8,9,10,11]. There is almost-periodic forcing in higher dimensions, because the reducibility is complex and doubtful All those KAM theorems fail to handle infinite-dimensional frequency ω = Eq (1.1) is a higher-dimensional equation, and the analysis of Birkhoff normal forms and more precise estimation of new perturbation is very difficult because of the effects of infinite-dimensional frequency, To state the main results of our paper, we need the following assumptions and sets. Remark 2.1 Property of (2.9) is important for higher-dimensional Hamiltonian systems It ensures the form of perturbations and the obtained normal form in the KAM iteration. The regularity of XR a,ρ ensures that XR sends a decaying qsequence to a faster decaying sequence

A theorem of reducibility

Rewrite the new normal form and new perturbation