Kolmogorov–Arnold–Moser theorem for nonlinear beam equations with almost-periodic forcing

Abstract

In this study, we construct a Kolmogorov–Arnold–Moser theorem regarding the existence of almost-periodic solutions for some infinitely dimensional Hamiltonian systems with almost-periodic forcing. This theorem is applied to an almost-periodically forced nonlinear beam equation with periodic boundary conditions to obtain the almost-periodic solutionsutt+(−∂xx+μ)2u+ψ(ωt)f(u)=0,μ>0,t∈R,x∈R, where ψ(ωt) is real analytic and almost periodic on t and the nonlinearity f is a real-analytic function near u=0 with f(0)=f′(0)=0.