Quasi-Periodic Solutions of the General Nonlinear Beam Equations

Abstract

In this paper, one-dimensional (1D) nonlinear beam equations of the form $$u_{tt} − u_{xx} + u_{xxxx} + mu = f(u)$$ with Dirichlet boundary conditions are considered, where the nonlinearity $f$ is an analytic, odd function and $f(u) = O(u^3)$. It is proved that for all $m ∈ (0, M^∗] ⊂ \boldsymbol{R}$    ($M^∗$ is a fixed large number), but a set of small Lebesgue measure, the above equations admit small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theory and a partial Birkhoff normal form technique.