Quasi-Periodic Solutions of the General Nonlinear Beam Equations (Communications in Mathematical Research, 2021)
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 ananalytic, 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 equationsadmit small-amplitude quasi-periodic solutions corresponding to finite dimensionalinvariant tori for an associated infinite dimensional dynamical system. The proof isbased on an infinite dimensional KAM theory and a partial Birkhoff normal formtechnique.
Keywords: beam equation; KAM theorem; quasi-periodic solution; partial Birkhoff normal form.