(untitled) (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 an analytic, odd function and f(u) = O(u^3). It is proved that for all m ∈ (0, M^∗] ⊂ 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.