THE CANTOR MANIFOLD THEOREM WITH SYMMETRY AND APPLICATIONS TO PDEs

Abstract

In this paper we introduce a new Cantor manifold theorem and then apply it to one new type of one-dimensional ($1d$) beam equations $$ u_{tt}+u_{xxxx}+mu-2u^2u_{xx}-2uu_x^2=0, m\gt 0,$$ with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invaraint tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry (cf. [19]).