Whiskered Tori for Forced Beam Equations with Multi-dimensional Liouvillean Frequency

Abstract

In this paper we develop a Kolmogorov–Arnold–Moser (KAM) theory close to two fixed points for quasi-periodically forced nonlinear beam equation $$\begin{aligned} \begin{aligned} y_{tt}+my+y_{xxxx}=y^{3}+ \varepsilon f(\omega t,x,y),\ \ x\in [0,\ \pi ], \end{aligned} \end{aligned}$$where the forcing frequency $$\omega $$ is a small dilatation of a fixed vector $${\overline{\omega }},$$ i.e., $$\omega =\xi {\overline{\omega }}\in {\mathbb {R}}^{d}$$ with $$\xi \in {\mathcal {O}}:=[1,2].$$ We will prove the existence of real analytic quasi-periodic solutions of the above equations under the hypothesis that the frequency $${\overline{\omega }}$$ is Liouvillean. The quasi-periodic solutions we obtain are around the equilibria, $$y(t,x)\equiv \pm \sqrt{m},\forall (t,x)\in {\mathbb {R}}\times [0,\ \pi ],$$ of the system $$\begin{aligned} \begin{aligned} y_{tt}+my+y_{xxxx}=y^{3}, \end{aligned} \end{aligned}$$and are whiskered, that is the linearized equation around $$\pm \,\sqrt{m}$$ owns the hyperbolic directions (the hyperbolic directions are finitely many, depending on m, and the elliptic directions are infinitely many in our case). The proof is based on a modified KAM iteration for infinite dimensional systems with finitely many hyperbolic directions, infinitely many elliptic directions and Liouvillean forcing frequency. We believe that the approach in this paper can be applied also to other integrable PDEs. For example, the same strategy should work for the non-linear wave equations and the non-linear Schrodinger equations.