Quasi-periodic solutions for a class of beam equation system

Abstract

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system \begin{document}$ \left\{ \begin{array}{lll} u_{1tt}+ \Delta^2 u_1 +\sigma u_1 +u_1u_2^2 & = & 0 \\ &&\\ u_{2tt}+ \Delta^2 u_2 +\mu u_2 +u_1^2 u_2 & = & 0 \end{array} \right. $\end{document} under periodic boundary conditions, where $ 0<\sigma \in [ \sigma_1,\sigma_2 ], $ $ 0<\mu\in [ \mu_1,\mu_2 ] $ are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.