Quasi-periodic Solutions for Two Dimensional Modified Boussinesq Equation

Abstract

In this paper, two dimensional modified Boussinesq equation $$\begin{aligned} u_{tt} +\Delta ^2u+ \Delta ( u^3 ) =0, \quad x\in {\mathbb {T}}^2,~t\in {\mathbb {R}} \end{aligned}$$ under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.