#### Min Zhang, Yonggang Chen, Zhe Hu, KAM tori for a two-dimensional Boussinesq equation with quasi-periodic forcing, Vol. 2021 (2021), Article ID 32, pp. 1-21

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DOI: 10.23952/jnfa.2021.32

Received July 20, 2021; Accepted September 12, 2021; Published October 10, 2021

Abstract. In this paper, a two-dimensional (2D) quasi-periodically forced Boussinesq equation $u_{tt}+\Delta^2 u+ \varepsilon\phi(t)\Delta(u+{u}^3)=0,$ $x\in\mathbb{T}^2,$ $t\in\mathbb{R}$ under periodic boundary conditions is considered, where $\varepsilon$ is a small positive parameter, and $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega=(\omega_1 ,\omega_2 \ldots,\omega_m)$. It is proved that the quasi-periodic solutions to the equation lie in a special subspace of $L^2(\mathbb{T}^2)$. By utilizing the measure estimation of infinitely many small divisors, for most values of the frequency vector $\omega$, the Hamiltonian of the linear part of the equation can be reduced to an autonomous system via a symplectic change of coordinates. And by some transformations of coordinates, the Hamiltonian of the equation can be transformed into an angle-dependent block-diagonal normal form, which can be achieved by choosing the appropriate tangential sites. By an abstract KAM theorem, the existence of a class of invariant tori is proved, which implies the existence of a class of small-amplitude quasi-periodic solutions for the above equation.