KAM for the nonlinear wave equation on the circle: small amplitude solution

In this paper we consider the nonlinear wave equation on the circle:\begin{equation} \nonumberu_{tt} - u_{xx} + m u = g(x,u), \quad t \in $\mathbb{R}$,: x \in $\mathbb{S}$^1,\end{equation}where $m\in [1,2]$ is a mass and $g(x,u)=4u^3+O(u^4)$. This equation will be treated as a perturbation of the integrable Hamiltonian:\begin{equation} \tag{$\ast$} \label{first equation}u_t= v, \quad v_t = - u_{xx} + m u.\end{equation}Near the origin and for generic $m$, we prove the existence of small amplitude quasi-periodic solutions close to the solution of the linear equation\eqref{first equation}. For the proof we use an abstract KAM theorem in infinite dimension and a Birkhoff normal form result.