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.