Stability of some waves in the Boussinesq system

$\partial_t u = -\partial_x v - \alpha \partial_{x x x} v - \epsilon \partial_x(u v), \quad \partial_t v = - \partial_x u - \epsilon v \partial_x v,$

where $\epsilon$ is an small parameter and $\alpha \in (0,1)$. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for some values of $\alpha$, it contains solutions that are defined for large values of time and they are very close (of order $O(\epsilon)$) to a linear torus for long times (of order $O(\epsilon^{-1})$). The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.