Long time behavior of the solutions of NLW on the d-dimensional torus
We consider the non linear wave equation (NLW) on the d-dimensional torus
$$u_{tt} - \Delta u + \mu + f(u) =0\quad x \in \mathbb T^d$$
where $f=\partial_u F$ is analytic on a neighborhood of the origin and which is at least of order 2 at the origin. Let $u(t)$ be a solution corresponding to a small initial datum $u(0)\in H^s(\mathbb T^d)$. We prove that we control $[u(t)]_s$ that mix the $H^s$ norm of the $\varepsilon^{-\beta(r)}$ lower Fourier modes of the solution $u$ and the energy norm of the remaining higher modes during long times of order $\varepsilon^{-r}$.
Our general strategy applies to any Hamiltonian PDEs whose linear frequencies satisfy only a first Melnikov condition. In particular it also applies to the Hamiltonian Boussinesq $abcd$ system and the Whitham-Boussinesq system in water waves theory. Joint work with Joackim Bernier and Erwan Faou.