**abstract:**
We consider the non linear wave equation (NLW) on the d-dimensional torus
$$u_{{tt}} - \Delta u + mu + f(u) =0\quad x\in\T^{d$$
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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 $u(0)\in H^{s}(\T^{d)$.
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We prove that we control $u(t)_{s$} that mix the $H^{s$} norm of the $\eps^{{}-\beta(r)}$ lower Fourier modes of the solution $u$ and the energy norm of the remaining higher modes during long times of order $\eps^{{}-r}$.
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.

Mon 20 May, 14:30 - 15:20, Aula Dini

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