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A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus.

We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation $$ iu_t=-Δu+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$ where $V$ is a typical smooth potential and $g$ is analytic in both variables. More precisely we prove that if the initial datum is analytic in a strip of width $ρ>0$ with a bound on this strip equals to $\eps$ then, if $\eps$ is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width $ρ/2$ and bounded on this strip by $C\eps$ during very long time of order $ \eps^{-α|\ln \eps|^β}$ for some constants $C> 0$, $α>0$ and $β<1$.