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

We prove a Nekhoroshev type theorem for the nonlinear Schrodinger equation iut = −�u + V ? u + @¯ ug(画 ¯ u) ; x2 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 � > 0 with a bound on this strip equals to " then, if " is small enough, the solution of the nonlinear Schrodinger equation above remains analytic in a strip of width ᨽ 2 and bounded on this strip by C" during very long time of order " −�jln"j � for some constants C > 0, � > 0 and � < 1.

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