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Section: New Results

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

Participant : Erwan Faou.

In [49] we prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation

iu t =-Δu+Vu+ u ¯ g(u,u ¯),xT d ,

where V is a typical smooth Fourier multiplier 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 whose norm on this strip is equal to ϵ then, if ϵ is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width ρ/2, with norm bounded on this strip by Cϵ over a very long time interval of order ϵ -α|lnϵ| β , where 0<β<1 is arbitrary and C>0 and α>0 are positive constants depending on β and ρ.