## 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

$i{u}_{t}=-\Delta u+V☆u+{\partial }_{\overline{u}}g\left(u,\overline{u}\right)\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}x\in {T}^{d},$

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 $\rho >0$ 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 $\rho /2$, with norm bounded on this strip by $Cϵ$ over a very long time interval of order ${ϵ}^{-{\alpha |lnϵ|}^{\beta }}$, where $0<\beta <1$ is arbitrary and $C>0$ and $\alpha >0$ are positive constants depending on $\beta$ and $\rho$.