#### Vol. 6, No. 6, 2013

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

### Erwan Faou and Benoît Grébert

Vol. 6 (2013), No. 6, 1243–1262
##### Abstract

We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation

$i{u}_{t}=-\Delta u+V\star u+{\partial }_{ū}g\left(u,ū\right),\phantom{\rule{1em}{0ex}}x\in {\mathbb{T}}^{d},$

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 $\epsilon$, then if $\epsilon$ 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 $C\epsilon$ over a very long time interval of order ${\epsilon }^{-\sigma |\phantom{\rule{0.3em}{0ex}}ln\epsilon {|}^{\beta }}$, where $0<\beta <1$ is arbitrary and $C>0$ and $\sigma >0$ are positive constants depending on $\beta$ and $\rho$.

##### Keywords
Nekhoroshev theorem, nonlinear Schrödinger equation, normal forms
##### Mathematical Subject Classification 2010
Primary: 35B40, 35Q55, 37K55