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

Abstract

We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation

$$i{u}_t=-\Delta u+V\star u+{\partial }_{ū}g\left(u,ū\right),x\in {\mathbb{T}}^d,$$

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 ${\epsilon }^{-\sigma |ln\epsilon {|}^{\beta }}$, where $0<\beta <1$ is arbitrary and $C>0$ and $\sigma >0$ are positive constants depending on $\beta$ and $\rho$.