Exponential stability of solutions to the Schr{ö}dinger-Poisson equation

We prove an exponential stability result for the small solutions of the Schr{ö}dinger-Poisson equation on the circle without exterior parameters in Gevrey class. More precisely we prove that for most of the initial data of Gevrey-norm smaller than $\varepsilon$ small enough, the solution of the Schr{ö}dinger-Poisson equation remains smaller than $2\varepsilon$ for times of order $exp(\alpha |\log \varepsilon|^2 / \log |\log \varepsilon|)$. We stress out that this is the optimal time expected for PDEs as conjectured by Jean Bourgain in [Bou04].