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 $\epsilon$ small enough, the solution of the Schr{ö}dinger-Poisson equation remains smaller than $2\epsilon$ for times of order $exp(\alpha \mathrm{\mid }\log \epsilon {\mathrm{\mid }}^2\mathrm{/}\log \mathrm{\mid }\log \epsilon \mathrm{\mid })$. We stress out that this is the optimal time expected for PDEs as conjectured by Jean Bourgain in [Bou04].