Exponential stability of solutions to the Schr{\"o}dinger-Poisson equation

TL;DR

This paper proves that small solutions to the Schrödinger-Poisson equation on the circle exhibit exponential stability over very long times, matching conjectured optimal stability durations for PDEs.

Contribution

It establishes the exponential stability of small solutions in Gevrey class for the Schrödinger-Poisson equation without exterior parameters, over near-optimal times.

Findings

Solutions remain small for exponentially long times

Stability holds for most initial data in Gevrey norm

Time scale matches conjectured optimal stability duration

Abstract

We prove an exponential stability result for the small solutions of the Schr{\"o}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{\"o}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].