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

Abstract

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 $exp(\alpha |\log \epsilon {|}^2/\log |\log \epsilon |)$. We stress out that this is the optimal time expected for PDEs as conjectured by Jean Bourgain in [Bou04].

References in corpus (4)

• Rational normal forms and stability of small solutions to nonlinear Schrödinger equations
• Exact Global Control of Small Divisors in Rational Normal Form
• Almost global existence for some nonlinear Schr{ö}dinger equations on ${\mathbb{T}}^d$ in low regularity
• Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori