Nekhoroshev type stability for non-local semilinear Schr\"odinger equations

TL;DR

This paper proves Nekhoroshev-type stability for non-local semilinear Schrödinger equations with ultra-differentiable regularity, achieving optimal stability times and introducing a new vector field norm for better analysis.

Contribution

It provides the first rigorous Nekhoroshev stability results for infinite-dimensional Hamiltonian systems with logarithmic ultra-differentiable regularity and introduces a novel norm to streamline the analysis.

Findings

Established Nekhoroshev stability for non-local Schrödinger equations.

Achieved stability times matching conjectured optimal times under Gevrey regularity.

Developed a new global vector field norm simplifying the normal form iteration.

Abstract

This paper investigates Nekhoroshev-type stability for solutions of ultra-differentiable regularity in Schr\"odinger equations with non-local nonlinear terms, employing the method of rational normal forms. We establish the first rigorous results for logarithmic ultra-differentiable regularity in infinite-dimensional Hamiltonian systems without external parameters. Under Gevrey class regularity assumptions, we achieve the stability times matching Bourgain's conjectured optimal stability time in \cite{B04}. Furthermore, we introduce a novel global vector field norm adapted to the rational normal form framework. This norm eliminate the need for degree tracking during the iteration process, thereby enabling a unified treatment of nonlinear terms.