Nekhoroshev type stability for non-local semilinear Schrödinger equations
Bingqi Yu, Li Yong
TL;DR
The paper advances Nekhoroshev-type stability for non-local Schrödinger equations by developing an internal-parameter framework via rational normal forms in an infinite-dimensional Hamiltonian setting. It introduces a fractional, vector-field norm adapted to this framework, enabling a unified treatment of nonlinear terms without degreed tracking and yielding sharp measure estimates for resonant sets across polynomial and exponential kernel decays. The authors establish stability results in Gevrey and logarithmic ultra-differentiable regimes, achieving Bourgain-like stability times for Gevrey regularity and analogous sub-exponential bounds in the ultra-differentiable case, with explicit time scales derived through a careful truncation and integrable normal-form analysis. The work combines a robust resonant-normal-form construction, integrable normal-form lemmas on finite-dimensional truncations, and precise bootstrap and measure-estimation arguments to extend Nekhoroshev-type results to non-local, infinite-dimensional PDEs, potentially impacting long-time dynamics in nonlinear dispersive systems.
Abstract
This paper investigates Nekhoroshev-type stability for solutions of ultra-differentiable regularity in Schrödinger 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.