Gevrey KAM equilibria for quasi-periodic long-range Frenkel-Kontorova models

TL;DR

This paper extends KAM theory for quasi-periodic long-range Frenkel-Kontorova models to Gevrey regularity, proving the persistence of quasi-periodic equilibria under certain conditions.

Contribution

It generalizes previous results from analytic to Gevrey regular potentials, establishing an a posteriori KAM theorem for these models.

Findings

Proves existence of true solutions near approximate solutions in Gevrey spaces.

Shows the preservation of quasi-periodicity and Gevrey regularity in solutions.

Develops a method combining quasi-Newton techniques with Gevrey estimates.

Abstract

We consider models of one-dimensional chains of non-nearest neighbor and many-body interacting particles subjected to quasi-periodic media. We extend the results in \cite{12Su&delaLlavelongrange} from analytic to Gevrey regularity potentials. More precisely, we establish an a posteriori KAM theorem showing that in the Gevrey topology, given an approximate solution of equilibrium equation, which satisfies some appropriate non-degeneracy conditions and decay property, then there is a true solution nearby and the solution preserves both the quasi-periodicity and Gevrey regularity. The method of proof is based on a combination of quasi-Newton methods and delicate estimates in spaces of Gevrey functions.