Anti-integrable limits for generalized Frenkel-Kontorova models on almost-periodic media

TL;DR

This paper develops an anti-integrable perturbation theory for generalized Frenkel-Kontorova models in almost-periodic media, demonstrating the existence of unique, hyperbolic equilibrium configurations with prescribed rotation parameters.

Contribution

It introduces an anti-integrable limit approach for chaotic systems, extending the analysis of Frenkel-Kontorova models beyond traditional KAM theory.

Findings

Existence of unique hyperbolic equilibria for large potentials

Applicable to both short-range and long-range models

Valid for multidimensional and almost-periodic media

Abstract

We study the equilibrium configurations for generalized Frenkel-Kontorova models subjected to almost-periodic media. By contrast with the spirit of the KAM theory, our approach consists in establishing the other perturbation theory for fully chaotic systems far away from the integrable, which is called "anti-integrable" limits. More precisely, we show that for large enough potentials, there exists a locally unique equilibrium with any prescribed rotation number/vector/plane, which is hyperbolic. The assumptions are general enough to satisfy both short-range and long-range Frenkel-Kontorova models and their multidimensional analogues.