Phonon gap analysis for equilibria of perturbed almost-periodic Frenkel-Kontorova models

TL;DR

This paper investigates phonon gaps in equilibria of perturbed almost-periodic Frenkel-Kontorova models, revealing that KAM equilibria lack phonon gaps while anti-integrable limit equilibria possess them.

Contribution

It provides a comparative analysis of phonon gaps in different types of equilibria derived from KAM and anti-integrable methods in almost-periodic Frenkel-Kontorova models.

Findings

KAM equilibria do not have phonon gaps.

Anti-integrable limit equilibria have phonon gaps.

The study clarifies the spectral properties of these equilibria.

Abstract

For generalized Frenkel-Kontorova models subjected to almost-periodic media, employing both the KAM method and the approach of `anti-integrable' limits, two different types of equilibria are obtained in \cite{an2024kamtheoryalmostperiodicequilibria} and \cite{du2024anti} respectively. We study the phonon gap around these equilibria and we find that the KAM equilibria do not have a phonon gap but the equilibria obtained by anti-integrable limits do have.