Resonant Equilibrium configurations in quasi-periodic media: perturbative expansions

TL;DR

This paper investigates the existence and properties of resonant equilibria in 1-D quasi-periodic Frenkel-Kontorova models, providing perturbative expansions and analyzing their dynamical and spectral characteristics.

Contribution

It develops perturbation theory for resonant equilibria in quasi-periodic media and explores their dynamical and spectral properties, including Lyapunov exponents and pinning behavior.

Findings

Existence of perturbative expansions for resonant equilibria to all orders.

At least two distinct perturbative solutions exist under general conditions.

Equilibria can be pinned even with zero phonon gap.

Abstract

We consider 1-D quasi-periodic Frenkel-Kontorova models. We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions to all orders for the quasi-periodic equilibria with resonant frequencies. Under very general conditions, we show that there are at least two such perturbative expansions for equilibria for small values of the parameter. We also develop a dynamical interpretation of the equilibria in these quasi-periodic media. We show that equilibria are orbits of a dynamical system which has very unusual properties. We obtain results on the Lyapunov exponents of the dynamical systems, i.e. the phonon gap of the resonant quasi-periodic equilibria. We show that the equilibria can be pinned even if the gap is zero.