On the existence of solutions for Frenkel-Kontorova models on quasi-crystals

Abstract

This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals. We present a specific one-dimensional model with an explicit potential driven by the Fibonacci quasi-crystal. For a given positive number $\theta$, we show that there are multiple equilibria with rotation number $\theta$, e.g., a minimal configuration and a non-minimal equilibrium configuration. Some numerical experiments verifying the existence of such equilibria are provided.