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

TL;DR

This paper investigates the existence of multiple equilibrium solutions in one-dimensional Frenkel-Kontorova models influenced by Fibonacci quasi-crystal potentials, demonstrating both theoretical and numerical evidence for diverse equilibria.

Contribution

It introduces a specific Fibonacci quasi-crystal potential model and proves the existence of multiple equilibria with the same rotation number, supported by numerical experiments.

Findings

Multiple equilibria exist for given rotation numbers.

Both minimal and non-minimal equilibria are demonstrated.

Numerical experiments verify theoretical results.

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.