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

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 \begin{document}$ \theta $\end{document} , we show that there are multiple equilibria with rotation number \begin{document}$ \theta $\end{document} , e.g., a minimal configuration and a non-minimal equilibrium configuration. Some numerical experiments verifying the existence of such equilibria are provided.