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

Abstract

<p style='text-indent:20px;'>This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals.</p><p style='text-indent:20px;'>We present a specific one-dimensional model with an explicit potential driven by the Fibonacci quasi-crystal. For a given positive number <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>, we show that there are multiple equilibria with rotation number <inline-formula><tex-math id="M2">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>, e.g., a minimal configuration and a non-minimal equilibrium configuration. Some numerical experiments verifying the existence of such equilibria are provided.</p>