# American Institute of Mathematical Sciences

December  2021, 29(6): 4177-4198. doi: 10.3934/era.2021078

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

 1 School of Mathematical Sciences, Beijing Normal University, No. 19, XinJieKouWai St., HaiDian District, Beijing 100875, China 2 Laboratory of Mathematics and Complex Systems (Ministry of Education)

* Corresponding author: Xifeng Su

Received  December 2020 Revised  August 2021 Published  December 2021 Early access  October 2021

Fund Project: The second author is supported by the National Natural Science Foundation of China (Grant No. 11971060, 11871242)

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.

Citation: Jianxing Du, Xifeng Su. On the existence of solutions for the Frenkel-Kontorova models on quasi-crystals. Electronic Research Archive, 2021, 29 (6) : 4177-4198. doi: 10.3934/era.2021078
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##### References:
the two possible position of $x$ in the quasi-crystal
the graph of $\zeta$
the graph of $V$
Branched Manifold $\mathcal{B}_l$
the map $\kappa_l$
The red points are the graph of $(\theta_{5,n})_{n\in[-50,50]}$ and the green line is $y = (3\tau+1)/2x$
The configuration $T_{2n+1}^{-1}y_{n}$ for $h$ and $h_1$ when $n = 500$
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