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2021, Issue 6: 4177-4198. doi: 10.3934/era.2021078
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# On the existence of solutions for the Frenkel-Kontorova models on quasi-crystals

• Received: 01 December 2020 Revised: 01 August 2021 Published: 08 October 2021
• Primary: 70K43, 52C23, 37A60; Secondary: 82B20

• 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[J]. Electronic Research Archive, 2021, 29(6): 4177-4198. doi: 10.3934/era.2021078

### Related Papers:

• 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.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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