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

Jianxing Du; Xifeng Su; Jianxing Du; Xifeng Su

doi:10.3934/era.2021078

Electronic Research Archive

2021, Volume 29, 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

Jianxing Du ^{1
,},
Xifeng Su ^{1,2
,
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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)

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.
- Quasi-crystals,
- quasi-periodic Frenkel-Kontorova models,
- minimal configurations,
- non-minimal equilibria
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

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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.

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