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
References:
[1]

B. Adamczewski and Y. Bugeaud, Transcendence and Diophantine approximation, in Combinatorics, Automata and Number Theory, vol. 135 of Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge, (2010), 410–451.  Google Scholar

[2]

R. ArtusoG. Casati and D. L. Shepelyansky, Breakdown of universality in renormalization dynamics for critical invariant torus, Europhys. Lett., 15 (1991), 381-386.  doi: 10.1209/0295-5075/15/4/003.  Google Scholar

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S. Aubry and G. Abramovici, Chaotic trajectories in the standard map. The concept of anti-integrability, Phys. D, 43 (1990), 199-219.  doi: 10.1016/0167-2789(90)90133-A.  Google Scholar

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R. BalasubramanianS. H. Kulkarni and R. Radha, Solution of a tridiagonal operator equation, Linear Algebra Appl., 414 (2006), 389-405.  doi: 10.1016/j.laa.2005.10.014.  Google Scholar

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T. Blass and R. de la Llave, The analyticity breakdown for Frenkel-Kontorova models in quasi-periodic media: Numerical explorations, J. Stat. Phys., 150 (2013), 1183-1200.  doi: 10.1007/s10955-013-0718-8.  Google Scholar

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O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10331-9.  Google Scholar

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J.-M. GambaudoP. Guiraud and S. Petite, Minimal configurations for the Frenkel-Kontorova model on a quasicrystal, Comm. Math. Phys., 265 (2006), 165-188.  doi: 10.1007/s00220-006-1531-x.  Google Scholar

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E. GaribaldiS. Petite and P. Thieullen, Calibrated configurations for Frenkel-Kontorova type models in almost periodic environments, Ann. Henri Poincaré, 18 (2017), 2905-2943.  doi: 10.1007/s00023-017-0589-7.  Google Scholar

[9]

Y. Huang and W. F. McColl, Analytical inversion of general tridiagonal matrices, J. Phys. A, 30 (1997), 7919-7933.  doi: 10.1088/0305-4470/30/22/026.  Google Scholar

[10]

P.-L. Lions and P. E. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting, Comm. Pure Appl. Math., 56 (2003), 1501-1524.  doi: 10.1002/cpa.10101.  Google Scholar

[11]

L. Sadun, Topology of Tiling Spaces, vol. 46 of University Lecture Series, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/ulect/046.  Google Scholar

[12]

A. A. Selvan and R. Radha, Invertibility of a tridiagonal operator with an application to a non-uniform sampling problem, Linear Multilinear Algebra, 65 (2017), 973-990.  doi: 10.1080/03081087.2016.1217978.  Google Scholar

[13]

X. Su and R. de la Llave, KAM theory for quasi-periodic equilibria in one-dimensional quasi-periodic media, SIAM J. Math. Anal., 44 (2012), 3901-3927.  doi: 10.1137/12087160X.  Google Scholar

[14]

X. Su and R. de la Llave, A continuous family of equilibria in ferromagnetic media are ground states, Comm. Math. Phys., 354 (2017), 459-475.  doi: 10.1007/s00220-017-2913-y.  Google Scholar

[15]

S. Tompaidis, Numerical study of invariant sets of a quasiperiodic perturbation of a symplectic map, Experiment. Math., 5 (1996), 211–230, http://projecteuclid.org/euclid.em/1047915102. doi: 10.1080/10586458.1996.10504589.  Google Scholar

[16]

R. Treviño, Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals, Comm. Math. Phys., 371 (2019), 1-17.  doi: 10.1007/s00220-019-03557-7.  Google Scholar

show all references

References:
[1]

B. Adamczewski and Y. Bugeaud, Transcendence and Diophantine approximation, in Combinatorics, Automata and Number Theory, vol. 135 of Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge, (2010), 410–451.  Google Scholar

[2]

R. ArtusoG. Casati and D. L. Shepelyansky, Breakdown of universality in renormalization dynamics for critical invariant torus, Europhys. Lett., 15 (1991), 381-386.  doi: 10.1209/0295-5075/15/4/003.  Google Scholar

[3]

S. Aubry and G. Abramovici, Chaotic trajectories in the standard map. The concept of anti-integrability, Phys. D, 43 (1990), 199-219.  doi: 10.1016/0167-2789(90)90133-A.  Google Scholar

[4]

R. BalasubramanianS. H. Kulkarni and R. Radha, Solution of a tridiagonal operator equation, Linear Algebra Appl., 414 (2006), 389-405.  doi: 10.1016/j.laa.2005.10.014.  Google Scholar

[5]

T. Blass and R. de la Llave, The analyticity breakdown for Frenkel-Kontorova models in quasi-periodic media: Numerical explorations, J. Stat. Phys., 150 (2013), 1183-1200.  doi: 10.1007/s10955-013-0718-8.  Google Scholar

[6]

O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-10331-9.  Google Scholar

[7]

J.-M. GambaudoP. Guiraud and S. Petite, Minimal configurations for the Frenkel-Kontorova model on a quasicrystal, Comm. Math. Phys., 265 (2006), 165-188.  doi: 10.1007/s00220-006-1531-x.  Google Scholar

[8]

E. GaribaldiS. Petite and P. Thieullen, Calibrated configurations for Frenkel-Kontorova type models in almost periodic environments, Ann. Henri Poincaré, 18 (2017), 2905-2943.  doi: 10.1007/s00023-017-0589-7.  Google Scholar

[9]

Y. Huang and W. F. McColl, Analytical inversion of general tridiagonal matrices, J. Phys. A, 30 (1997), 7919-7933.  doi: 10.1088/0305-4470/30/22/026.  Google Scholar

[10]

P.-L. Lions and P. E. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting, Comm. Pure Appl. Math., 56 (2003), 1501-1524.  doi: 10.1002/cpa.10101.  Google Scholar

[11]

L. Sadun, Topology of Tiling Spaces, vol. 46 of University Lecture Series, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/ulect/046.  Google Scholar

[12]

A. A. Selvan and R. Radha, Invertibility of a tridiagonal operator with an application to a non-uniform sampling problem, Linear Multilinear Algebra, 65 (2017), 973-990.  doi: 10.1080/03081087.2016.1217978.  Google Scholar

[13]

X. Su and R. de la Llave, KAM theory for quasi-periodic equilibria in one-dimensional quasi-periodic media, SIAM J. Math. Anal., 44 (2012), 3901-3927.  doi: 10.1137/12087160X.  Google Scholar

[14]

X. Su and R. de la Llave, A continuous family of equilibria in ferromagnetic media are ground states, Comm. Math. Phys., 354 (2017), 459-475.  doi: 10.1007/s00220-017-2913-y.  Google Scholar

[15]

S. Tompaidis, Numerical study of invariant sets of a quasiperiodic perturbation of a symplectic map, Experiment. Math., 5 (1996), 211–230, http://projecteuclid.org/euclid.em/1047915102. doi: 10.1080/10586458.1996.10504589.  Google Scholar

[16]

R. Treviño, Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals, Comm. Math. Phys., 371 (2019), 1-17.  doi: 10.1007/s00220-019-03557-7.  Google Scholar

Figure 1.  the two possible position of $ x $ in the quasi-crystal
Figure 2.  the graph of $ \zeta $
Figure 3.  the graph of $ V $
Figure 4.  Branched Manifold $ \mathcal{B}_l $
Figure 5.  the map $ \kappa_l $
Figure 6.  The red points are the graph of $ (\theta_{5,n})_{n\in[-50,50]} $ and the green line is $ y = (3\tau+1)/2x $
Figure 7.  The configuration $ T_{2n+1}^{-1}y_{n} $ for $ h $ and $ h_1 $ when $ n = 500 $
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