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Resonant Equilibrium Configurations in Quasi-periodic Media: Perturbative Expansions

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Abstract

We consider 1-D quasi-periodic Frenkel–Kontorova models. We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions to all orders for the quasi-periodic equilibria with resonant frequencies. Under very general conditions, we show that there are at least two such perturbative expansions for equilibria for small values of the parameter. We also develop a dynamical interpretation of the equilibria in these quasi-periodic media. We show that equilibria are orbits of a dynamical system which has very unusual properties. We obtain results on the Lyapunov exponents of the dynamical systems, i.e. the phonon gap of the resonant quasi-periodic equilibria. We show that the equilibria can be pinned even if the gap is zero.

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Acknowledgments

We thank Dr. T. Blass and Mr. A.H. Salahshoor for discussions. R. L. and L. Z. have been supported by DMS-1500943. The hospitality of JLU-GT Joint institute for Theoretical Sciences for the three authors was instrumental in finishing the work. R.L also acknowledges the hospitality of the Chinese Acad. of Sciences and Beijing Normal Univ. X. Su is supported by both National Natural Science Foundation of China (Grant No. 11301513) and “the Fundamental Research Funds for the Central Universities”.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA, 30332, USA

    Rafael de la Llave & Lei Zhang

  2. JLU-GT Joint Institute for Theoretical Science, Jilin University, Changchun, 130012, People’s Republic of China

    Rafael de la Llave

  3. School of Mathematical Sciences, Beijing Normal University, No. 19, XinJieKouWai St., HaiDian District, Beijing, 100875, People’s Republic of China

    Xifeng Su

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  1. Rafael de la Llave
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  2. Xifeng Su
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Correspondence to Xifeng Su.

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de la Llave, R., Su, X. & Zhang, L. Resonant Equilibrium Configurations in Quasi-periodic Media: Perturbative Expansions. J Stat Phys 162, 1522–1538 (2016). https://doi.org/10.1007/s10955-016-1464-5

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