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Journal of Statistical Physics

, Volume 162, Issue 6, pp 1522–1538 | Cite as

Resonant Equilibrium Configurations in Quasi-periodic Media: Perturbative Expansions

  • Rafael de la Llave
  • Xifeng SuEmail author
  • Lei Zhang
Article

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.

Keywords

Quasi-periodic Frenkel–Kontorova models Resonant frequencies Equilibria Quasicrystals Lindstedt series  Counterterms 

Mathematics Subject Classification

70K43 37J50 37J40 52C23 

Notes

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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.JLU-GT Joint Institute for Theoretical ScienceJilin UniversityChangchunPeople’s Republic of China
  3. 3.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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