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Bezuglova G. S., Goncharov P. P., Gurov J. V., Chechin G. M. Discrete breathers in scalar dynamical models on the plane square lattice. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 3, pp. 89-103. DOI: 10.18500/0869-6632-2011-19-3-89-103

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Discrete breathers in scalar dynamical models on the plane square lattice

Bezuglova Galina Sergeevna, Southern Federal University
Goncharov Petr Petrovich, Southern Federal University
Gurov Jurij Vladimirovich, Southern Federal University
Chechin Georgij Mihajlovich, Southern Federal University

All symmetry related invariant manifolds, admitting localized vibrations, for dynamical models on plane square lattice were found by group­theoretical methods. Discrete breathers were constructed on these manifolds for the model with homogeneous potentials of interparticle interactions and their stability was studied. Nontrivial breather solutions which are not nonlinear normal modes by Rosenberg have been revealed for the above model despite it admits space­time separation of dynamical variables. Discrete breathers of the same type were also found in the system of linear coupled Duffing oscillators situated in sites of square lattice. Our approach for studying discrete breathers can be spread to different two­ and three­dimensional space­periodic dynamical models.

  1. Aubry S. Breathers in nonlinear lattices: existence, linear stability and quantization. Physica D. 1997;103(1–4):201–250. DOI: 10.1016/S0167-2789(96)00261-8.
  2. Flach S, Willis CR. Discrete breathers. Physics Reports. 1997;295(5):181–264. DOI: 10.1016/S0370-1573(97)00068-9.
  3. Aubry S. Discrete breathers: Localization and transfer of energy in discrete Hamiltonian nonlinear systems. Physica D. 2006;216(1):1–30. DOI: 10.1016/j.physd.2005.12.020.
  4. Flach S, Gorbach A. Discrete breathers – Advances in theory and applications. Physics Reports. 2007;467(1–3):1–116. DOI: 10.1016/j.physrep.2008.05.002.
  5. Fischer F. Self-localized single-anharmonic vibrational modes in two-dimensional lattices. Ann. Physik. 1993;505(3):296–307. DOI: 10.1002/andp.19935050308.
  6. Flach F, Kladko K, Willis CR. Acoustic breathers in two-dimensional lattices. Phys. Rev. Lett. 1997;79(24):4838–4841. DOI: 10.1103/PhysRevLett.79.4838.
  7. Kiselev SA, Sievers AJ. Generation of intrinsic vibrational gap modes in three-dimensional ionic crystals. Phys. Rev. B. 1997;55(9):5755–5758. DOI: 10.1103/PhysRevB.55.5755.
  8. Kevrekidis PG, Rasmussen KO, Bishop AR. Two-dimensional discrete breathers: Construction, stability, and bifurcations. Phys. Rev. E. 2000;61(2):2006–2009. DOI: 10.1103/physreve.61.2006.
  9. Doi Y, Nakatani A. Structures of discrete breathers in two-dimensional Fermi–Pasta–Ulam lattices. Theor. Appl. Mech. Jpn. 2006;55:103–110. DOI: 10.11345/nctam.55.103.
  10. Butt IA, Wattis JAD. Discrete breathers in a two-dimensional Fermi–Pasta–Ulam lattice. J. Phys. A. Math. Gen. 2006;39(18):4955. DOI: 10.1088/0305-4470/39/18/013.
  11. Butt IA, Wattis JAD. Discrete breathers in a two-dimensional hexagonal Fermi–Pasta–Ulam lattice. J. Phys. A. Math. Gen. 2006;39(18):4955. DOI: 10.1088/0305-4470/39/18/013.
  12. Feng BF, Kawahara T. Discrete breathers in two-dimensional nonlinear lattices. Wave Motion. 2007;45(1–2):68–82. DOI: 10.1016/j.wavemoti.2007.04.002.
  13. Xu Q, Qiang T. Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein–Gordon lattice. Chin. Phys. Lett. 2009;26(7):070501. DOI: 10.1088/0256-307X/26/7/070501.
  14. Yi X, Wattis JAD, Susanto H, Cummings LJ. Discrete breathers in a two-dimensional spring-mass lattice. J. Phys. A. Math. Theor. 2009;42(35):355207–355226. DOI: 10.1088/1751-8113/42/35/355207.
  15. Dmitriev SV, Khadeeva LZ, Pshenichnyuk AI, Medvedev NN. Slot discrete breathers in two-component three-dimensional and two-dimensional crystals with interatomic Morse potentials. Physics of the Solid State. 2010;52(7):1398–1403 (in Russian).
  16. Koukouloyannis V, Kevrekidis PG, Law KJH, Kourakis I, Frantzeskakis DJ. Existence and stability of multisite breathers in honeycomb and hexagonal lattices. J. Phys. A. Math. Theor. 2010;43(23):235101. DOI: 10.1088/1751-8113/43/23/235101.
  17. Sakhnenko VP, Chechin GM. Symmetry selection rules in nonlinear dynamics of atomic systems. Proc. Acad. Sci. 1993;330(3):308–310 (in Russian).
  18. Chechin GM, Sakhnenko VP. Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results. Physica D. 1998;117(1–4):43–76. DOI: 10.1016/S0167-2789(98)80012-2.
  19. Chechin GM, Ryabov DS, Sakhnenko VP. Bushes of normal modes as exact excitations in nonlinear dynamical systems with discrete symmetry. In: Wang CW, editor. Nonlinear Phenomena Research Perspectives. Nova Science Publishers, NY; 2007. P. 225.
  20. Bokiy GB. Crystal Vhemistry. Moscow: MSU Publishing; 1960. 357 p. (in Russian).
  21. Chechin GM. Computers and group-theoretical methods for studying structural phase transitions. Comput. Math. Appl. 1989;17(1–3):255–278.
  22. Rosenberg RM. On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 1966;9:155–242. DOI: 10.1016/S0065-2156(08)70008-5.
  23. Chechin GM, Dzhelauhova GS, Mehonoshina EA. Quasibreathers as a generalization of the concept of discrete breathers. Phys. Rev. E. 2006;74(3):036608. DOI: 10.1103/physreve.74.036608.
  24. Goncharov PP, Dzhelauhova GS, Chechin GM. Discrete breathers and quasibreathers in nonlinear monoatomic chains. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(6):57–74 (in Russian). DOI: 10.18500/0869-6632-2007-15-6-57-74.
  25. Chechin GM, Dzhelauhova GS. Discrete breathers and nonlinear normal modes in monoatomic chains. Journal of Sound and Vibration. 2009;322(3):490–512.
  26. Avramov KV, Mikhlin YV. Nonlinear Dynamics of Elastic Systems. Models, Methods, Phenomena. Vol. 1. Izhevsk: RCD; 2010. 704 p. (in Russian). 
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