ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


Cite this article as:

Kanakov O. I., Flah S. ., Shalfeev V. D. Introduction to discrete breathers theory. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 3, pp. 112-128. DOI: https://doi.org/10.18500/0869-6632-2008-16-3-112-128

Language: 
Russian

Introduction to discrete breathers theory

Autors: 
Kanakov Oleg Igorevich, Federal state budgetary educational institution of higher professional education "Nizhny Novgorod state University named N. And.Lobachevsky"
Flah Sergej , Institute of Complex Systems of the Max Planck Society
Shalfeev Vladimir Dmitrievich, Federal state budgetary educational institution of higher professional education "Nizhny Novgorod state University named N. And.Lobachevsky"
Abstract: 

We make a basic review of the theory of discrete breathers – spatially localized solutions in nonlinear lattices. We describe the mathematical conditions and physical prerequisites of their existence and methods of their study by example of one-dimensional lattices. We consider localized solutions with infinite and finite lifetimes. We include some new results within the problems of discrete breather generation resulting from harmonic wave destruction and controlling the formation of rotational breather solutions by external forcing.

Key words: 
DOI: 
10.18500/0869-6632-2008-16-3-112-128
References: 

1. Flach S., Willis C.R. Discrete breathers // Phys. Reports. 1998. Vol. 295. P. 181. 2. Flach S., Gorbach A.V. Computational studies of discrete breathers – from basics to competing length scales // Int. J. Bif. Chaos. 2006. Vol. 16. P. 1645. 3. Flach S., Gorbach A. Discrete breathers: advances in theory and applications // принято к печати в Physics Reports, 2008. 4. MacKay R.S., Aubry S. Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators // Nonlinearity. 1994. Vol. 7. P. 1623. 5. Sato M., Sievers A.J. Direct observation of the discrete character of intrinsic localized modes in an antiferromagnet // Nature. 2004. Vol. 432. P. 486. 6. Fleischer J.W., Carmon T., Segev M., Efremidis N.K. Christodoulides D.N. Observation of discrete solitons in optically induced real time waveguide arrays // Phys. Rev. Lett. 2003. Vol. 90, No 2. P. 023902. 7. Sato M., Hubbard B.E., Sievers A.J., Ilic B., Czaplewski D.A., Craighead H.G. Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array // Phys. Rev. Lett. 2003. Vol. 90, No 4. P. 044102. 8. Zalalutdinov M.K., Baldwin J.W., Marcus M.H., Reichenbach R.B., Parpia J.M., Houston B.H. Two-dimensional array of coupled nanomechanical resonators // Applied Physics Letters. 2006. Vol. 88. P. 143504. 9. Takeno S., Peyrard M. Nonlinear modes in coupled rotator models // Physica D. 1996. Vol. 92. P. 140. 10. Miroshnichenko A., Flach S., Fistul M., Zolotaryuk Y., Page J.B. Breathers in Josephson junction ladders: resonances and electromagnetic waves spectroscopy // Phys. Rev. E. 2001. Vol. 64. P. 066601. 11. Binder P., Abraimov D., Ustinov A.V. Diversity of discrete breathers observed in a Josephson ladder // Phys. Rev. E. 2000. Vol. 62, No 2. P. 2858. 12. Веников В.А. Переходные электромеханические процессы в электрических системах. М.: Высшая школа, 1978. 13. Афраймович В.С., Некоркин В.И., Осипов Г.В., Шалфеев В.Д. Устойчивость, структуры и хаос в нелинейных сетях синхронизации. Горький: ИПФ АН СССР, 1989. 14. Островский Л.А., Потапов А.И. Введение в теорию модулированных волн. М.: Физматлит, 2003. C. 359. 15. Denzler J. Nonpersistence of breather families for the perturbed sine Gordon equation // Commun. Math. Phys. 1993. Vol. 158. P. 397. 16. Овчинников А.А. Локализованные долгоживущие колебательные состояния в молекулярных кристаллах // ЖЭТФ. 1969. Т. 57, вып. 1(7). C. 263. 17. Takeno S., Kisoda K., Sievers A.J. Intrinsic localized vibrational modes in anharmonic crystals // Prog. Theor. Phys. Suppl. // 1988. Vol. 94. P. 242. 18. Campbell D.K., Peyrard M. Chaos and order in nonintegrable model field theories // CHAOS – Soviet-American Perspectives on Nonlinear Science / Ed. D.K. Campbell. New York: American Institute of Physics, 1990. P. 305. 19. Aubry S. Breathers in nonlinear lattices: existence, linear stability and quantization // Physica D. 1997. Vol. 103. P. 201. 20. Marin J.L., Aubry S. Breathers in nonlinear lattices: Numerical calculation from the anticontinuous limit // Nonlinearity. 1996. Vol. 9. P. 1501. 21. Flach S., Willis C.R., Olbrich E. Integrability and localized excitations in nonlinear discrete systems // Phys. Rev. E. 1994. Vol. 49. P. 836. 22. Flach S., Kladko K., MacKay R.S. Energy thresholds of discrete breathers in one-, two- and three-dimensional lattices // Phys. Rev. Lett. 1997. Vol. 78. P. 1207. 23. Косевич А.М., Ковалев А.С. Самолокализация колебаний в одномерной ангармонической цепочке // ЖЭТФ. 1974. Т. 67. Вып. 5(11). C. 1793. 24. Kivshar Yu.S., Peyrard M. Modulational instabilities in discrete lattices // Phys. Rev. A. 1992. Vol. 46. P. 3198. 25. Daumont I., Dauxois T., Peyrard M. Modulational instability: first step towards energy localization in nonlinear lattices // Nonlinearity. 1997. Vol. 10. P. 617. 26. Peyrard M. The pathway to energy localization in nonlinear lattices // Physica D. 1998. Vol. 119. P. 184. 27. Cretegny T., Dauxois T., Ruffo S., Torcini A. Localization and equipartition of energy in the beta-FPU chain: Chaotic breathers // Physica D. 1998. Vol. 121. P. 109. 28. Johansson M., Morgante A.M., Aubry S., Kopidakis G. Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice // Eur. Phys. J. B. 2002. Vol. 29. P. 279. 29. Ivanchenko M.V., Kanakov O.I., Shalfeev V.D., Flach S. Discrete Breathers in Transient Processes and Thermal Equilibrium // Physica D. 2004. Vol. 198. P. 120. 30. Tsironis G.P., Aubry S. Slow relaxation phenomena induced by breathers in nonlinear lattices // Phys. Rev. Lett. 1996. Vol. 77. P. 5225. 31. Bikaki A., Voulgarakis N.K., Aubry S., Tsironis G.P. Energy relaxation in discrete nonlinear lattices // Phys. Rev. E. 1999. Vol. 59. P. 1234. 32. Flach S., Miroshnichenko A.E., Fistul M.V. Wave scattering by discrete breathers // CHAOS. 2003. Vol. 13. P. 596. 33. Miroshnichenko A.E., Schuster M., Flach S., Fistul M.V., Ustinov A.V. Resonant plasmon scattering by discrete breathers in Josephson junction ladders // Phys. Rev. B. 2005. Vol. 71. P. 174306. 34. Flach S., Kladko K., Willis C.R. Localized excitations in two-dimensional lattices // Phys. Rev. E. 1994. Vol. 50. P. 2293. 35. MacKay R.S. Discrete breathers: classical and quantum // Physica A. 2000. Vol. 288. P. 174. 36. Flach S., Fleurov V., Ovchinnikov A.A. Tunneling of localized excitations: Giant enhancement due to fluctuations // Phys. Rev. B. 2001. Vol. 63. P. 094304. 37. Flach S., Ivanchenko M.V., Kanakov O.I. q-Breathers and the Fermi–Pasta–Ulam problem // Phys. Rev. Lett. 2005. Vol. 95. P. 064102. 38. Flach S., Ivanchenko M.V., Kanakov O.I., Mishagin K.G. Periodic orbits, localization in normal mode space and the Fermi–Pasta–Ulam problem // American Journal of Physics. 2008. Vol. 76, No 4/5. P. 453.

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