Abstract
As indicated in the introductory chapter, discrete breathers are one of the most prototypical excitations that can arise in nonlinear dynamical lattices. They are periodic in time and strongly localized in space (Flach, Gorbach in Phys Rep 467:1, 2008, Aubry in Physica D 216:1, 2006). In integrable systems such as the sine-Gordon equation, such solutions were long known (Dodd, Eilbeck, Gibbon, Morris in Solitons and nonlinear wave equations. Academic Press, San Diego, 1982, Cuevas, Kevrekidis, Williams in The sine-Gordon model and its applications: from pendula and Josephson junctions to gravity and high energy physics. Springer, Heidelberg, 2014) but were also believed to be rather special. However, a series of investigations by Sievers, Takeno, Page, and others (Sievers, Takeno in Rev Lett 61:970–973, 1988, Page in Phys Rev B 41:7835–7838, 1990) in the late 1980s and early 1990s suggested that they might be generic for nonlinear lattices of the Klein–Gordon (i.e., onsite nonlinearity) and FPUT (i.e., intersite nonlinearity) types. Then, the work of (MacKay, Aubry in Nonlinearity 7:1623, 1994) provided a mathematical foundation for this expectation illustrating that under (fairly generic) non-resonance conditions, discrete breathers should be expected to persist in such nonlinear, spatially extended discrete dynamical systems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
At a much more refined level of analysis involving exponential asymptotics, it can be seen that only these two cases (\(x_0=0\) and \(x_0=0.5\)) persist as solutions of the genuinely discrete problem. A relevant qualitative discussion for a different model can be found, e.g., in [15].
References
C. Chong, P.G. Kevrekidis, G. Theocharis, C. Daraio, Dark breathers in granular crystals. Phys. Rev. E 87, 042202 (2013)
S. Flach, A. Gorbach, Discrete breathers: advances in theory and applications. Phys. Rep. 467, 1 (2008)
S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems. Physica D 216, 1 (2006)
R. Dodd, J. Eilbeck, J. Gibbon, H. Morris, Solitons and Nonlinear Wave Equations (Academic Press, San Diego, 1982)
J. Cuevas, P. Kevrekidis, F. Williams, The sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High Energy Physics (Springer, Heidelberg, 2014)
A.J. Sievers, S. Takeno, Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970–973 (1988)
J.B. Page, Asymptotic solutions for localized vibrational modes in strongly anharmonic periodic systems. Phys. Rev. B 41, 7835–7838 (1990)
R.S. MacKay, S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623 (1994)
N. Boechler, G. Theocharis, S. Job, P.G. Kevrekidis, M.A. Porter, C. Daraio, Discrete breathers in one-dimensional diatomic granular crystals. Phys. Rev. Lett. 104, 244302 (2010)
G. Theocharis, N. Boechler, P.G. Kevrekidis, S. Job, M.A. Porter, C. Daraio, Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Phys. Rev. E 82, 056604 (2010)
Y.S. Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)
M. Remoissenet, Waves Called Solitons (Springer, Berlin, 1999)
G. James, P.G. Kevrekidis, J. Cuevas, Breathers in oscillator chains with Hertzian interactions. Physica D 251, 39 (2013)
P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, The Defocusing Nonlinear Schrödinger Equation (SIAM, Philadelphia, 2015)
Y.S. Kivshar, D.K. Campbell, Peierls-Nabarro potential barrier for highly localized nonlinear modes. Phys. Rev. E 48, 3077–3081 (1993)
G. James, Centre manifold reduction for quasilinear discrete systems. J. Nonlin. Sci. 13, 27 (2003)
C. Chong, F. Li, J. Yang, M.O. Williams, I.G. Kevrekidis, P.G. Kevrekidis, C. Daraio, Damped-driven granular chains: An ideal playground for dark breathers and multibreathers. Phys. Rev. E 89, 032924 (2014)
M. Beck, J. Knobloch, D.J.B. Lloyd, B. Sandstede, T. Wagenknecht, Snakes, ladders, and isolas of localized patterns. SIAM J. Math. Anal. 41, 936–972 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 The Author(s)
About this chapter
Cite this chapter
Chong, C., Kevrekidis, P.G. (2018). Discrete (Dark) Breathers. In: Coherent Structures in Granular Crystals. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-77884-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-77884-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77883-9
Online ISBN: 978-3-319-77884-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)