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Physics Reports Discrete Breathers Physics Reports 467 (2008) 1–116 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Discrete breathers — Advances in theory and applications Sergej Flach a,∗, Andrey V. Gorbach b a Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany b Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath BA2 7AY, UK article info a b s t r a c t Article history: Nonlinear classical Hamiltonian lattices exhibit generic solutions — discrete breathers. Accepted 7 May 2008 They are time-periodic and (typically exponentially) localized in space. The lattices have Available online 20 May 2008 discrete translational symmetry. Discrete breathers are not confined to certain lattice editor: I. Procaccia dimensions. We will introduce the concept of these localized excitations and review their basic properties including dynamical and structural stability. We then focus on advances PACS: in the theory of discrete breathers in three directions — scattering of waves by these 02.00.00 excitations, persistence of discrete breathers in long transient processes and thermal 05.00.00 05.10.-a equilibrium, and their quantization. The second part of this review is devoted to a detailed 45.00.00 discussion of recent experimental observations and studies of discrete breathers, including 45.10.-b theoretical modelling of these experimental situations on the basis of the general theory 03.75.Lm of discrete breathers. In particular we will focus on their detection in Josephson junction 67.85.Hj networks, arrays of coupled nonlinear optical waveguides, Bose–Einstein condensates 42.65.-k loaded on optical lattices, antiferromagnetic layered structures, PtCl based single crystals 63.00.00 and driven micromechanical cantilever arrays. 63.20.Ry © 2008 Elsevier B.V. All rights reserved. Keywords: Anharmonic lattice dynamics Discrete breathers Localization Contents 1. Introduction............................................................................................................................................................................................. 3 1.1. Nonlinearity and symmetry — the dimer.................................................................................................................................. 4 1.2. Spatial discreteness and nonlinearity........................................................................................................................................ 5 1.3. Why only time-periodic orbits? ................................................................................................................................................ 6 1.4. Examples of discrete breather solutions ................................................................................................................................... 7 1.5. The discrete nonlinear Schrödinger equation........................................................................................................................... 8 2. Existence proofs ...................................................................................................................................................................................... 9 2.1. Networks of weakly interacting anharmonic oscillators ......................................................................................................... 10 2.2. Anharmonic interactions............................................................................................................................................................ 10 2.3. Breathers with frequencies close to the linear spectrum ........................................................................................................ 10 2.4. Applicability ................................................................................................................................................................................ 10 3. Computational methods for obtaining and analyzing discrete breathers........................................................................................... 11 Abbreviations: BEC, Bose–Einstein condensate; BEM, band edge mode; DB, discrete breather; DNLS, discrete nonlinear Schrödinger equation; DS, discrete soliton; FPU, Fermi, Pasta and Ulam; ILM, intrinsic localized mode; NLS, nonlinear Schrödinger equation; PDE, partial differential equation; PO, periodic orbit; QB, quantum breather; RWA, rotating wave approximation. ∗ Corresponding author. Tel.: +49 3518712103; fax: +49 3518712199. E-mail address: [email protected] (S. Flach). 0370-1573/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2008.05.002 2 S. Flach, A.V. Gorbach / Physics Reports 467 (2008) 1–116 3.1. Targeted initial conditions ......................................................................................................................................................... 11 3.2. Obtaining DB solutions with high precision ............................................................................................................................. 12 3.2.1. Periodic orbits in phase space..................................................................................................................................... 12 3.2.2. Periodic orbits from Fourier space.............................................................................................................................. 15 3.2.3. Use of symmetries ....................................................................................................................................................... 15 3.2.4. Dissipative DBs ............................................................................................................................................................ 15 3.3. Perturbing discrete breathers .................................................................................................................................................... 15 4. Basic properties of discrete breathers ................................................................................................................................................... 17 4.1. Spatial localization...................................................................................................................................................................... 17 4.1.1. Short-range interactions ............................................................................................................................................. 18 4.1.2. Long-range interactions .............................................................................................................................................. 18 4.1.3. Almost compact discrete breathers............................................................................................................................ 20 4.1.4. Resonances with Goldstone modes ............................................................................................................................ 23 4.1.5. Nonlinear corrections.................................................................................................................................................. 24 4.2. Dynamical stability of perturbed discrete breathers................................................................................................................ 25 4.2.1. Linear stability ............................................................................................................................................................. 25 4.2.2. Going beyond linearization......................................................................................................................................... 29 4.3. Modulational instability of extended states and discrete breathers ....................................................................................... 29 4.3.1. Tangent bifurcations of band edge plane waves ....................................................................................................... 29 4.3.2. Tangent bifurcations of the in-phase mode ............................................................................................................... 32 4.3.3. Tangent bifurcations of the out-of-phase mode........................................................................................................ 32 4.3.4. Symmetry breaking ..................................................................................................................................................... 34 4.3.5. Energy thresholds for tangent bifurcations of BEMs................................................................................................. 34 4.4. Energy thresholds of discrete breathers.................................................................................................................................... 34 4.4.1. Short range interaction case ....................................................................................................................................... 35 4.4.2. Long range interactions............................................................................................................................................... 36 4.4.3. No bifurcation but finite thresholds ........................................................................................................................... 37 4.4.4. Some concluding remarks..........................................................................................................................................
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