Discrete breathers

Flach, Sergej; Willis, C. R.

doi:10.1016/s0370-1573(97)00068-9

Cited by 1,195 publications

(996 citation statements)

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“…The existence of ILMs (also called breathers) on periodic chains and the complex behavior of more arbitrary high-frequency initial conditions has led to extensive study of these structures to understand their stability. A comprehensive review of these studies would lead us far from the main topic of this review (see [84] for a review and further references). The breathers can be stationary or moving, and, like low-frequency solitons, can pass through one another.…”

Section: Dynamics At Short Wavelengths: Chaotic Breathersmentioning

confidence: 99%

“…For "strong spring" potentials the breather frequency is above the optical band, so a more subtle energy interchange must occur [84,87,95,96]. A beat phenomenon has been postulated as the energy interchange mechanism, and used to calculate an ε-scaling that agrees with numerics [92].…”

Section: Dynamics At Short Wavelengths: Chaotic Breathersmentioning

confidence: 99%

“…Similarly, for a strong spring (β > 0) but with an optical branch, a nonlinear acoustic mode can be shifted into the optical branch where it can dissipate by loosing energy to that branch. The various treatments of these phenomena are a major area for study (see, e.g., [84]), beyond the subject matter of this review. However, the phenomena will reappear in various subsequent sections.…”

Section: Representation In Modes Of the Linear Systemmentioning

confidence: 99%

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Dynamics of Oscillator Chains

Lichtenberg

Livi

Pettini

et al.

The Fermi-Pasta-Ulam Problem

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Abstract. The Fermi-Pasta-Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of

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Section: Dynamics At Short Wavelengths: Chaotic Breathersmentioning

confidence: 99%

Section: Dynamics At Short Wavelengths: Chaotic Breathersmentioning

confidence: 99%

Section: Representation In Modes Of the Linear Systemmentioning

confidence: 99%

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Dynamics of Oscillator Chains

Lichtenberg

Livi

Pettini

et al.

The Fermi-Pasta-Ulam Problem

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“…Despite the given translational invariance of a lattice, nonlinearity may trap initially localized excitations. The generic existence and properties of discrete breathers -time-periodic and spatially localized solutions of the underlying classical equations of motion -allow us to describe and understand these localization phenomena [1,2,3,4]. Discrete breathers were observed in many different systems like bond excitations in molecules, lattice vibrations and spin excitations in solids, electronic currents in coupled Josephson junctions, light propagation in interacting optical waveguides, cantilever vibrations in micromechanical arrays, cold atom dynamics in Bose-Einstein condensates loaded on optical lattices, among others (for references see [1,2]).…”

Section: Introductionmentioning

confidence: 99%

“…In many cases quantum effects are important. Quantum breathers are nearly degenerate many-quanta bound states which, when superposed, form a spatially localized excitation with a very long time to tunnel from one lattice site to another (for references see [1,2,4]). …”

Section: Introductionmentioning

confidence: 99%

Boundary effects on quantum -breathers in a Bose–Hubbard chain

Pinto

Nguenang

Flach

2009

Physica D: Nonlinear Phenomena

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We investigate the spectrum and eigenstates of a Bose-Hubbard chain containing two bosons with fixed boundary conditions. In the noninteracting case the eigenstates of the system define a two-dimensional normal-mode space. For the interacting case weight functions of the eigenstates are computed by perturbation theory and numerical diagonalization. We identify paths in the two-dimensional normal-mode space which are rims for the weight functions. The decay along and off the rims is algebraic. Intersection of two paths (rims) leads to a local enhancement of the weight functions. We analyze nonperturbative effects due to the degeneracies and the formation of two-boson bound states.

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