Intrinsic localized modes: Discrete breathers. Existence and linear stability

Marín, J. L.; Aubry, S.; Floría, L. M.

doi:10.1016/s0167-2789(97)00280-7

Cited by 73 publications

(54 citation statements)

References 19 publications

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“…for describing energy and charge transport and storage in biological macromolecules [5]. Recently, discrete breathers have been experimentally observed in coupled optical waveguides [6], in charge-density wave systems [7], in magnetic systems [8], and in arrays of coupled Josephson junctions [9].Although a discrete breather under quite general conditions is linearly stable [1,3,10] (and thus no perturbations grow exponentially), there are many questions remaining concerning the long-time fate of perturbed breathers. In a previous paper [11], some of these questions were addressed considering a particular model, the discrete nonlinear Schrödinger (DNLS) equation.…”

mentioning

confidence: 99%

“…Although a discrete breather under quite general conditions is linearly stable [1,3,10] (and thus no perturbations grow exponentially), there are many questions remaining concerning the long-time fate of perturbed breathers. In a previous paper [11], some of these questions were addressed considering a particular model, the discrete nonlinear Schrödinger (DNLS) equation.…”

mentioning

confidence: 99%

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Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Johansson¹

2001

Phys. Rev. E

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We consider the long-time evolution of weakly perturbed discrete nonlinear Schrödinger breathers. While breather growth can occur through nonlinear interaction with one single initial linear mode, breather decay is found to require excitation of at least two independent modes. All growth and decay processes of lowest order are found to disappear for breathers larger than a threshold value.PACS numbers: 63.20. Pw, 45.05.+x, 42.65.Tg The study of intrinsically localized modes in anharmonic lattices, discrete breathers, has yielded much attention during the last decade (see e.g. [1,2]). In particular, their existence as time-periodic solutions of nonlinear lattice-equations was proven under quite general conditions [3], and numerical schemes were developed for their explicit calculation [4]. The generality of the concept of discrete breathers, which provide very efficient means to localize energy, has lead to numerous suggestions to its application in contexts where anharmonicity and discreteness are important, e.g. for describing energy and charge transport and storage in biological macromolecules [5]. Recently, discrete breathers have been experimentally observed in coupled optical waveguides [6], in charge-density wave systems [7], in magnetic systems [8], and in arrays of coupled Josephson junctions [9].Although a discrete breather under quite general conditions is linearly stable [1,3,10] (and thus no perturbations grow exponentially), there are many questions remaining concerning the long-time fate of perturbed breathers. In a previous paper [11], some of these questions were addressed considering a particular model, the discrete nonlinear Schrödinger (DNLS) equation. The interaction between stationary breathers and small perturbations corresponding to time-periodic eigensolutions to the linearized equations of motion around the breather was investigated using a multiscale perturbational approach, and it was found that the nonlinear interaction between the breather and single-mode small-amplitude perturbations could lead to breather growth through generation of radiating higher harmonics, but not to breather decay. It is the purpose of this Report to extend these results to more general perturbations of stationary DNLS breathers. In particular, we find that while the simplest growth process can be described as an inelastic scattering process with a one-frequency incoming wave and an additional outgoing higher-harmonic wave, the description of a decay process requires at least two incoming modes yielding outgoing modes with frequencies being linear combinations of the original ones.In order to make this Report self-contained, we first recapitulate the main formalism from [11] (to which the reader is referred for details; see also [12] for a similar approach in continuous NLS models). With canonical conjugated variables {iψ n }, {ψ * n }, the DNLS equation can be derived from the Hamiltonian H ({iψ n } , {ψ * n }) = n

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confidence: 99%

mentioning

confidence: 99%

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Johansson¹

2001

Phys. Rev. E

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“…These regions appear to happen near bifurcations which connect the site-centered and bond-centered stationary breathers (see Ref. [12]), and the subject is being investigated in greater detail [6]. It is difficult to estimate how wide these regions are, other than by means of numerical simulation.…”

Section: Longitudinal Moving Breathers In 2d Latticesmentioning

confidence: 99%

“…Just like in 1D models, it was observed that a too strong on-site potential hinders mobility, and favours the pinning of the excitations into stationary breathers (intrinsic localized modes). A too weak on-site potential usually destroys the breather by broadening and radiation into the background, due to resonances with the phonon band [12]. In between these extremes, there exists an ample region where it is easy to obtain moving breathers, with a more or less wide range of energies and velocities.…”

Section: Longitudinal Moving Breathers In 2d Latticesmentioning

confidence: 99%

2-D Breathers and Applications

Marín¹,

Eilbeck²,

Russell³

Nonlinear Science at the Dawn of the 21st Century

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In this chapter we show how a new type of nonlinear lattice excitation is helping to understand the long-standing puzzle of unexplained dark lines in crystals of muscovite mica. In fact, it was the conjecture that some kind of quasi-one-dimensional lattice soliton was responsible for those lines which led to the discovery of this new family of lattice excitations: mobile localized breathers of longitudinal type. We explore several properties of these moving breathers, both by numerical methods and by experimenting with analogue models. The results suggest a much broader application than just the mica problem.

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“…In the linear stability problem, oscillatory instabilities arise through Hamiltonian Hopf bifurcations yielding eigenvalues with nonzero real as well as imaginary parts. Well-known examples for one-dimensional lattices are, e.g., two-site localized "twisted" modes [2,3], discrete dark solitons ("dark breathers") [4,5], spatially periodic or quasiperiodic nonlinear standing waves [6], and gap modes in diatomic chains [7,8]. In the latter contexts, oscillatory instabilities may play an important role in the initial stages of breather formation and thermalization processes [6,8].…”

Section: Introductionmentioning

confidence: 99%

Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer

2012

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We study the Bose-Hubbard model for three sites in a symmetric, triangular configuration and search for quantum signatures of the classical regime of oscillatory instabilities, known to appear through Hamiltonian Hopf bifurcations for the "single-depleted-well" family of stationary states in the discrete nonlinear Schrödinger equation. In the regimes of classical stability, single quantum eigenstates with properties analogous to those of the classical stationary states can be identified already for rather small particle numbers. On the other hand, in the instability regime the interaction with other eigenstates through avoided crossings leads to strong mixing, and no single eigenstate with classical-like properties can be seen. We compare the quantum dynamics resulting from initial conditions taken as perturbed quantum eigenstates and SU(3) coherent states, respectively, in a quantum-semiclassical transitional regime of 10-100 particles. While the perturbed quantum eigenstates do not show a classical-like behavior in the instability regime, a coherent state behaves analogously to a perturbed classical stationary state, and exhibits initially resonant oscillations with oscillation frequencies well described by classical internal-mode oscillations.

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Intrinsic localized modes: Discrete breathers. Existence and linear stability

Cited by 73 publications

References 19 publications

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

Decay of discrete nonlinear Schrödinger breathers through inelastic multiphonon scattering

2-D Breathers and Applications

Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer

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