Integrability and localized excitations in nonlinear discrete systems

Flach, Sergej; Willis, C. R.; Olbrich, Eckehard

doi:10.1103/physreve.49.836

Cited by 58 publications

(94 citation statements)

References 26 publications

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“…At the classical level, both analytical and numerical experience support an infinite lifetime, at least for the one-frequency breather [4]. In a diatomic lattice there are two breather frequencies (in the phonon gap and above the optical phonons) and although there are indications that combinations of the modes can allow energy to leave via the phonons, in our simulations this does not take place in any perceptible way.…”

Section: Introductionmentioning

confidence: 47%

Structure and time-dependence of quantum breathers

Schulman¹,

Tolkunov²,

Mihóková³

2006

Chemical Physics

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Quantum states of a discrete breather are studied in two ways. One method involves numerical diagonalization of the Hamiltonian, the other uses the path integral to examine correlations in the eigenstates. In both cases only the central nonlinearity is retained. To reduce truncation effects in the numerical diagonalization, a basis is used that involves a quadratic local mode. A similar device is used in the path integral method for deducing localization. Both approaches lead to the conclusion that aside from quantum tunneling the quantized discrete breather is stable.

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Section: Introductionmentioning

confidence: 47%

Structure and time-dependence of quantum breathers

Schulman¹,

Tolkunov²,

Mihóková³

2006

Chemical Physics

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“…The reason why the RWA can give quite reasonable estimations even when being far away from the usual perturbation region of amplitudes has been explained in [90] by studying a single nonlinear oscillator. We apply the RWA to the one particle problemQ…”

Section: Rotating Wave Approximationmentioning

confidence: 99%

Discrete breathers

1998

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Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices -quantum breathers.Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.

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“…[9,10] in the light of canonical perturbation theory. With reference to the system (5), let us initially displace from the equilibrium position = 0 only a finite number n Ӷ N of consecutive particles.…”

Section: The Correspondence Conjecture Revisitedmentioning

confidence: 99%

“…We define a localized motion as one in which the amplitude of oscillation is small outside a group of n max oscillators for all times, with n Ͻ n max Ӷ N, and we introduce a numerical criterion to quantify localization based on the information entropy. We use the correspondence conjecture (CC) of Flach et al [9,10] that the nonlinear dynamics of the isolated group of n oscillators must have frequencies inside the reactive band of the linearized chain for localization to be possible. Within this conjecture, we show that there is a minimum value for the Morse parameter (the only parameter of the model) for a localized excitation to be possible.…”

Section: Introductionmentioning

confidence: 99%

Energy localization in the Peyrard-Bishop DNA model

et al. 2004

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We study energy localization on the oscillator chain proposed by Peyrard and Bishop to model DNA. We search numerically for conditions with initial energy in a small subgroup of consecutive oscillators of a finite chain and such that the oscillation amplitude is small outside this subgroup on a long time scale. We use a localization criterion based on the information entropy and verify numerically that such localized excitations exist when the nonlinear dynamics of the subgroup oscillates with a frequency inside the reactive band of the linear chain. We predict a mimium value for the Morse parameter ͑ Ͼ 2.25͒ (the only parameter of our normalized model), in agreement with the numerical calculations (an estimate for the biological value is = 6.3). For supercritical masses, we use canonical perturbation theory to expand the frequencies of the subgroup and we calculate an energy threshold in agreement with the numerical calculations.

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Integrability and localized excitations in nonlinear discrete systems

Cited by 58 publications

References 26 publications

Structure and time-dependence of quantum breathers

Structure and time-dependence of quantum breathers

Discrete breathers

Energy localization in the Peyrard-Bishop DNA model

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