Asymmetric gap soliton modes in diatomic lattices with cubic and quartic nonlinearity

Huang, Guoxiang; Hu, Bambi

doi:10.1103/physrevb.57.5746

Cited by 46 publications

(36 citation statements)

References 56 publications

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“…An "inversion of stability" regime was found characterized by practically radiationless mobility. The authors emphasize that discrete breathers studied in [30][31][32] are spatially-localized and thus in contrast to the plane waves of infinite extent studied herein. This paper builds upon recent studies of plane waves in nonlinear monoatomic and diatomic chains and other discrete systems by developing a higher-order multiple scales procedure to inform dispersion, stability, and waveform invariance.…”

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confidence: 87%

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Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Fronk

Leamy

2017

Journal of Vibration and Acoustics

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Recent studies have presented first-order multiple time scale approaches for exploring amplitudedependent plane-wave dispersion in weakly nonlinear chains and lattices characterized by cubic stiffness. These analyses have yet to assess solution stability, which requires an analysis incorporating damping. Furthermore, due to their first-order dependence, they make an implicit assumption that cubic stiffness influences dispersion shifts to a greater degree than quadratic stiffness, and they thus ignore quadratic shifts. This paper addresses these limitations by carryingout higher-order, multiple scales perturbation analyses of linearly-damped nonlinear monoatomic and diatomic chains. The study derives higher-order dispersion corrections informed by both quadratic and cubic stiffness, and quantifies plane wave stability using evolution equations resulting from the multiple scales analysis and numerical experiments. Additionally, by reconstructing plane waves using both homogeneous and particular solutions at multiple orders, the study introduces a new interpretation of multiple scales results in which predicted waveforms are seen to exist over all space and time, constituting an invariant, multi-harmonic wave of infinite A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dDownloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jvacek/0/ on 04/24/2017 Terms of Use: http://www.asme.org/aboutVIB-16-1265, Leamy 2 extent analogous to cnoidal waves in continuous systems. Using example chains characterized by dimensionless parameters, numerical studies confirm that the spectral content of the predicted waveforms exhibits less growth/decay over time as higher-order approximations are used in defining the simulations' initial conditions. Thus the study results suggest that higher-order multiple scales perturbation analysis captures long-term, non-localized invariant plane waves, which have the potential for propagating coherent information over long distances.

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confidence: 87%

“…In [30], Flach and Gorbach uncover a threshold for tangent bifurcations in discrete breathers. Huang and Hu [31] investigated the stability of diatomic lattices using a quasi-discreteness approach. They derive evolution equations and fixed points for acoustic and optical modes and identify the presence of asymmetrical gap solitons.…”

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confidence: 99%

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Fronk

Leamy

2017

Journal of Vibration and Acoustics

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“…It is relevant to mention that the defocusing NLS equation can also be derived in the displacement variable formulation [26]. If ν 3 > 0 (which is not possible here), then one would need κ > 0, which, in turn, would result in the existence of homoclinic solutions and thus of bright breathers with frequencies above the phonon band.…”

Section: Derivation Of the Defocusing Nls Equation In The Strainmentioning

confidence: 99%

Dark breathers in granular crystals

et al. 2013

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We present a study of the existence, stability, and bifurcation structure of families of dark breathers in a one-dimensional uniform chain of spherical beads under static load. A defocusing nonlinear Schrödinger equation (NLS) is derived for frequencies that are close to the edge of the phonon band and is used to construct targeted initial conditions for numerical computations. Salient features of the system include the existence of large amplitude solutions that emerge from the small amplitude solutions described by the NLS equation, and the presence of a nonlinear instability that, to the best of the authors' knowledge, has not been observed in classical Fermi-Pasta-Ulam lattices. Finally, it is also demonstrated that these dark breathers can be detected in a physically realistic experimental settings by merely actuating the ends of an initially at rest chain of beads and inducing destructive interference between their signals.

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“…In that same limit (of τ → ∞ and a → 0) the nonlinear Schrödinger (NLS) equation can be derived from Eq. (1) written in terms of the negative strain variable y n = u n−1 − u n [26,53]. In particular, one defines the multiple-scale ansatz,…”

Section: Nonlinear Schrödinger Approximationmentioning

confidence: 99%

Nonlinear vibrational-state excitation and piezoelectric energy conversion in harmonically driven granular chains

et al. 2016

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This article explores the excitation of different vibrational states in a spatially extended dynamical system through theory and experiment. As a prototypical example, we consider a one-dimensional packing of spherical particles (a so-called granular chain) that is subject to harmonic boundary excitation. The combination of the multimodal nature of the system and the strong coupling between the particles due to the nonlinear Hertzian contact force leads to broad regions in frequency where different vibrational states are possible. In certain parametric regions, we demonstrate that the nonlinear Schrödinger equation predicts the corresponding modes fairly well. The electromechanical model we apply predicts accurately the conversion from the obtained mechanical energy to the electrical energy observed in experiments.

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Asymmetric gap soliton modes in diatomic lattices with cubic and quartic nonlinearity

Cited by 46 publications

References 56 publications

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Dark breathers in granular crystals

Nonlinear vibrational-state excitation and piezoelectric energy conversion in harmonically driven granular chains

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