Dispersive evolution of pulses in oscillator chains with general
interaction potentials

Giannoulis, Johannes; Mielke, Alexander

doi:10.3934/dcdsb.2006.6.493

Cited by 29 publications

(34 citation statements)

References 37 publications

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“…The precompression effectively suppresses the fully nonlinear character of the Hertzian interactions and leads to a weakly nonlinear system in the small-amplitude limit. Following the approach developed in [7,10] for two limiting cases of the present model and adopting a multiscale asymptotic technique [31,36,37], we derived modulation equations that reduce to the NLS equation at finite mass ratio.…”

Section: Discussionmentioning

confidence: 99%

“…denotes complex conjugate. More precisely, following [36,37] (see also [31]), we substitute the multiple-scale ansatz…”

Section: Derivation Of the Nonlinear Schrödinger Equationmentioning

confidence: 99%

“…We now use a Newton-type algorithm (Algorithm 2 in [40]) with the initial seed (48) to compute numerically exact stationary dark breather solutions of system (37). Computations are performed with periodic boundary conditions.…”

Section: Numerical Continuation Of Stationary Dark Breathersmentioning

confidence: 99%

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Breathers in a locally resonant granular chain with precompression

Liu

James

Kevrekidis

et al. 2016

Physica D: Nonlinear Phenomena

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We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schrödinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, time-periodic states mounted on top of a non-vanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.

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Section: Discussionmentioning

confidence: 99%

“…denotes complex conjugate. More precisely, following [36,37] (see also [31]), we substitute the multiple-scale ansatz…”

Section: Derivation Of the Nonlinear Schrödinger Equationmentioning

confidence: 99%

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Breathers in a locally resonant granular chain with precompression

Liu

James

Kevrekidis

et al. 2016

Physica D: Nonlinear Phenomena

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“…To simplify analysis, it is better to introduce the parametrization E = Q 2 and rewrite (20) in the equivalent form…”

Section: Justification Of the Dnls Equation On The Dnls Time Scalementioning

confidence: 99%

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

2016

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Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrödinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss applications of the discrete nonlinear Schrödinger equation in the context of existence and stability of breathers of the Klein-Gordon lattice.

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“…For this purpose we consider a simpler initial condition where all cantilevers with index n ≥ 2 are initially at rest and the first cantilever has initial velocity V and zero deflection. This corresponds to fixing the initial condition (28), which yields (29) in rescaled form.…”

Section: A Lattice Model For Cantilevers Decorated By Spherical Beadsmentioning

confidence: 99%

Breathers in oscillator chains with Hertzian interactions

James

Kevrekidis

Cuevas

2013

Physica D: Nonlinear Phenomena

115

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We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schrödinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors.

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Dispersive evolution of pulses in oscillator chains with general interaction potentials

Cited by 29 publications

References 37 publications

Breathers in a locally resonant granular chain with precompression

Breathers in a locally resonant granular chain with precompression

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

Breathers in oscillator chains with Hertzian interactions

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