Stability and motion of intrinsic localized modes in nonlinear periodic lattices

Sandusky, K. W.; Page, J. B.; Schmidt, Kevin

doi:10.1103/physrevb.46.6161

Cited by 175 publications

(119 citation statements)

References 10 publications

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“…Although self-localized solitons in anharmonic lattices without impurities were predicted quite a long time ago in Refs. [123,124], only recently the existence of breathers in FPU lattices has been proved rigorously (see, e.g., Refs. [125]).…”

Section: Discussionmentioning

confidence: 99%

The Fermi–Pasta–Ulam problem: Fifty years of progress

Berman

Izrailev

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

296

292

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A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Boseparticles are also considered.

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Section: Discussionmentioning

confidence: 99%

The Fermi–Pasta–Ulam problem: Fifty years of progress

Berman

Izrailev

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

296

292

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“…For a classical nonlinear oscillator array, there are a number of characteristic ILM properties, probed theoretically, such as their interaction with an ac driver, 14,34 -36 their propagation 5,[37][38][39][40] and amplitude dependent mobility 4,6,[40][41][42] in a discrete lattice potential, 43,44 as well as their interactions with impurities, [45][46][47][48][49][50] that still need to be examined experimentally. Note that strongly excited ILMs 42 can be trapped anywhere in the lattice, so they also could approach impurity mode behavior.…”

Section: Introductionmentioning

confidence: 99%

Study of intrinsic localized vibrational modes in micromechanical oscillator arrays

Satō

Hubbard

English

et al. 2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

103

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Intrinsic localized modes ͑ILMs͒ have been observed in micromechanical cantilever arrays, and their creation, locking, interaction, and relaxation dynamics in the presence of a driver have been studied. The micromechanical array is fabricated in a 300 nm thick silicon-nitride film on a silicon substrate, and consists of up to 248 cantilevers of two alternating lengths. To observe the ILMs in this experimental system a line-shaped laser beam is focused on the 1D cantilever array, and the reflected beam is captured with a fast charge coupled device camera. The array is driven near its highest frequency mode with a piezoelectric transducer. Numerical simulations of the nonlinear Klein-Gordon lattice have been carried out to assist with the detailed interpretation of the experimental results. These include pinning and locking of the ILMs when the driver is on, collisions between ILMs, low frequency excitation modes of the locked ILMs and their relaxation behavior after the driver is turned off. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1540771͔An advance of the theory of nonlinear excitations in discrete lattices was the discovery that some localized vibrations in perfectly periodic but nonintegrable lattices could be stabilized by lattice discreteness. The modulational instability of extended large amplitude vibrational modes has been proposed as a mechanism for the realization of dynamical localization on the scale of the lattice constant. Although theoretically a variety of methods to excite the instability of a homogeneous vibrational mode have been proposed, these ideas have yet to be tested experimentally. Since the observation of nanoscale localized vibrational modes still cannot be achieved there is definite advantage to examining a macroscopic array, which is small enough so that the entire time dependence of the instability dynamics occurs in a practical measurement interval. This has been accomplished by using micromechanical silicon technology to fabricate up to 248 identical cantilevers with a 40 micron lattice constant. Optical techniques have been used to track the motion of individual cantilevers in the presence of an inertial driver. In addition to experimentally characterizing the modulational instability and identifying the best method for producing intrinsic localized modes a new discovery is the locking of the local mode amplitude with the driver frequency. Numerical simulations have been used to better understand the nature of this synchronization effect.

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“…To convince oneself that this is true, we consider now the simple case µ = (a, φ 0 , P, Q), and will write the effective Hamiltonian dynamics for (P, Q), deduced from (6). The tangent vectors to the loops are decomposed along these coordinates µ.…”

Section: The Methods Of the Effective Hamiltonianmentioning

confidence: 99%

“…Nevertheless, spatio-temporal structures which resemble travelling DB have been observed in numerous numerical simulations, leading to several theoretical analyses which have attempted to describe these solutions [6]- [11]. For instance in [7] the problem of moving breather is dealt with by perturbation of the AL system; in [9] numerics are compared with high order multiple scale expansions of a Klein-Gordon model, and in [6,10] the mobility of DB is related to the stability analysis of DB. In the present work, we would like to approach this problem from the general point of view sketched in [11]: if we forget about its internal DOF of vibration, a solution which looks like a travelling discrete breathers should be governed by a 1-DOF effective Hamiltonian that could be constructed, at least perturbatively.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, most likely travelling DB are nongeneric phenomena [5]. Nevertheless, spatio-temporal structures which resemble travelling DB have been observed in numerous numerical simulations, leading to several theoretical analyses which have attempted to describe these solutions [6]- [11]. For instance in [7] the problem of moving breather is dealt with by perturbation of the AL system; in [9] numerics are compared with high order multiple scale expansions of a Klein-Gordon model, and in [6,10] the mobility of DB is related to the stability analysis of DB.…”

Section: Introductionmentioning

confidence: 99%

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Effective Hamiltonian for travelling discrete breathers

MacKay¹,

Sepulchre²

2002

J. Phys. A: Math. Gen.

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Hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers. However solutions which look like moving discrete breathers for some time are not difficult to observe in numerics. In this paper we propose an abstract framework for the description of approximate travelling discrete breathers in Hamiltonian chains of oscillators. The method is based on the construction of an effective Hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers. Error estimate on the approximate dynamics is also studied. The concept of Peierls-Nabarro barrier can be made clear in this framework. We illustrate the method on two simple examples, namely the Salerno model which interpolates between the Ablowitz-Ladik lattice and the discrete nonlinear Schrödinger system, and the Fermi-Pasta-Ulam chain.

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Stability and motion of intrinsic localized modes in nonlinear periodic lattices

Cited by 175 publications

References 10 publications

The Fermi–Pasta–Ulam problem: Fifty years of progress

The Fermi–Pasta–Ulam problem: Fifty years of progress

Study of intrinsic localized vibrational modes in micromechanical oscillator arrays

Effective Hamiltonian for travelling discrete breathers

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