Localized states in discrete nonlinear Schrödinger equations

Cai, David; Bishop, A. R.; Grønbech‐Jensen, Niels

doi:10.1103/physrevlett.72.591

Cited by 174 publications

(138 citation statements)

References 15 publications

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“…42 Although the nonlinear KG lattice does not support a Peierls-Nabarro potential 43,44 due to the presence of internal degrees of freedom, 40,69 a pinning potential of some sort still appears to be a valid concept.…”

Section: Discussionmentioning

confidence: 99%

“…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%

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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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Section: Discussionmentioning

confidence: 99%

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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“…We can test stability against other perturbations by applying Peierls-Nabarro potential ideas. 2 According to this concept, intersite and onsite modes with the same topology can be seen as constituents of the same soliton. Assuming equal intensity, the component with the highest energy will be unstable and evolve into the stable component with the lowest energy.…”

Section: Self-localized Waveguidesmentioning

confidence: 99%

“…The self-localized waveguides can be seen as a variant of gap solitons 7,8 because they overcome the longitudinal bandgap. But they can also be thought of as kinds of intrinsic localized modes, 1,2 which are localized structures in discrete nonlinear lattices without defects, e.g. discrete solitons in waveguide arrays.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, not only is it possible to adjust the linear properties but there are also new effects, such as localized nonlinear modes and gap solitons, to be studied. [1][2][3][4][5][6][7][8] Along with theoretical and experimental progress, there is a need for efficient numerical tools with which to explore nonlinear devices and phenomena. We recently expanded the mode expansion technique for twodimensional Kerr nonlinear structures of finite size.…”

Section: Introductionmentioning

confidence: 99%

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Bloch modes and self-localized waveguides in nonlinear photonic crystals

2005

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We present a modeling technique that uses eigenmode expansion to simulate infinite periodic structures with Kerr nonlinearity. Using a unit cell with Bloch boundary conditions, our iterative algorithm efficiently calculates self-consistent two-dimensional Bloch modes. We show how it can be used to study the band structure of nonlinear photonic crystals and to gain rapid insight in the operation of devices. Furthermore, we present nonlinear transversely localized guided modes, which are kinds of gap solitons or intrinsic localized modes, that induce their own waveguide through a photonic crystal without linear defects.

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Localized Modes in Nonlinear Fractional Systems with Deep Lattices

Liu

Malomed

Zeng

2022

Advcd Theory and Sims

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Solitons in the fractional space, supported by lattice potentials, have recently attracted much interest. The limit of deep 1D and 2D lattices in this system is considered, featuring finite bandgaps separated by nearly flat Bloch bands. Such spectra are also a subject of great interest in current studies. The existence, shapes, and stability of various localized modes, including fundamental gap and vortex solitons, are investigated by means of numerical methods; some results are also obtained with the help of analytical approximations. In particular, the 1D and 2D gap solitons, belonging to the first and second finite bandgaps, are tightly confined around a single cell of the deep lattice. Vortex gap solitons are constructed as four-peak "squares" and "rhombuses" with imprinted winding number S = 1. Stability of the solitons is explored by means of the linearization and verified by direct simulations.

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Localized states in discrete nonlinear Schrödinger equations

Cited by 174 publications

References 15 publications

Study of intrinsic localized vibrational modes in micromechanical oscillator arrays

Study of intrinsic localized vibrational modes in micromechanical oscillator arrays

Bloch modes and self-localized waveguides in nonlinear photonic crystals

Localized Modes in Nonlinear Fractional Systems with Deep Lattices

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