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Ma L, Fang M, Song H, Zhou J. Spatial structure of the non-integrable discrete defocusing Hirota equation. CHAOS (WOODBURY, N.Y.) 2023; 33:083123. [PMID: 37549119 DOI: 10.1063/5.0151473] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2023] [Accepted: 07/21/2023] [Indexed: 08/09/2023]
Abstract
In this paper, we investigate the spatial property of the non-integrable discrete defocusing Hirota equation utilizing a planar nonlinear discrete dynamical map method. We construct the periodic orbit solutions of the stationary discrete defocusing Hirota equation. The behavior of the orbits in the vicinity of the special periodic solution is analyzed by taking advantage of the named residue. We characterize the effects of the parameters on the aperiodic orbits with the aid of numerical simulations. A comparison with the non-integrable discrete defocusing nonlinear Schrödinger equation case reveals that the non-integrable discrete defocusing Hirota equation has more abundant spatial properties. Rather an interesting and novel thing is that for any initial value, there exists triperiodic solutions for a reduced map.
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Affiliation(s)
- Liyuan Ma
- Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China
| | - Miaoshuang Fang
- Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China
| | - Haifang Song
- Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China
| | - Jiali Zhou
- Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China
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2
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Sato M, Furusawa H, Soga Y, Sievers AJ. Propagating intrinsic localized mode in a cyclic, dissipative, self-dual one-dimensional nonlinear transmission line. Phys Rev E 2023; 107:034202. [PMID: 37072939 DOI: 10.1103/physreve.107.034202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2022] [Accepted: 02/14/2023] [Indexed: 04/20/2023]
Abstract
A well-known feature of a propagating localized excitation in a discrete lattice is the generation of a backwave in the extended normal mode spectrum. To quantify the parameter-dependent amplitude of such a backwave, the properties of a running intrinsic localized mode (ILM) in electric, cyclic, dissipative, nonlinear 1D transmission lines, containing balanced nonlinear capacitive and inductive terms, are studied via simulations. Both balanced and unbalanced damping and driving conditions are treated. The introduction of a unit cell duplex driver, with a voltage source driving the nonlinear capacitor and a synchronized current source, the nonlinear inductor, provides an opportunity to design a cyclic, dissipative self-dual nonlinear transmission line. When the self-dual conditions are satisfied, the dynamical voltage and current equations of motion within a cell become the same, the strength of the fundamental, resonant coupling between the ILM and the lattice modes collapses, and the associated fundamental backwave is no longer observed.
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Affiliation(s)
- M Sato
- Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan
| | - H Furusawa
- Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan
| | - Y Soga
- Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan
| | - A J Sievers
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA
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3
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Mithun T, Maluckov A, Mančić A, Khare A, Kevrekidis PG. How close are integrable and nonintegrable models: A parametric case study based on the Salerno model. Phys Rev E 2023; 107:024202. [PMID: 36932573 DOI: 10.1103/physreve.107.024202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Accepted: 01/09/2023] [Indexed: 06/18/2023]
Abstract
In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for "generic" initial data, how close are the integrable to the nonintegrable models? Our more precise formulation of this question is, How well is the constancy of formerly conserved quantities preserved in the nonintegrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.
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Affiliation(s)
- Thudiyangal Mithun
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - Aleksandra Maluckov
- COHERENCE, Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, University of Belgrade, P.O.B. 522, 11001 Belgrade, Republic of Serbia
| | - Ana Mančić
- COHERENCE, Department of Physics, Faculty of Sciences and Mathematics, University of Niš, P.O.B. 224, 18000 Niš, Serbia
| | - Avinash Khare
- Department of Physics, Savitribai Phule Pune University, Pune 411007, India
| | - Panayotis G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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4
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Liu X, Malomed BA, Zeng J. Localized Modes in Nonlinear Fractional Systems with Deep Lattices. ADVANCED THEORY AND SIMULATIONS 2022. [DOI: 10.1002/adts.202100482] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Xiuye Liu
- State Key Laboratory of Transient Optics and Photonics Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences Xi'an 710119 China
- University of Chinese Academy of Sciences Beijing 100049 China
| | - Boris A. Malomed
- Department of Physical Electronics School of Electrical Engineering Faculty of Engineering, and the Center for Light‐Matter Interaction Tel Aviv University Ramat Aviv Tel Aviv P.O.B. 39040 Israel
- Instituto de Alta Investigación Universidad de Tarapacá Casilla 7D Arica Chile
| | - Jianhua Zeng
- State Key Laboratory of Transient Optics and Photonics Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences Xi'an 710119 China
- University of Chinese Academy of Sciences Beijing 100049 China
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5
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Mithun T, Maluckov A, Manda BM, Skokos C, Bishop A, Saxena A, Khare A, Kevrekidis PG. Thermalization in the one-dimensional Salerno model lattice. Phys Rev E 2021; 103:032211. [PMID: 33862787 DOI: 10.1103/physreve.103.032211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 03/04/2021] [Indexed: 11/07/2022]
Abstract
The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.
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Affiliation(s)
- Thudiyangal Mithun
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - Aleksandra Maluckov
- Vinca Institute of Nuclear Sciences, University of Belgrade, National Institute of the Republic of Serbia, P.O.B. 522, 11001 Belgrade, Serbia.,Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, S. Korea
| | - Bertin Many Manda
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - Charalampos Skokos
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - Alan Bishop
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Avadh Saxena
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Avinash Khare
- Department of Physics, Savitribai Phule Pune University, Pune 411007, India
| | - Panayotis G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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6
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Yu F. Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz-Ladik equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032914. [PMID: 25871179 DOI: 10.1103/physreve.91.032914] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2014] [Indexed: 06/04/2023]
Abstract
We present the nonautonomous discrete bright soliton solutions and their interactions in the discrete Ablowitz-Ladik (DAL) equation with variable coefficients, which possesses complicated wave propagation in time and differs from the usual bright soliton waves. The differential-difference similarity transformation allows us to relate the discrete bright soliton solutions of the inhomogeneous DAL equation to the solutions of the homogeneous DAL equation. Propagation and interaction behaviors of the nonautonomous discrete solitons are analyzed through the one- and two-soliton solutions. We study the discrete snaking behaviors, parabolic behaviors, and interaction behaviors of the discrete solitons. In addition, the interaction management with free functions and dynamic behaviors of these solutions is investigated analytically, which have certain applications in electrical and optical systems.
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Affiliation(s)
- Fajun Yu
- School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China
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7
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Ma LY, Zhu ZN. Nonintegrable semidiscrete Hirota equation: gauge-equivalent structures and dynamical properties. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:033202. [PMID: 25314554 DOI: 10.1103/physreve.90.033202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2014] [Indexed: 06/04/2023]
Abstract
In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations.
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Affiliation(s)
- Li-Yuan Ma
- Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P. R. China
| | - Zuo-Nong Zhu
- Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P. R. China
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8
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Han D, Westley M, Sen S. Mechanical energy fluctuations in granular chains: the possibility of rogue fluctuations or waves. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032904. [PMID: 25314501 DOI: 10.1103/physreve.90.032904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2014] [Indexed: 06/04/2023]
Abstract
The existence of rogue or freak waves in the ocean has been known for some time. They have been reported in the context of optical lattices and the financial market. We ask whether such waves are generic to late time behavior in nonlinear systems. In that vein, we examine the dynamics of an alignment of spherical elastic beads held within fixed, rigid walls at zero precompression when they are subjected to sufficiently rich initial conditions. Here we define such waves generically as unusually large energy fluctuations that sustain for short periods of time. Our simulations suggest that such unusually large fluctuations ("hot spots") and occasional series of such fluctuations through space and time ("rogue fluctuations") are likely to exist in the late time dynamics of the granular chain system at zero dissipation. We show that while hot spots are common in late time evolution, rogue fluctuations are seen in purely nonlinear systems (i.e., no precompression) at late enough times. We next show that the number of such fluctuations grows exponentially with increasing nonlinearity whereas rogue fluctuations decrease superexponentially with increasing precompression. Dissipation-free granular alignment systems may be possible to realize as integrated circuits and hence our observations may potentially be testable in the laboratory.
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Affiliation(s)
- Ding Han
- Department of Physics, State University of New York, Buffalo, New York 14260-1500, USA
| | - Matthew Westley
- Department of Physics, State University of New York, Buffalo, New York 14260-1500, USA
| | - Surajit Sen
- Department of Physics, State University of New York, Buffalo, New York 14260-1500, USA
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9
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Maluckov A, Hadzievski L, Lazarides N, Tsironis GP. Extreme events in discrete nonlinear lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:025601. [PMID: 19391797 DOI: 10.1103/physreve.79.025601] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2008] [Indexed: 05/27/2023]
Abstract
We perform statistical analysis on discrete nonlinear waves generated through modulational instability in the context of the Salerno model that interpolates between the integrable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schrödinger equation. We focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes. We find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation. A characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the AL equation and the accompanied transition from the local to the global stochasticity monitored through the positive Lyapunov exponent of a nonlinear map.
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Affiliation(s)
- A Maluckov
- Faculty of Sciences and Mathematics, Department of Physics, P.O. Box 224, 18001 Nis, Serbia
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10
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Van der Sande G, Maes B, Bienstman P, Danckaert J, Baets R, Veretennicoff I. Nonlinear lattice model for spatially guided solitons in nonlinear photonic crystals. OPTICS EXPRESS 2005; 13:1544-1554. [PMID: 19495030 DOI: 10.1364/opex.13.001544] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Numerical simulations have shown the existence of transversely localized guided modes in nonlinear two-dimensional photonic crystals. These soliton-like Bloch waves induce their own waveguide in a photonic crystal without the presence of a linear defect. By applying a Green's function method which is limited to within a strip perpendicular to the propagation direction, we are able to describe these Bloch modes by a nonlinear lattice model that includes the long-range site-to-site interaction between the scattered fields and the non-local nonlinear response of the photonic crystal. The advantages of this semi-analytical approach are discussed and a comparison with a rigorous numerical analysis is given in different configurations. Both monoatomic and diatomic nonlinear photonic crystals are considered.
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11
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Gómez-Gardeñes J, Floría LM, Peyrard M, Bishop AR. Nonintegrable Schrodinger discrete breathers. CHAOS (WOODBURY, N.Y.) 2004; 14:1130-1147. [PMID: 15568927 DOI: 10.1063/1.1811991] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
In an extensive numerical investigation of nonintegrable translational motion of discrete breathers in nonlinear Schrödinger lattices, we have used a regularized Newton algorithm to continue these solutions from the limit of the integrable Ablowitz-Ladik lattice. These solutions are shown to be a superposition of a localized moving core and an excited extended state (background) to which the localized moving pulse is spatially asymptotic. The background is a linear combination of small amplitude nonlinear resonant plane waves and it plays an essential role in the energy balance governing the translational motion of the localized core. Perturbative collective variable theory predictions are critically analyzed in the light of the numerical results.
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Affiliation(s)
- J Gómez-Gardeñes
- Departamento de Teoría y Simulación de Sistemas Complejos, Instituto de Ciencia de Materiales de Aragón, C.S.I.C.-Universidad de Zaragoza, 50009 Zaragoza, Spain.
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12
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Xu Z, Kartashov YV, Crasovan LC, Mihalache D, Torner L. Spatiotemporal discrete multicolor solitons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:066618. [PMID: 15697537 DOI: 10.1103/physreve.70.066618] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2004] [Revised: 07/27/2004] [Indexed: 05/24/2023]
Abstract
We have found various families of two-dimensional spatiotemporal solitons in quadratically nonlinear waveguide arrays. The families of unstaggered odd, even, and twisted stationary solutions are thoroughly characterized and their stability against perturbations is investigated. We show that the twisted and even solitons display instability, while most of the odd solitons show remarkable stability upon evolution.
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Affiliation(s)
- Zhiyong Xu
- ICFO-Institut de Ciencies Fotoniques, and Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain.
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13
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Doktorov EV, Matsuka NP, Rothos VM. Perturbation-induced radiation by the Ablowitz-Ladik soliton. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 68:066610. [PMID: 14754339 DOI: 10.1103/physreve.68.066610] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2003] [Indexed: 11/07/2022]
Abstract
An efficient formalism is elaborated to analytically describe dynamics of the Ablowitz-Ladik soliton in the presence of perturbations. This formalism is based on using the Riemann-Hilbert problem and provides the means of calculating evolution of the discrete soliton parameters, as well as shape distortion and perturbation-induced radiation effects. As an example, soliton characteristics are calculated for linear damping and quintic perturbations.
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Affiliation(s)
- E V Doktorov
- B.I. Stepanov Institute of Physics, 68 F. Skaryna Avenue, 220072 Minsk, Belarus.
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14
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Kevrekidis PG, Quintero NR. Using the finite domain remnant of the continuous spectrum to examine integrability: effect of boundary conditions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:036612. [PMID: 14524917 DOI: 10.1103/physreve.68.036612] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2002] [Revised: 06/30/2003] [Indexed: 05/24/2023]
Abstract
The aim of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE), by examining it in a finite but large domain. For monoparametric families of PDE/DDE's, that are known to possess isolated integrable points, we find that very special features occur in the finite domain remnant of the continuous ("phonon") spectrum at these "singular" points. We identify these features in the case example of a PDE and a DDE (that sustain front and pulselike solutions, respectively) for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular, integrable points and hence we propose to explore a given PDE/DDE in a finite but large domain for such traits, as a means of assessing its potential integrability.
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Affiliation(s)
- Panayotis G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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15
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Sato M, Hubbard BE, English LQ, Sievers AJ, Ilic B, Czaplewski DA, Craighead HG. Study of intrinsic localized vibrational modes in micromechanical oscillator arrays. CHAOS (WOODBURY, N.Y.) 2003; 13:702-715. [PMID: 12777135 DOI: 10.1063/1.1540771] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Intrinsic localized modes (ILMs) 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.
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Affiliation(s)
- M Sato
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA
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16
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17
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Kevrekidis PG, Kevrekidis IG, Bishop AR, Titi ES. Continuum approach to discreteness. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046613. [PMID: 12006053 DOI: 10.1103/physreve.65.046613] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2001] [Indexed: 05/23/2023]
Abstract
We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls-Nabarro barriers and Bloch oscillations). We analyze the role of such methods in providing an effective "homogenization" of spatially discrete, as well as of heterogeneous continuum equations. Finally, we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations, as well as to compare them with other semicontinuum approximations.
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Affiliation(s)
- P G Kevrekidis
- Theoretical Division and Center for Nonlinear Studies, MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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18
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Ablowitz MJ, Musslimani ZH, Biondini G. Methods for discrete solitons in nonlinear lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:026602. [PMID: 11863673 DOI: 10.1103/physreve.65.026602] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2001] [Indexed: 05/23/2023]
Abstract
A method to find discrete solitons in nonlinear lattices is introduced. Using nonlinear optical waveguide arrays as a prototype application, both stationary and traveling-wave solitons are investigated. In the limit of small wave velocity, a fully discrete perturbative analysis yields formulas for the mode shapes and velocity.
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Affiliation(s)
- Mark J Ablowitz
- Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, Colorado 80309-0526, USA
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19
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Christodoulides DN, Eugenieva ED. Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays. PHYSICAL REVIEW LETTERS 2001; 87:233901. [PMID: 11736450 DOI: 10.1103/physrevlett.87.233901] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2001] [Indexed: 05/23/2023]
Abstract
It is shown that discrete solitons can be navigated in two-dimensional networks of nonlinear waveguide arrays. This can be accomplished via vector interactions between two classes of discrete solitons: signals and blockers. Discrete solitons in such two-dimensional networks can exhibit a rich variety of functional operations, e.g., blocking, routing, logic functions, and time gating.
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Affiliation(s)
- D N Christodoulides
- Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, USA
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20
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Trombettoni A, Smerzi A. Discrete solitons and breathers with dilute Bose-Einstein condensates. PHYSICAL REVIEW LETTERS 2001; 86:2353-2356. [PMID: 11289927 DOI: 10.1103/physrevlett.86.2353] [Citation(s) in RCA: 70] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2000] [Indexed: 05/23/2023]
Abstract
We study the dynamical phase diagram of a dilute Bose-Einstein condensate (BEC) trapped in a periodic potential. The dynamics is governed by a discrete nonlinear Schrödinger equation: intrinsically localized excitations, including discrete solitons and breathers, can be created even if the BEC's interatomic potential is repulsive. Furthermore, we analyze the Anderson-Kasevich experiment [Science 282, 1686 (1998)], pointing out that mean field effects lead to a coherent destruction of the interwell Bloch oscillations.
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Affiliation(s)
- A Trombettoni
- Istituto Nazionale di Fisica per la Materia and International School for Advanced Studies, via Beirut 2/4, I-34014, Trieste, Italy
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21
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Kevrekidis PG, Bishop AR, Rasmussen KØ. Twisted localized modes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:036603. [PMID: 11308784 DOI: 10.1103/physreve.63.036603] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2000] [Indexed: 05/23/2023]
Abstract
In a number of recent papers the so-called twisted localized mode of the discrete nonlinear Schrödinger equation has been proposed. Herein, we study the existence and stability properties of such modes. We analyze the persistence of quasiperiodic modes and study the domains of existence and numerical stability of the exact form of such solutions. We identify the bifurcations through which they lose their stability and follow the behavior of the intrinsic localized modes and their eigenmodes even in the unstable regime.
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Affiliation(s)
- P G Kevrekidis
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Hennig D. Electron-vibron-breather interaction. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:2846-2857. [PMID: 11088767 DOI: 10.1103/physreve.62.2846] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2000] [Revised: 04/14/2000] [Indexed: 05/23/2023]
Abstract
We study the interaction of breathers in the context of a coupled electron-vibron lattice system. Starting with single-site excitations, it is demonstrated that constellations exist for which the coexistence of electronic and vibronic breathers is assured. The energy exchange between the vibrational and electronic subsystems and its impact on the breather formation are discussed in detail. The coupled electron-vibron dynamics shows a tendency toward energy redistribution into the vibronic degrees of freedom at the expense of the electronic energy content. Attention is paid to the relaxation dynamics in the energy exchange and we discuss the attainment of a steady regime for the coupled electron-vibron dynamics starting from a nonequilibrium state. It is demonstrated that the presence of breathers has a strong impact on the relaxation dynamics. Breathers can assist the relaxation process. With the help of a linear stability analysis, we show why the electronic subsystem acts as an energy donor while the vibron system serves as the energy acceptor. To this end we investigate the existence and stability of localized breathing eigenmodes capable of energy trapping. A frequency analysis reveals that strong exchange also occurs due to a temporal transition from single-frequency breathers to those oscillating with two frequencies and their temporal resonance interaction. Finally, the self-stabilized electron-vibron system relaxes to a combined electron-vibron breather. On increasing the electron-vibron coupling strength, only a vibronic phonobreather of large amplitude survives, whereas the electronic subsystem tends to energy equipartition.
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Affiliation(s)
- D Hennig
- Freie Universitat Berlin, Fachbereich Physik, Institut fur Theoretische Physik, Arnimallee 14, 14195 Berlin, Germany
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Kundu K. Perturbative study of classical Ablowitz-Ladik type soliton dynamics in relation to energy transport in alpha-helical proteins. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:5839-5851. [PMID: 11031645 DOI: 10.1103/physreve.61.5839] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/1999] [Revised: 12/03/1999] [Indexed: 05/23/2023]
Abstract
Classical Ablowitz-Ladik type soliton dynamics from three closely related classical nonlinear equations is studied using a perturbative method. Model nonintegrable equations are derived by assuming nearest neighbor hopping of an exciton(vibron) in the presence of a full exciton(vibron)-phonon interaction in soft molecular chains in general and spines of alpha-helices in particular. In all cases, both trapped and moving solitons are found implying activation energy barrier for propagating solitons. Analysis further shows that staggered and nearly staggered trapped solitons will have a negative effective mass. In some models the exciton(vibron)-phonon coupling affects the hopping. For these models, when the conservation of probability is taken into account, only propagating solitons with a broad profile are found to be acceptable solutions. Of course, for the soliton to be a physically meaningful entity, total nonlinear coupling strength should exceed a critical value. On the basis of the result, a plausible modification in the mechanism for biological energy transport involving conformational change in alpha-helix is proposed. Future directions of the work are also mentioned.
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Affiliation(s)
- K Kundu
- Institute of Physics, Bhubaneswar, India
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Hennig D. Solitonic energy transfer in a coupled exciton-vibron system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:4550-5. [PMID: 11088255 DOI: 10.1103/physreve.61.4550] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/1999] [Indexed: 11/07/2022]
Abstract
We consider the exciton transfer along a one-dimensional molecular chain. The exciton motion is influenced by longitudinal vibrations evolving in a Toda lattice potential. It is shown how the soliton solutions of the vibron system coupled to the exciton system induce solitonic exciton transfer. To this aim the existence of a regime of suppressed energy exchange between the coupled excitonic and vibrational degrees of freedom is established in the case of which a nonlinear Schrodinger equation for the exciton variable is derived. The nonlinear Schrodinger equation possesses soliton solutions corresponding to coherent transfer of the localized exciton.
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Affiliation(s)
- D Hennig
- Freie Universitat Berlin, Fachbereich Physik, Institut fur Theoretische Physik, Arnimallee 14, 14195 Berlin, Germany
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Affiliation(s)
- V. M. Kenkre
- Center for Advanced Studies and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131
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Lai R, Kiselev SA, Sievers AJ. Intrinsic localized spin-wave modes in antiferromagnetic chains with single-ion easy-axis anisotropy. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:R12665-R12668. [PMID: 9985214 DOI: 10.1103/physrevb.54.r12665] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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27
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Hennig D, Rasmussen KO, Gabriel H, Bülow A. Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5788-5801. [PMID: 9965768 DOI: 10.1103/physreve.54.5788] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Boiti M, Leon J, Pempinelli F. Nonlinear spectral characterization of discrete data. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5739-5742. [PMID: 9965761 DOI: 10.1103/physreve.54.5739] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Laedke EW, Kluth O, Spatschek KH. Existence of solitary solutions in nonlinear chains. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4299-4311. [PMID: 9965578 DOI: 10.1103/physreve.54.4299] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Raghavan S, Kenkre VM, Dunlap DH, Bishop AR, Salkola MI. Relation between dynamic localization in crystals and trapping in two-level atoms. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1996; 54:R1781-R1784. [PMID: 9913763 DOI: 10.1103/physreva.54.r1781] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Christiansen PL, Gaididei YB, Rasmussen KO, Mezentsev VK, Rasmussen JJ. Dynamics in discrete two-dimensional nonlinear Schrödinger equations in the presence of point defects. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:900-912. [PMID: 9985357 DOI: 10.1103/physrevb.54.900] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Datta PK, Kundu K. Time evolution of models described by a one-dimensional discrete nonlinear Schrödinger equation. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:14929-14936. [PMID: 9983286 DOI: 10.1103/physrevb.53.14929] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Bilbault JM, Marquié P. Energy localization in a nonlinear discrete system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:5403-5408. [PMID: 9964873 DOI: 10.1103/physreve.53.5403] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cai D, Bishop AR, Gronbech-Jensen N. Perturbation theories of a discrete, integrable nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:4131-4136. [PMID: 9964726 DOI: 10.1103/physreve.53.4131] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Aceves AB, Peschel T, Muschall R, Lederer F, Trillo S, Wabnitz S. Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1172-1189. [PMID: 9964354 DOI: 10.1103/physreve.53.1172] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cai D, Bishop AR, Gronbech-Jensen N. Discrete lattice effects on breathers in a spatially linear potential. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1202-1205. [PMID: 9964356 DOI: 10.1103/physreve.53.1202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hennig D, Rasmussen KO, Tsironis GP, Gabriel H. Breatherlike impurity modes in discrete nonlinear lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:R4628-R4631. [PMID: 9964091 DOI: 10.1103/physreve.52.r4628] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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38
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Hennig D, Sun NG, Gabriel H, Tsironis GP. Spatial properties of integrable and nonintegrable discrete nonlinear Schrödinger equations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:255-269. [PMID: 9963429 DOI: 10.1103/physreve.52.255] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Marquié P, Bilbault JM, Remoissenet M. Observation of nonlinear localized modes in an electrical lattice. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:6127-6133. [PMID: 9963352 DOI: 10.1103/physreve.51.6127] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cai D, Bishop AR, Gronbech-Jensen N, Salerno M. Electric-field-induced nonlinear bloch oscillations and dynamical localization. PHYSICAL REVIEW LETTERS 1995; 74:1186-1189. [PMID: 10058956 DOI: 10.1103/physrevlett.74.1186] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Flytzanis N, Malomed BA, Neuper A. Resonances in driven dynamical lattices. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:3498-3502. [PMID: 9979159 DOI: 10.1103/physrevb.51.3498] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Severs A, Page J. Chapter 3 Unusual anharmonic local mode systems. DYNAMICAL PROPERTIES OF SOLIDS 1995. [DOI: 10.1016/s1874-5628(06)80005-7] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
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Kivshar YS, Królikowski W, Chubykalo OA. Dark solitons in discrete lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:5020-5032. [PMID: 9962586 DOI: 10.1103/physreve.50.5020] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Laedke EW, Spatschek KH, Turitsyn SK. Stability of discrete solitons and quasicollapse to intrinsically localized modes. PHYSICAL REVIEW LETTERS 1994; 73:1055-1059. [PMID: 10057613 DOI: 10.1103/physrevlett.73.1055] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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45
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Cai D, Bishop AR, Gronbech-Jensen N, Malomed BA. Moving solitons in the damped Ablowitz-Ladik model driven by a standing wave. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:R694-R697. [PMID: 9962178 DOI: 10.1103/physreve.50.r694] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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46
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Kivshar YS, Salerno M. Modulational instabilities in the discrete deformable nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3543-3546. [PMID: 9961633 DOI: 10.1103/physreve.49.3543] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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