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Ben Zion MY, Oppenheimer N. Compact Expansion of a Repulsive Suspension. PHYSICAL REVIEW LETTERS 2024; 132:238201. [PMID: 38905695 DOI: 10.1103/physrevlett.132.238201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2023] [Accepted: 04/30/2024] [Indexed: 06/23/2024]
Abstract
Short-range repulsion governs the dynamics of matter from atoms to animals. Using theory, simulations, and experiments, we find that an ensemble of repulsive particles spreads compactly with a sharp boundary, in contrast to the diffusive spreading of Brownian particles. Starting from the pair interactions, at high densities, the many-body dynamics follow nonlinear diffusion with a self-similar expansion, growing as t^{1/4}; At longer times, thermal motion dominates with the classic t^{1/2} expansion. A logarithmic growth controlled by nearest-neighbor interactions connects the two self-similar regimes.
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Affiliation(s)
- Matan Yah Ben Zion
- School of Physics and Astronomy and the Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
- Department of Machine Learning and Natural Computing, Donders Institute for Brain, Cognition and Behavior, Thomas van Aquinostraat 4, Nijmegen, 6525GD, The Netherlands
| | - Naomi Oppenheimer
- School of Physics and Astronomy and the Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
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2
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Ecke RE. The center for nonlinear studies: A personal history. CHAOS (WOODBURY, N.Y.) 2024; 34:042102. [PMID: 38639345 DOI: 10.1063/5.0204698] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2024] [Accepted: 03/08/2024] [Indexed: 04/20/2024]
Abstract
The Center for Nonlinear Studies (CNLS) was an integral part of my scientific career starting as a Postdoctoral Fellow in 1983 up to my tenure as CNLS Director from 2004 to 2015. As such, I experienced a number of scientific phases of CNLS through almost four decades of foundation, evolution, and transition. Throughout this entire interval, the inspiration and influence of David Campbell guided my way. A proper history of CNLS encompassing all of the many contributors to the CNLS story is beyond my means or purpose here. Instead, I present the history as I experienced it. I emphasize the main scientific accomplishments achieved at CNLS over more than 40 years, but I will also attempt to describe and quantify the attributes that made and continue to make the Center for Nonlinear Studies a special institution of remarkable impact and longevity. Throughout its existence, CNLS owes much to the enduring legacy of David Campbell who laid down the foundations and operating principles that have made it so successful.
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Affiliation(s)
- Robert E Ecke
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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ABBAS TASAWAR, UL HASSAN QAZIMAHMOOD, HUSSAIN ANWAR, FATIMA MAHEEN, AHMAD BILAL. COMPACTION IN A CLASS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS. JOURNAL OF SCIENCE AND ARTS 2022. [DOI: 10.46939/j.sci.arts-22.4-a13] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
Abstract
We inspect the compaction structure in a class of nonlinear dispersive conditions in this article. The compaction sort of lone waves free of exponential tails and width self-sufficient of abundance is formally created. We further set up particular examples of answers for the defocusing parts of these models.
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Affiliation(s)
- TASAWAR ABBAS
- University of Wah, Department of Mathematics, 47040 Punjab, Pakistan
| | | | - ANWAR HUSSAIN
- National University of Sciences and Technology, School of Mechanical and Manufacturing Engineering, Department of Mechanical Engineering, Islamabad, Pakistan
| | - MAHEEN FATIMA
- Comsats University, Lahore Campus, Department of Mathematics, Pakistan
| | - BILAL AHMAD
- University of Wah, Department of Mathematics, 47040 Punjab, Pakistan
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4
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New Solutions of Nonlinear Dispersive Equation in Higher-Dimensional Space with Three Types of Local Derivatives. FRACTAL AND FRACTIONAL 2022. [DOI: 10.3390/fractalfract6040202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study of differential equations with local derivatives using the reduction technique. This higher-dimensional equation is considered with three types of local derivatives in the temporal sense. Different types of exact solutions in five cases are reported. Furthermore, with the help of the Maple package, the solutions found in this study are verified. Finally, several interesting 3D, 2D and density plots are demonstrated to visualize the nonlinear wave structures more efficiently.
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Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method. MATHEMATICS 2022. [DOI: 10.3390/math10060954] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation.
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Akinfe TK, Loyinmi AC. A solitary wave solution to the generalized Burgers-Fisher's equation using an improved differential transform method: A hybrid scheme approach. Heliyon 2021; 7:e07001. [PMID: 34136674 PMCID: PMC8180614 DOI: 10.1016/j.heliyon.2021.e07001] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Revised: 03/02/2021] [Accepted: 04/28/2021] [Indexed: 11/19/2022] Open
Abstract
In this research, an unrivalled hybrid scheme which involves the coupling of the new Elzaki integral transform (an improved version of Laplace transform) and a modified differential transform called the projected differential transform (PDTM) have been implemented to solve the generalized Burgers-Fisher's equation; which springs up due to the fusion of the Burgers' and the Fisher's equation; describing convective effects, diffusion transport or interaction between reaction mechanisms, traffic flows; and turbulence; consequently finding meaningful applicability in the applied sciences viz: gas dynamics, fluid dynamics, turbulence theory, reaction-diffusion theory, shock-wave formation, traffic flows, financial mathematics, and so on. Using the proposed Elzaki projected differential transform method (EPDTM), a generalized exact solution (Solitary solution) in form of a Taylor multivariate series has been obtained; of which the highly nonlinear terms and derivatives handled by PDTM have been decomposed without expansion, computation of Adomian or He's polynomials, discretization, restriction of parameters, and with less computational work whilst achieving a highly convergent results when compared to other existing analytical/exact methods in the literature, via comparison tables, 3D plots, convergence plots and fluid-like plots. Thus showing the distinction, novelty and huge advantage of the proposed method as an asymptotic alternative, in providing generalized or solitary wave solution to a wider class of differential equations.
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El-Ajou A, Oqielat MN, Al-Zhour Z, Kumar S, Momani S. Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative. CHAOS (WOODBURY, N.Y.) 2019; 29:093102. [PMID: 31575153 DOI: 10.1063/1.5100234] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2019] [Accepted: 08/13/2019] [Indexed: 06/10/2023]
Abstract
In this paper, the time-fractional nonlinear dispersive (TFND) partial differential equations (PDEs) in the sense of conformable fractional derivative (CFD) are proposed and analyzed. Three types of TFND partial differential equations are considered in the sense of CFD, which are the TFND Boussinesq, TFND Klein-Gordon, and TFND B(2, 1, 1) PDEs. Solitary pattern solutions for this class of TFND partial differential equations based on the residual fractional power series method is constructed and discussed. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. The results suggest that the algorithm presented here offers solutions to problems in a rapidly convergent series leading to ideal solutions. Furthermore, the results obtained are like those in previous studies that used other types of fractional derivatives. In addition, the calculations used were much easier and shorter compared with other types of fractional derivatives.
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Affiliation(s)
- Ahmad El-Ajou
- Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan
| | - Moa'ath N Oqielat
- Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan
| | - Zeyad Al-Zhour
- Department of Basic Engineering Sciences, College of Engineering, ImamAbdulrahman Bin Faisal University, P. O. Box 1982, Dammam 31441, Saudi Arabia
| | - Sunil Kumar
- Department of Mathematics, National Institute of Technology, Jamshedpur 831014, Jharkhand, India
| | - Shaher Momani
- College of Humanities and Sciences, Ajman University, Ajman, United Arab Emirates
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Gninzanlong CL, Ndjomatchoua FT, Tchawoua C. Taming intrinsic localized modes in a DNA lattice with damping, external force, and inhomogeneity. Phys Rev E 2019; 99:052210. [PMID: 31212565 DOI: 10.1103/physreve.99.052210] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2018] [Indexed: 11/07/2022]
Abstract
The dynamics of DNA in the presence of uniform damping and periodic force is studied. The damped and driven Joyeux-Buyukdagli model is used to investigate the formation of intrinsic localized modes (ILMs). Branches of ILMs are identified as well as their orbital stabilities. A study of the effect of inhomogeneity introduced into the DNA lattice and its ability to control chaotic behavior is conducted. It is seen that a single defect in the chain can induce synchronized spatiotemporal patterns, despite the fact that the entire set of oscillators and the impurity are chaotic when uncoupled. It is also shown that the periodic excitation applied on a specific site can drive the whole lattice into chaotic or regular spatial and temporal patterns.
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Affiliation(s)
| | - Frank Thomas Ndjomatchoua
- Sustainable Impact Platform, Adaptive Agronomy and Pest Ecology Cluster, International Rice Research Institute (IRRI), DAPO Box 7777-1301, Metro Manila, Philippines
| | - Clément Tchawoua
- Department of Physics, Faculty of Science, University of Yaoundé 1, P.O. Box 812, Yaoundé, Cameroon
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10
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Nesterenko VF. Waves in strongly nonlinear discrete systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0130. [PMID: 30037928 PMCID: PMC6077858 DOI: 10.1098/rsta.2017.0130] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 05/21/2018] [Indexed: 06/08/2023]
Abstract
The paper presents the main steps in the development of the strongly nonlinear wave dynamics of discrete systems. The initial motivation was prompted by the challenges in the design of barriers to mitigate high-amplitude compression pulses caused by impact or explosion. But this area poses a fundamental mathematical and physical problem and should be considered as a natural step in developing strongly nonlinear wave dynamics. Strong nonlinearity results in a highly tunable behaviour and allows design of systems with properties ranging from a weakly nonlinear regime, similar to the classical case of the Fermi-Pasta-Ulam lattice, or to a non-classical case of sonic vacuum. Strongly nonlinear systems support periodic waves and one of the fascinating results was a discovery of a strongly nonlinear solitary wave in sonic vacuum (a limiting case of a periodic wave) with properties very different from the Korteweg de Vries solitary wave. Shock-like oscillating and monotonous stationary stress waves can also be supported if the system is dissipative. The paper discusses the main theoretical and experimental results, focusing on travelling waves and possible future developments in the area of strongly nonlinear metamaterials.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.
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Affiliation(s)
- Vitali F Nesterenko
- Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA
- Materials Science and Engineering Program, University of California at San Diego, La Jolla, CA 92093-0411, USA
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11
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Shi J, Zeng J, Malomed BA. Suppression of the critical collapse for one-dimensional solitons by saturable quintic nonlinear lattices. CHAOS (WOODBURY, N.Y.) 2018; 28:075501. [PMID: 30070511 DOI: 10.1063/1.5015933] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The stabilization of one-dimensional solitons by a nonlinear lattice against the critical collapse in the focusing quintic medium is a challenging issue. We demonstrate that this purpose can be achieved by combining a nonlinear lattice and saturation of the quintic nonlinearity. The system supports three species of solitons, namely, fundamental (even-parity) ones and dipole (odd-parity) modes of on- and off-site-centered types. Very narrow fundamental solitons are found in an approximate analytical form, and systematic results for very broad unstable and moderately broad partly stable solitons, including their existence and stability areas, are produced by means of numerical methods. Stability regions of the solitons are identified by means of systematic simulations. The stability of all the soliton species obeys the Vakhitov-Kolokolov criterion.
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Affiliation(s)
- Jincheng Shi
- State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics of CAS, Xi'an 710119, China
| | - Jianhua Zeng
- State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics of CAS, Xi'an 710119, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Forni RJ, Shockley RC. Simplest breather. Phys Rev E 2018; 97:052210. [PMID: 29906859 DOI: 10.1103/physreve.97.052210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2017] [Indexed: 06/08/2023]
Abstract
The simplest Klein-Gordon (KG) breather is a compacton on a string subject to the force of gravity in a frictionless V-shaped trough. Its dynamics, spectrum, and energy are discussed and it is compared to sine-Gordon breathers. A generalization of this problem consists of a charged string subject to the electrostatic force of two semi-infinite coplanar charged planes separated by a gap of constant width. For motion in the midplane between these planes, the string's displacement u(x,t) satisfies the nonlinear KG equation (∂_{t}^{2}-∂_{x}^{2})u=-tan^{-1}u in dimensionless form. Simulations of this equation reveal long-lived, breatherlike states, or "pseudobreathers," which preserve shape and speed to high accuracy when Lorentz-transformed to simulate collisions.
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Affiliation(s)
- Ronald J Forni
- Agilent Technologies Vacuum Products Division 121 Hartwell Avenue, Lexington, Massachusetts 02421, USA
| | - Richard C Shockley
- Space & Naval Warfare Systems Center Pacific Maritime Systems Divison, Code 56490 53560 Hull Street, San Diego, California 92152, USA
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Kevrekidis PG, Siettos CI, Kevrekidis YG. To infinity and some glimpses of beyond. Nat Commun 2017; 8:1562. [PMID: 29146925 PMCID: PMC5691182 DOI: 10.1038/s41467-017-01502-7] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/29/2016] [Accepted: 09/22/2017] [Indexed: 11/08/2022] Open
Abstract
When mathematical and computational dynamic models reach infinity in finite time, extending analysis and numerics beyond it becomes a notorious challenge. We suggest how, upon suitable transformations, it may become possible to go beyond infinity with the solution becoming again well behaved and the computations continuing normally. In our Ordinary Differential Equation examples the crossing of infinity occurs instantaneously. For Partial Differential Equations, the crossing of infinity may persist for finite time, necessitating the introduction of buffer zones, within which an appropriate transformation is adaptively identified. Along the path of our analysis, we present a regularization process via complexification and explore its impact on the dynamics; we also discuss a set of compactification transformations and their intuitive implications. This methodology could be useful toward a systematic approach to bypassing infinity and thus going beyond it in a broader range of evolution equation models. Certain physical problems such as the rupture of a thin sheet can be difficult to solve as computations breakdown at the point of rupture. Here the authors propose a regularization approach to overcome this breakdown which could help dealing with mathematical models that have finite time singularities.
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Affiliation(s)
- Panayotis G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, 01003-4515, USA
| | - Constantinos I Siettos
- School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Athens, GR, 15780, Greece
| | - Yannis G Kevrekidis
- Department of Chemical and Biological Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA. .,Institute for Advanced Study, TUM, Munich, 85748, Germany. .,Zuse Institute, Berlin, 14195, Germany.
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Perchikov N, Gendelman OV. Flat bands and compactons in mechanical lattices. Phys Rev E 2017; 96:052208. [PMID: 29347743 DOI: 10.1103/physreve.96.052208] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2017] [Indexed: 11/07/2022]
Abstract
Local configurational symmetry in lattice structures may give rise to stationary, compact solutions, even in the absence of disorder and nonlinearity. These compact solutions are related to the existence of flat dispersion curves (bands). Nonlinearity can destabilize such compactons. One common flat-band-generating system is the one-dimensional cross-stitch model, in which compactons were shown to exist for the photonic lattice with Kerr nonlinearity. The compactons exist there already in the linear regime and are not generally destructed by that nonlinearity. Smooth nonlinearity of this kind does not permit performing complete stability analysis for this chain. We consider a discrete mechanical system with flat dispersion bands, in which the nonlinearity exists due to impact constraints. In this case, one can use the concept of the saltation matrix for the analytic construction of the monodromy matrix. Besides, we consider a smooth nonlinear lattice with linearly connected massless boxes, each containing two symmetric anharmonic oscillators. In this model, the flat bands and discrete compactons also readily emerge. This system also permits performing comprehensive stability analysis, at least in the anticontinuum limit, due to the reduced number of degrees of freedom. In both systems, there exist two types of localization. The first one is the complete localization, and the second one is the more common exponential localization. The latter type is associated with discrete breathers (DBs). Two principal mechanisms for the loss of stability are revealed. The first one is the possible internal instability of the symmetric and/or antisymmetric solution in the individual unit cell of the chain. One can interpret this instability pattern as internal resonance between the compacton and the DB. The other mechanism is global instability related to resonance of the stationary solution with the propagation frequencies. Different instability mechanisms lead to different bifurcations at the stability threshold.
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Affiliation(s)
- Nathan Perchikov
- Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
| | - O V Gendelman
- Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
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Chong C, Porter MA, Kevrekidis PG, Daraio C. Nonlinear coherent structures in granular crystals. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:413003. [PMID: 28877033 DOI: 10.1088/1361-648x/aa7672] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The study of granular crystals, which are nonlinear metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures-which include traveling solitary waves, dispersive shock waves, and discrete breathers-have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research to date has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.
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Affiliation(s)
- C Chong
- Department of Mathematics, Bowdoin College, Brunswick, Maine 04011, United States of America
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Pokam Nguewawe C, Fewo SI, Yemélé D. Higher-order effects on the properties of the optical compact bright pulse: Collective variable approach. Phys Rev E 2017; 95:012215. [PMID: 28208352 DOI: 10.1103/physreve.95.012215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2016] [Indexed: 06/06/2023]
Abstract
The effects of higher-order (HO) terms on the properties of the compact bright (CB) pulse described by the dispersionless nonlocal nonlinear Schrödinger (DNNLS) equation are investigated. These effects include third-order dispersion (TOD), the Raman term, and the time derivative of the pulse envelope. By means of the collective variable method, the dynamical behavior of the pulse amplitude, width, frequency, velocity, phase, and chirp during propagation is pointed out. The results indicate that the CB pulse experiences a self-frequency shift and self-steepening, respectively, in the presence of an isolated Raman term and the time derivative of the pulse envelope and acquires a velocity as the result of the TOD effect. In addition, TOD may also induce the breathing mode inside the variation of the pulse parameters when the width of the input pulse is slightly less than that of the unperturbed CB pulse. The combination of these terms, indispensable for describing ultrashort pulses, reproduces all these phenomena in the CB pulse behavior. Further, other properties are observed, namely, the pulse decay, the breathing mode even when the unperturbed CB pulse is taken as the input signal, and the attenuated pulse. These results are in good agreement with the results of the direct numerical simulations of the DNNLS equation with HO terms.
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Affiliation(s)
- Chancelor Pokam Nguewawe
- Laboratory of Mechanics and Modeling of Physics Systems, Faculty of Science, University of Dschang, P.O. Box 067, Dschang, Cameroon
| | - Serge I Fewo
- Laboratory of Mechanics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon
| | - David Yemélé
- Laboratory of Mechanics and Modeling of Physics Systems, Faculty of Science, University of Dschang, P.O. Box 067, Dschang, Cameroon
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Ahsan Z, Jayaprakash KR. Evolution of a primary pulse in the granular dimers mounted on a linear elastic foundation: An analytical and numerical study. Phys Rev E 2016; 94:043001. [PMID: 27841495 DOI: 10.1103/physreve.94.043001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2016] [Indexed: 06/06/2023]
Abstract
In this exposition we consider the wave dynamics of a one-dimensional periodic granular dimer (diatomic) chain mounted on a damped and an undamped linear elastic foundation (otherwise called the on-site potential). It is very well known that periodic granular dimers support solitary wave propagation (similar to that in the homogeneous granular chains) for a specific discrete set of mass ratios. In this work we present the analytical investigation of the evolution of solitary waves and primary pulses in granular dimers when they are mounted on on-site potential with and without velocity proportional foundation damping. We invoke a methodology based on the multiple time-scale asymptotic analysis and partition the dynamics of the perturbed dimer chain into slow and fast components. The dynamics of the dimer chain in the limit of large mass mismatch (auxiliary chain) mounted on on-site potential and foundation damping is used as the basis for the analysis. A systematic analytical procedure is then developed for the slowly varying response of the beads and in estimating primary pulse amplitude evolution resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The methodology is applicable for arbitrary mass ratios between the beads. We present several examples to demonstrate the efficacy of the proposed method. It is observed that the amplitude evolution predicted by the described methodology is in good agreement with the numerical simulation of the original system. This work forms a basis for further application of the considered methodology to weakly coupled granular dimers which finds practical relevance in designing shock mitigating granular layers.
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Affiliation(s)
- Zaid Ahsan
- Department of Aerospace Engineering, Indian Institute of Science, Bengaluru-560012, India
| | - K R Jayaprakash
- Department of Aerospace Engineering, Indian Institute of Science, Bengaluru-560012, India
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Cirillo ENM, Ianiro N, Sciarra G. Compacton formation under Allen–Cahn dynamics. Proc Math Phys Eng Sci 2016. [DOI: 10.1098/rspa.2015.0852] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non–constant higher order stiffness. We analytically solve the stationary problem and deduce the existence of so-called compactons, namely, connections on a finite interval between the two phases. The dynamics problem is numerically solved and compacton formation is described.
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Affiliation(s)
- E. N. M. Cirillo
- Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, Roma 00161, Italy
| | - N. Ianiro
- Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, Roma 00161, Italy
| | - G. Sciarra
- Dipartimento di Ingegneria Chimica Materiali Ambiente, Sapienza Università di Roma, via Eudossiana 18, Roma 00184, Italy
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Zarmi Y. Spatially Extended Relativistic Particles Out of Traveling Front Solutions of Sine-Gordon Equation in (1+2) Dimensions. PLoS One 2016; 11:e0148993. [PMID: 26930077 PMCID: PMC4773000 DOI: 10.1371/journal.pone.0148993] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2015] [Accepted: 12/22/2015] [Indexed: 11/25/2022] Open
Abstract
Slower-than-light multi-front solutions of the Sine-Gordon in (1+2) dimensions, constructed through the Hirota algorithm, are mapped onto spatially localized structures, which emulate free, spatially extended, massive relativistic particles. A localized structure is an image of the junctions at which the fronts intersect. It propagates together with the multi-front solution at the velocity of the latter. The profile of the localized structure obeys the linear wave equation in (1+2) dimensions, to which a term that represents interaction with a slower-than-light, Sine-Gordon-multi-front solution has been added. This result can be also formulated in terms of a (1+2)-dimensional Lagrangian system, in which the Sine-Gordon and wave equations are coupled. Expanding the Euler-Lagrange equations in powers of the coupling constant, the zero-order part of the solution reproduces the (1+2)-dimensional Sine-Gordon fronts. The first-order part is the spatially localized structure. PACS 02.30.Ik, 03.65.Pm, 05.45.Yv, 02.30.Ik.
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Affiliation(s)
- Yair Zarmi
- The Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Midreshet Ben-Gurion, 8499000, Israel
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Pozharskiy D, Zhang Y, Williams MO, McFarland DM, Kevrekidis PG, Vakakis AF, Kevrekidis IG. Nonlinear resonances and antiresonances of a forced sonic vacuum. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:063203. [PMID: 26764846 DOI: 10.1103/physreve.92.063203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Indexed: 06/05/2023]
Abstract
We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force ("antiresonances") between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) "nonlinear spectrum" in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). We rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.
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Affiliation(s)
- D Pozharskiy
- Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA
| | - Y Zhang
- Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, Illinois 61822, USA
| | - M O Williams
- Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA
| | - D M McFarland
- Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61822, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA
| | - A F Vakakis
- Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, Illinois 61822, USA
| | - I G Kevrekidis
- Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA
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22
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Liu SW, Yang YY, Duan WS, Yang L. Pulse reflection and transmission due to impurities in a granular chain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:013202. [PMID: 26274296 DOI: 10.1103/physreve.92.013202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2014] [Indexed: 06/04/2023]
Abstract
Reflection and transmission due to the incident wave in one-dimensional bead chains when there are impurities have been studied. The impurities can be any kind of material, any size, and their numbers are arbitrary. The dependence of the transmission and the reflection on the numbers and the material parameters of the impurities are given. The analytical results are given by using the inverse scattering method. Substantial reflection is observed if there is only one steel bead. However, the reflection is negligible if there are two steel beads. The reflection monotonously increases as the numbers of the steel beads increase. The reflection remains a constant when the numbers of the steel beads are so many that the length composed by the steel beads is larger than the width of the solitary wave. It can be used to detect the impurities in the beads' chain by measuring the reflection of a pulse.
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Affiliation(s)
- Shi Wei Liu
- Institute of Modern Physics & College of Physics and Electronic Engineering and Joint Laboratory of Atomic and Molecular Physics of NWNU, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, and Northwest Normal University, Lanzhou 730070, China
| | - Yang Yang Yang
- Institute of Modern Physics & College of Physics and Electronic Engineering and Joint Laboratory of Atomic and Molecular Physics of NWNU, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, and Northwest Normal University, Lanzhou 730070, China
| | - Wen Shan Duan
- Institute of Modern Physics & College of Physics and Electronic Engineering and Joint Laboratory of Atomic and Molecular Physics of NWNU, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, and Northwest Normal University, Lanzhou 730070, China
| | - Lei Yang
- Institute of Modern Physics & College of Physics and Electronic Engineering and Joint Laboratory of Atomic and Molecular Physics of NWNU, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, and Northwest Normal University, Lanzhou 730070, China
- Department of Physics, Lanzhou University, Lanzhou 730000, China
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23
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Kenmogne F, Yemélé D, Kengne J, Ndjanfang D. Transverse compactlike pulse signals in a two-dimensional nonlinear electrical network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052921. [PMID: 25493872 DOI: 10.1103/physreve.90.052921] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2014] [Indexed: 06/04/2023]
Abstract
We investigate the compactlike pulse signal propagation in a two-dimensional nonlinear electrical transmission network with the intersite circuit elements (both in the propagation and transverse directions) acting as nonlinear resistances. Model equations for the circuit are derived and can reduce from the continuum limit approximation to a two-dimensional nonlinear Burgers equation governing the propagation of the small amplitude signals in the network. This equation has only the mass as conserved quantity and can admit as solutions cusp and compactlike pulse solitary waves, with width independent of the amplitude, according to the sign of the product of its nonlinearity coefficients. In particular, we show that only the compactlike pulse signal may propagate depending on the choice of the realistic physical parameters of the network, and next we study the dissipative effects on the pulse dynamics. The exactness of the analytical analysis is confirmed by numerical simulations which show a good agreement with results predicted by the Rosenau and Hyman K(2,2) equation.
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Affiliation(s)
- Fabien Kenmogne
- Laboratory of Modelling and Simulation in Engineering and Biological Physics, Faculty of Science, University of Yaoundé I, Po Box 812, Yaoundé, Cameroon
| | - David Yemélé
- Laboratoire de Mécanique et de Modélisation des Systèmes Physiques L2MSP, Faculté des Sciences, Université de Dschang, B. P. 067, Dschang, Cameroon
| | - Jacques Kengne
- Laboratoire d' Automatique et Informatique Apliquée (LAIA), IUT-FV Bandjoun, University of Dschang, Cameroon
| | - Désiré Ndjanfang
- Laboratoire de Mécanique et de Modélisation des Systèmes Physiques L2MSP, Faculté des Sciences, Université de Dschang, B. P. 067, Dschang, Cameroon
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James G, Pelinovsky D. Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials. Proc Math Phys Eng Sci 2014; 470:20130462. [PMID: 24808748 PMCID: PMC3973387 DOI: 10.1098/rspa.2013.0462] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2013] [Accepted: 01/24/2014] [Indexed: 11/12/2022] Open
Abstract
We consider a class of fully nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order α>1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyse the propagation of localized waves when α is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic Korteweg-de Vries (KdV) equation and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When [Formula: see text], we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.
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Affiliation(s)
- Guillaume James
- INRIA Grenoble Rhône-Alpes and Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, Grenoble Cedex 9 38041, France
| | - Dmitry Pelinovsky
- Department of Mathematics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
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26
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Rosenau P, Pikovsky A. Breathers in strongly anharmonic lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022924. [PMID: 25353562 DOI: 10.1103/physreve.89.022924] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2013] [Indexed: 06/04/2023]
Abstract
We present and study a family of finite amplitude breathers on a genuinely anharmonic Klein-Gordon lattice embedded in a nonlinear site potential. The direct numerical simulations are supported by a quasilinear Schrodinger equation (QLS) derived by averaging out the fast oscillations assuming small, albeit finite, amplitude vibrations. The genuinely anharmonic interlattice forces induce breathers which are strongly localized with tails evanescing at a doubly exponential rate and are either close to a continuum, with discrete effects being suppressed, or close to an anticontinuum state, with discrete effects being enhanced. Whereas the D-QLS breathers appear to be always stable, in general there is a stability threshold which improves with spareness of the lattice.
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Affiliation(s)
- Philip Rosenau
- School of Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel
| | - Arkady Pikovsky
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
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27
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Yulin AV, Konotop VV. Conservative and PT-symmetric compactons in waveguide networks. OPTICS LETTERS 2013; 38:4880-4883. [PMID: 24322156 DOI: 10.1364/ol.38.004880] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Stable discrete compactons in interconnected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time (PT)-symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e., gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.
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28
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Kenmogne F, Yemélé D. Bright and peaklike pulse solitary waves and analogy with modulational instability in an extended nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:043204. [PMID: 24229297 DOI: 10.1103/physreve.88.043204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2013] [Revised: 09/16/2013] [Indexed: 06/02/2023]
Abstract
The modulational instability (MI) phenomenon in the nonlinear Schrödinger equation (NLSE) extended by two different nonlinear dispersion terms and the gradient term is investigated. We find that the possibility of instability of plane waves depends on the sign of the nonlinear dispersion parameters with regard to the linear dispersion coefficient. In contrast to the basic NLSE, the system may exhibit instability in the defocusing media for amplitude exceeding a critical value depending on the magnitude of the nonlinear dispersion. An additional feature, namely the higher order or the infinite gain band, absent in the NLSE case, may appear and in which MI induces the birth of the nonlinear localized wave (NLW) of different carrier wave numbers. The result of the qualitative investigations of the system's dynamics indicates the existence of the NLW, such as peak, bright, dark, and compact dark solitary waves which can be well predicted by the MI criteria. In addition the nonlinear dispersion induces the existence of a pair of bright-dark solitary waves which is usually exhibited by the coupled NLSEs only, and the pairs of peak-dark and compact dark-bright solitary waves.
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Affiliation(s)
- Fabien Kenmogne
- Laboratory of Modelling and Simulation in Engineering and Biological Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
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29
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Bazeia D, Lobão AS, Losano L, Menezes R. First-order formalism for flat branes in generalizedN-field models. Int J Clin Exp Med 2013. [DOI: 10.1103/physrevd.88.045001] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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30
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Starosvetsky Y, Ben-Meir Y. Nonstationary regimes of homogeneous Hamiltonian systems in the state of sonic vacuum. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062919. [PMID: 23848760 DOI: 10.1103/physreve.87.062919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2013] [Revised: 04/22/2013] [Indexed: 06/02/2023]
Abstract
In the present paper we study the mechanism that leads to the formation of regular patterns of energy localization and complete recurrent energy transport in the homogeneous systems of anharmonic oscillators and oscillatory chains subjected to a state of sonic vacuum. The basic model under investigation comprises a system of purely anharmonic oscillators as well as oscillatory chains given to a localized excitation where the initial energy is imported to one of the oscillators or oscillatory chains. The results of numerical simulations reveal the existence of a strong classical beating phenomenon, characterized by complete, recurrent, resonant energy exchanges between the oscillators and oscillatory chain and this in the state of sonic vacuum where no regular resonant frequencies can be defined. In this study we show that formation of the recurrent energy exchanges in this highly degenerate model is strictly stipulated by the system parameters. Thus, for instance, choosing the parameter of coupling below a certain threshold leads to significant energy localization on one of the oscillators or oscillatory chains. However, increasing the strength of coupling above the threshold leads to the formation of a strong beating response. The analytical study pursued in this paper predicts the origin of formation of a strong beating phenomenon and provides the necessary conditions on the system parameter for its excitation. Moreover, careful analysis of the beating phenomenon reveals the qualitatively different global bifurcation undergone by this type of highly nonstationary regime. The theoretical study is further extended to the system of coupled purely anharmonic lattices. Thus we show analytically and numerically that excitation of some particular solutions (e.g., spatially periodic standing waves and standing breathers) on one of the lattices results in the formation of similar patterns of energy (wave) localization as well as the regime of complete recurrent interchain energy transport. In particular we demonstrate that the formation of these regimes is solely affected by a particular choice of system parameters. The results of the analytical study are found to be in very good agreement with those of numerical simulations.
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Affiliation(s)
- Y Starosvetsky
- Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel.
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31
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Theocharis G, Boechler N, Daraio C. Nonlinear Periodic Phononic Structures and Granular Crystals. ACOUSTIC METAMATERIALS AND PHONONIC CRYSTALS 2013. [DOI: 10.1007/978-3-642-31232-8_7] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
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32
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Mulansky M, Pikovsky A. Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:056214. [PMID: 23214864 DOI: 10.1103/physreve.86.056214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2012] [Revised: 09/26/2012] [Indexed: 06/01/2023]
Abstract
In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder.
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Affiliation(s)
- Mario Mulansky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24, D-14476 Potsdam-Golm, Germany
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33
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Jordan PM, Saccomandi G. Compact acoustic travelling waves in a class of fluids with nonlinear material dispersion. Proc Math Phys Eng Sci 2012. [DOI: 10.1098/rspa.2012.0321] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We apply a phenomenological theory of continua put forth by Rubin, Rosenau and Gottlieb in 1995 to an important class of compressible media. Regarding the material characteristic length coefficient,
α
, not as constant, but instead as a quadratic function of the velocity gradient, we carry out an in-depth analysis of one-dimensional acoustic travelling waves in inviscid, non-thermally conducting fluids. Analytical and numerical methods are employed to study the resulting waveforms, a special case of which exhibits compact support. In particular, a phase plane analysis is performed; simplified approximate/asymptotic expressions are presented; and a weakly nonlinear, KdV-like model that admits compact travelling wave solutions (TWSs), but which is not of the class
K
(
m
,
n
), is derived and analysed. Most significantly, our formulation allows for compact, pulse-type, acoustic waveforms in both gases and liquids.
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Affiliation(s)
- P. M. Jordan
- Acoustics Division, Naval Research Laboratory, Stennis Space Center, MS 39529, USA
| | - G. Saccomandi
- Dipartimento di Ingegneria Industriale, Università degli Studi di Perugia, 06125 Perugia, Italy
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Roy S, Pikovsky A. Spreading of energy in the Ding-Dong model. CHAOS (WOODBURY, N.Y.) 2012; 22:026118. [PMID: 22757577 DOI: 10.1063/1.3695369] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We study the properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice, the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.
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Affiliation(s)
- S Roy
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany
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35
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Starosvetsky Y. Evolution of the primary pulse in one-dimensional granular crystals subject to on-site perturbations: analytical study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051306. [PMID: 23004749 DOI: 10.1103/physreve.85.051306] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2012] [Indexed: 06/01/2023]
Abstract
The propagation of the primary pulse through an uncompressed granular chain subject to external on-site perturbation is studied. An analytical procedure predicting the evolution of the primary pulse is devised for the general form of the on-site perturbation applied on the chain. The validity of the analytical model is confirmed with several specific granular setups, such as chains mounted on the nonlinear elastic foundation, chains perturbed by the dissipative forces, as well as randomly perturbed chains. An additional interesting finding made in the present paper corresponds to the chains subject to a special type of perturbation, including the terms leading to dissipation and those acting as an energy source. In the paper, it is shown that an application of such a perturbation may lead to the formation of stable stationary primary pulses propagating with constant amplitudes and acting as attractors for the initially unperturbed Nesterenko solitary waves. Interestingly enough, the developed analytical procedure provides an extremely close estimation for the amplitudes of these stationary primary pulses as well as predicts zones of their stability. In conclusion, we would like to stress that the developed analytical model has demonstrated spectacular agreement with the results of the direct numerical simulations, and this is for various configurations considered in the current paper.
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Affiliation(s)
- Yuli Starosvetsky
- Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel.
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36
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Kenmogne F, Yemélé D, Woafo P. Electrical dark compacton generator: theory and simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056606. [PMID: 23004894 DOI: 10.1103/physreve.85.056606] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2011] [Revised: 03/06/2012] [Indexed: 06/01/2023]
Abstract
A modified Colpitts oscillator (MCO) associated with a nonlinear transmission line (NLTL) with intersite nonlinearity is introduced as a self-sustained generator of a train of modulated dark signals with compact shape. Equations of state describing the dynamics of the MCO part are derived and the stationary state is obtained. Using the Routh-Hurwitz criterion, the result of a stability analysis indicates the existence of a limit cycle in certain parameter regimes and there the oscillation of the circuit delivers pulselike electrical signals. The train of generated signals is then transformed into a train of compact modulated dark voltage solitons by the NLTL. The exactness of this analytical analysis is confirmed by numerical simulations performed on the circuit equations. Finally, simulations show the capacity of this circuit to work as a generator of compactlike dark voltage solitons. The performance of the generator, namely, the pulse width and the repetition rate, is controlled by the magnitude of the characteristic parameters of the electronic components of the device.
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Affiliation(s)
- Fabien Kenmogne
- Laboratory of Modelling and Simulation in Engineering and Biological Physics, Faculty of Science, University of Yaoundé I, Cameroon.
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37
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Ndjoko PB, Bilbault JM, Binczak S, Kofane TC. Compact-envelope bright solitary wave in a DNA double strand. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:011916. [PMID: 22400600 DOI: 10.1103/physreve.85.011916] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2011] [Revised: 12/08/2011] [Indexed: 05/31/2023]
Abstract
We study the nonlinear dynamics of a homogeneous DNA chain that is based on site-dependent finite stacking and pairing enthalpies. We introduce an extended nonlinear Schrödinger equation describing the dynamics of modulated wave in DNA model. We obtain envelope bright solitary waves with compact support as a solution. Analytical criteria of existence of this solution are derived. The stability of bright compactons is confirmed by numerical simulations of the exact equations of the lattice. The impact of the finite stacking energy is investigated, and we show that some of these compact bright solitary waves are robust, while others decompose very quickly depending on the finite stacking parameters.
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Affiliation(s)
- P B Ndjoko
- Laboratoire LE2I, Université de Bourgogne UMR CNRS 5158, B.P. 47870, 21078 Dijon Cedex, France
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39
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Ndjoko PB, Bilbault JM, Binczak S, Kofané TC. Nonlinear mechanics of DNA doule strand: existence of the compact-envelope bright solitary wave. ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. IEEE ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY. ANNUAL INTERNATIONAL CONFERENCE 2012; 2012:5428-5431. [PMID: 23367157 DOI: 10.1109/embc.2012.6347222] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We study the nonlinear dynamics of a homogeneous DNA chain which is based on site-dependent finite stacking and pairing enthalpies. We introduce an extended nonlinear Schrödinger equation describing the dynamics of modulated wave in DNA model. We obtain envelope bright solitary waves with compact support as a solution. Analytical criteria of existence of this solution are derived. The stability of bright compactons is confirmed by numerical simulations of the exact equations of the lattice. The impact of the finite stacking energy is investigated and we show that some of these compact bright solitary waves are robust, while others decompose very quickly depending on the finite stacking parameters.
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Affiliation(s)
- Paul Berlin Ndjoko
- LE2I Laboratory , University of Burgundy, UMR CNRS 5158, B.P. 47870, 21078 Dijon Cedex, France
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40
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Adam C, Queiruga JM, Sanchez-Guillen J, Wereszczynski A. Supersymmetrickfield theories and defect structures. Int J Clin Exp Med 2011. [DOI: 10.1103/physrevd.84.065032] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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42
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Al-Akhaly GA, Dey B. Gap compactonlike solutions of coupled Kortweg-de Vries equations with linear and nonlinear dispersions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036607. [PMID: 22060522 DOI: 10.1103/physreve.84.036607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2011] [Indexed: 05/31/2023]
Abstract
We show the existence of a type of excitation, which we term as "gap compactonlike," within the gap of the linear spectrum of a system of coupled Kortweg-de Vries equations with linear and nonlinear dispersions. Since the solutions lie in the gap region of the spectra, they avoid resonance with the linear oscillatory wave and, therefore, do not decay into radiations. These types of solutions are important in energy localization and transport in polymers and biopolymers, optical systems, etc.
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Affiliation(s)
- Galal A Al-Akhaly
- Department of Physics, University of Pune, Ganeshkhind, Pune 411007, India
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43
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Adam C, Queiruga JM, Sanchez-Guillen J, Wereszczynski A. N=1supersymmetric extension of the baby Skyrme model. Int J Clin Exp Med 2011. [DOI: 10.1103/physrevd.84.025008] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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44
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Cardenas A, Mihaila B, Cooper F, Saxena A. Properties of compacton-anticompacton collisions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:066705. [PMID: 21797514 DOI: 10.1103/physreve.83.066705] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2011] [Indexed: 05/31/2023]
Abstract
We study the properties of compacton-anticompacton collision processes. We compare and contrast results for the case of compacton-anticompacton solutions of the K(l,p) Rosenau-Hyman (RH) equation for l = p = 2, with compacton-anticompacton solutions of the L(l,p) Cooper-Shepard-Sodano (CSS) equation for p = 1 and l = 3. This study is performed using a Padé discretization of the RH and CSS equations. We find a significant difference in the behavior of compacton-anticompacton scattering. For the CSS equation, the scattering can be interpreted as "annihilation" as the wake left behind dissolves over time. In the RH equation, the numerical evidence is that multiple shocks form after the collision, which eventually lead to "blowup" of the resulting wave form.
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Affiliation(s)
- Andres Cardenas
- Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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Straube AV. Small-scale particle advection, manipulation and mixing: beyond the hydrodynamic scale. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2011; 23:184122. [PMID: 21508483 DOI: 10.1088/0953-8984/23/18/184122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
In this paper we discuss the problems of particle advection, manipulation and mixing at small scales. We start by considering reaction-advection-diffusion systems with the focus on mixing. We show how mixing advection affects the processes of reaction-diffusion and discuss mixing-induced instabilities. Further, we consider the problem of particle manipulation and discuss collective effects in systems comprising solid and compressible particles. We particularly discuss mechanisms of particle entrapment, the role of compressibility in the dynamics of bubbly liquids and nonequilibrium colloidal explosion. Finally, we address two issues related to the problem of wetting. First, we study the role of contact line motion for a sessile droplet (or a bubble) on an oscillating substrate. Second, we discuss an instability of a thin film leading to the formation of a fractal structure of droplets.
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Affiliation(s)
- Arthur V Straube
- Department of Physics, Humboldt University of Berlin, Berlin, Germany.
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Benzi R, Sbragaglia M, Bernaschi M, Succi S. Phase-field model of long-time glasslike relaxation in binary fluid mixtures. PHYSICAL REVIEW LETTERS 2011; 106:164501. [PMID: 21599369 DOI: 10.1103/physrevlett.106.164501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2010] [Indexed: 05/30/2023]
Abstract
We present a new phase-field model for binary fluids, exhibiting typical signatures of soft-glassy behavior, such as long-time relaxation, aging, and long-term dynamical arrest. The present model allows the cost of building an interface to vanish locally within the interface, while preserving positivity of the overall surface tension. A crucial consequence of this property, which we prove analytically, is the emergence of free-energy minimizing density configurations, hereafter named "compactons," to denote their property of being localized to a finite-size region of space and strictly zero elsewhere (no tails). Thanks to compactness, any arbitrary superposition of compactons still is a free-energy minimizer, which provides a direct link between the complexity of the free-energy landscape and the morphological complexity of configurational space.
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Affiliation(s)
- R Benzi
- Department of Physics and INFN, University of Tor Vergata, Rome, Italy
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Zhirov OV, Pikovsky AS, Shepelyansky DL. Quantum vacuum of strongly nonlinear lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:016202. [PMID: 21405755 DOI: 10.1103/physreve.83.016202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2010] [Revised: 10/04/2010] [Indexed: 05/30/2023]
Abstract
We study the properties of classical and quantum strongly nonlinear chains by means of extensive numerical simulations. Due to strong nonlinearity, the classical dynamics of such chains remains chaotic at arbitrarily low energies. We show that the collective excitations of classical chains are described by sound waves whose decay rate scales algebraically with the wave number with a generic exponent value. The properties of the quantum chains are studied by the quantum Monte Carlo method and it is found that the low-energy excitations are well described by effective phonon modes with the sound velocity dependent on an effective Planck constant. Our results show that at low energies the quantum effects lead to a suppression of chaos and drive the system to a quasi-integrable regime of effective phonon modes.
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Affiliation(s)
- O V Zhirov
- Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
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Mihaila B, Cardenas A, Cooper F, Saxena A. Stability and dynamical properties of Cooper-Shepard-Sodano compactons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:066702. [PMID: 21230749 DOI: 10.1103/physreve.82.066702] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2010] [Revised: 11/04/2010] [Indexed: 05/30/2023]
Abstract
Extending a Padé approximant method used for studying compactons in the Rosenau-Hyman (RH) equation, we study the numerical stability of single compactons of the Cooper-Shepard-Sodano (CSS) equation and their pairwise interactions. The CSS equation has a conserved Hamiltonian which has allowed various approaches for studying analytically the nonlinear stability of the solutions. We study three different compacton solutions and find they are numerically stable. Similar to the collisions between RH compactons, the CSS compactons re-emerge with same coherent shape when scattered. The time evolution of the small-amplitude ripple resulting after scattering depends on the values of the parameters l and p characterizing the corresponding CSS equation. The simulation of the CSS compacton scattering requires a much smaller artificial viscosity to obtain numerical stability than in the case of RH compacton propagation.
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Affiliation(s)
- Bogdan Mihaila
- Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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