P. G. Kevrekidis

University of Massachusetts Amherst, Amherst Center, Massachusetts, United States

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Publications (592)1122.25 Total impact

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    ABSTRACT: In this work we study spherical shell dark soliton states in three-dimensional atomic Bose-Einstein condensates. Their symmetry is exploited in order to analyze their existence, as well as that of topologically charged variants of the structures, and, importantly, to identify their linear stability Bogolyubov-de Gennes spectrum. We compare our effective 1D spherical and 2D cylindrical computations with the full 3D numerics. An important conclusion is that such spherical shell solitons can be stable sufficiently close to the linear limit of the isotropic condensates considered herein. We have also identified their instabilities leading to the emergence of vortex line and vortex ring cages. In addition, we generalize effective particle pictures of lower dimensional dark solitons and ring dark solitons to the spherical shell solitons concerning their equilibrium radius and effective dynamics around it. In this case too, we favorably compare the resulting predictions such as the shell equilibrium radius, qualitatively and quantitatively, with full numerical solutions in 3D.
    No preview · Article · Jan 2016
  • I K Mylonas · V M Rothos · P G Kevrekidis · D J Frantzeskakis
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    ABSTRACT: We develop a direct perturbation theory for dark-bright solitons and derive evolution equations for the soliton parameters. In particular, first the linearization equation around the solitons is solved by expanding its solution into a set of complete eigenfunctions of the linearization operator. Then, suppression of secular growth in the linearized solution leads to the evolution equations of soliton parameters. The results are applied to a number of case examples motivated by the physics of atomic Bose-Einstein condensates, where dark-bright solitons have recently been studied both in theory and in experiments. We thus consider perturbations corresponding to (a) finite temperature-induced thermal losses, and (b) the presence of localized (delta-function) impurities. In these cases, relevant equations of motion for the dark-bright soliton center are in agreement with ones previously obtained via alternative methods, including energy-based methods, as well as numerical linear stability analysis and direct simulations.
    No preview · Article · Jan 2016 · Journal of Physics A Mathematical and Theoretical
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    ABSTRACT: We consider a two-component, two-dimensional nonlinear Schr\"odinger system with unequal dispersion coefficients and self-defocusing nonlinearities. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component. We show that the potential well may support not only the fundamental bound state, which forms a vortex-bright (VB) soliton, but also multi-ring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution in the case examples considered leads to transformation of the multi-ring complexes into stable VB soliton ones with the fundamental state in the second component.
    Full-text · Article · Dec 2015
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    P. G. Kevrekidis · D. J. Frantzeskakis
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    ABSTRACT: In this review we try to capture some of the recent excitement induced by experimental developments, but also by a large volume of theoretical and computational studies addressing multi-component nonlinear Schrodinger models and the localized structures that they support. We focus on some prototypical structures, namely the dark-bright and dark-dark solitons. Although our focus will be on one-dimensional, two-component Hamiltonian models, we also discuss variants, including three (or more)-component models, higher-dimensional states, as well as dissipative settings. We also offer an outlook on interesting possibilities for future work on this theme.
    Preview · Article · Dec 2015
  • J. Cuevas-Maraver · P. G. Kevrekidis · A. Saxena · A. Comech · R. Lan
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    ABSTRACT: We explore a prototypical two-dimensional model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions consisting of a soliton in one component and a vortex in the other to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These results pave the way for a systematic stability and dynamics analysis of higher dimensional waveforms in a broad class of nonlinear Dirac models and a comparison revealing nontrivial differences with respect to their better understood non-relativistic analogue, the nonlinear Schr\"odinger equation.
    No preview · Article · Dec 2015
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    ABSTRACT: Performing a systematic Bogoliubov-de Gennes spectral analysis, we illustrate that stationary vortex lines, vortex rings, and more exotic states, such as hopfions, are robust in three-dimensional atomic Bose-Einstein condensates, for large parameter intervals. Importantly, we find that the hopfion can be stabilized in a simple parabolic trap, without the need for trap rotation or inhomogeneous interactions. We supplement our spectral analysis by studying the dynamics of such stationary states; we find them to be robust against significant perturbations of the initial state. In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode, but we also observe the corresponding instability dynamics. Furthermore, deep in the Thomas-Fermi regime, we investigate the particlelike behavior of vortex rings and hopfions.
    No preview · Article · Dec 2015 · Physical Review A
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    Full-text · Article · Dec 2015 · Physica D Nonlinear Phenomena
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    Haitao Xu · Panayotis G. Kevrekidis · Dmitry E. Pelinovsky
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    ABSTRACT: Vortices symmetric with respect to simultaneous parity and time reversing transformations are considered on the square lattice in the framework of the discrete nonlinear Schr\"{o}dinger equation. The existence and stability of vortex configurations is analyzed in the limit of weak coupling between the lattice sites, when predictions on the elementary cell of a square lattice (i.e., a single square) can be extended to a large (yet finite) array of lattice cells. Our analytical predictions are found to be in good agreement with numerical computations.
    Full-text · Article · Nov 2015
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    Mason A. Porter · Panayotis G. Kevrekidis · Chiara Daraio
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    ABSTRACT: The freedom to choose the size, stiffness, and spatial distribution of macroscopic particles in a lattice makes granular crystals easily tailored building blocks for shock-absorbing materials, sound-focusing devices, acoustic switches, and other exotica.
    Preview · Article · Nov 2015 · Physics Today
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    ABSTRACT: The study of the dynamics in granular systems has witnessed much attention from diverse research fields such as condensed matter physics, geophysics, and material science. In particular, such systems are at the focal of intense research in acoustic and mechanical metamaterial community due to the ease of their tailorability from the dynamic standpoint. Moreover, the inherent nonlinearity, discreteness, and periodicity of such systems can be exploited to enable a variety of engineering devices (e.g., shock and energy absorbing layers, acoustic diodes and switches, and sensing and actuation devices) with novel physical characteristics. We study one-dimensional wave propagation in granular periodic dimer chains with no pre-compression. Here, the dimer system refers to a chain consisting of alternating ‘heavy’ and ‘light’ masses of beads, or alternating ‘stiff’ and ‘soft’ contacts. The beads interact according to the Hertzian contact law, which makes the system nonlinear. However, in the absence of pre-compression, the system becomes even more nonlinear, i.e., ‘strongly nonlinear’ as the beads can lose contact with each other. Nesterenko described it as ‘Sonic Vacuum’ which refers to no propagation of sound wave in the medium. In this presentation, we discuss that the system displays interesting dynamics at different dimer mass and stiffness ratios under impact loading. In particular, anti-resonance of dimer beads results in wave localization, i.e., solitary wave formation, whereas the resonance phenomenon leads to strong wave dispersion and attenuation. We first predict the existence of such resonance and anti-resonance mechanisms at different mass and stiffness ratios by numerical simulations. We then experimentally verify the existence of the same using laser Doppler vibrometry. We focus our study on investigating the resonance mechanism which offers a unique way to attenuate impact energy without solely relying on system damping. In such scenario, we report two important characteristics of the system. First, the primary pulse of the propagating wave transfers its energy to the near-field tailing pulse in the form of higher frequency oscillations. This low to high frequency (LF-HF) scattering is of immense significance as it facilitates wave attenuation even more effectively in the presence of system damping. We envision novel impact attenuation systems which would make use of this adaptive capacity of the granular dimer for nonlinear scattering and redistribution of energy. The second characteristic of the system is that the primary pulse energy relocates itself to a smaller length scale in the near-field, and to a wide range of length scales in the far-field tailing pulse. Thus, the presence of such energy cascading across various length-scales hints at mechanical turbulence in the system, which has not been reported in granular systems so far. Overall, these unique wave propagation mechanisms in both temporal and spatial domains (i.e., LF-HF scattering and turbulence-like cascading) can be highly useful in manipulating stress waves for impact mitigation purposes. Thus, we envision that the findings in this study can open new avenues to designing and fabricating a new type of impact mitigating and wave filtering devices for engineering applications.
    No preview · Conference Paper · Nov 2015
  • C. Chong · E. Kim · E. G. Charalampidis · H. Kim · F. Li · P. G. Kevrekidis · J. Lydon · C. Daraio · J. Yang
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    ABSTRACT: This article explores the excitation of different vibrational states in a spatially extended dynamical system through theory and experiment. As a prototypical example, we consider a one-dimensional packing of spherical particles (a so-called granular chain) that is subject to harmonic boundary excitation. The combination of the multi-modal nature of the system and the strong coupling between the particles due to the nonlinear Hertzian contact force leads to broad regions in frequency where different vibrational states are possible. In certain parametric regions, we demonstrate that the Nonlinear Schrodinger (NLS) equation predicts the corresponding modes fairly well. We propose that nonlinear multi-modal systems can be useful in vibration energy harvest- ing and discuss a prototypical framework for its realization. The electromechanical model we derive predicts accurately the conversion from mechanical to electrical energy observed in the experiments.
    No preview · Article · Oct 2015
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    ABSTRACT: Performing a systematic Bogoliubov-de Gennes spectral analysis, we illustrate that stationary vortex lines, vortex rings and more exotic states, such as hopfions, are robust in three-dimensional atomic Bose-Einstein condensates, for large parameter intervals. Importantly, we find that the hopfion can be stabilized in a simple parabolic trap, without the need for trap rotation or inhomogeneous interactions. We supplement our spectral analysis by studying the dynamics of such stationary states; we find them to be robust against significant perturbations of the initial state. In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode, but we also observe the corresponding instability dynamics. Furthermore, deep in the Thomas-Fermi regime, we investigate the particle-like behavior of vortex rings and hopfions.
    No preview · Article · Oct 2015
  • C. Chong · P. G. Kevrekidis · M. J. Ablowitz · Yi-Ping Ma
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    ABSTRACT: Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wavepacket, as well as via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a non-oscillatory nature, resulting from the complex interplay between the discreteness, nonlinearity and geometry of the packing. The transition between these two types of propagation is explored.
    No preview · Article · Oct 2015
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    Panayotis G. Kevrekidis · Jesús Cuevas–Maraver · Avadh Saxena · Fred Cooper · Avinash Khare
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    ABSTRACT: In the present work, we combine the notion of parity-time (PT) symmetry with that of supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the so-called Pöschl-Teller potential which is real. Not only are we able to identify and numerically confirm the eigenvalues of the relevant problem, but we also show that the corresponding nonlinear problem, in the presence of an arbitrary power-law nonlinearity, has an exact bright soliton solution that can be analytically identified and has intriguing stability properties, such as an oscillatory instability, which is absent for the corresponding solution of the regular nonlinear Schrödinger equation with arbitrary power-law nonlinearity. The spectral properties and dynamical implications of this instability are examined. We believe that these findings may pave the way toward initiating a fruitful interplay between the notions of PT symmetry, supersymmetric partner potentials, and nonlinear interactions.
    Full-text · Article · Oct 2015 · Physical Review E
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    ABSTRACT: We study dark solitons near potential and nonlinearity steps and combinations thereof, forming rectangular barriers. This setting is relevant to the contexts of atomic Bose-Einstein condensates (where such steps can be realized by using proper external fields) and nonlinear optics (for beam propagation near interfaces separating optical media of different refractive indices). We use perturbation theory to develop an equivalent particle theory, describing the matter-wave or optical soliton dynamics as the motion of a particle in an effective potential. This Newtonian dynamical problem provides information for the soliton statics and dynamics, including scenarios of reflection, transmission, or quasi-trapping at such steps. The case of multiple such steps and its connection to barrier potentials is also touched upon. Our analytical predictions are found to be in very good agreement with the corresponding numerical results.
    No preview · Article · Sep 2015
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    DESCRIPTION: The dynamical behavior of a higher-order nonlinear Schr\"{o}dinger equation is found to include a very wide range of scenarios due to the interplay of higher-order physically relevant terms. The dynamics extends from Poincar\'{e}-Bendixson--type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. Suitable low-dimensional phase space diagnostics are developed and are used to illustrate the different possibilities and to identify their respective parametric intervals.
    Full-text · Research · Sep 2015
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    ABSTRACT: The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a very wide range of scenarios due to the interplay of higher-order physically relevant terms. The dynamics extend from Poincar\'e-Bendixson--type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. Suitable low-dimensional phase space diagnostics are developed and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.
    No preview · Article · Sep 2015
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    ABSTRACT: For a dissipative variant of the two-dimensional Gross-Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas-Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one-dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. We show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations.
    Preview · Article · Sep 2015
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    F. Palmero · J. Han · L. Q. English · T. J. Alexander · P. G. Kevrekidis
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    ABSTRACT: We consider a chain of torsionally-coupled, planar pendula shaken horizontally by an external sinusoidal driver. It has been known that in such a system, theoretically modeled by the discrete sine-Gordon equation, intrinsic localized modes, also known as discrete breathers, can exist. Recently, the existence of multifrequency breathers via subharmonic driving has been theoretically proposed and numerically illustrated by Xu {\em et al.} in Phys. Rev. E {\bf 90}, 042921 (2014). In this paper, we verify this prediction experimentally. Comparison of the experimental results to numerical simulations with realistic system parameters (including a Floquet stability analysis), and wherever possible to analytical results (e.g. for the subharmonic response of the single driven-damped pendulum), yields good agreement. Finally, we report on period-1 and multifrequency edge breathers which are localized at the open boundaries of the chain, for which we have again found good agreement between experiments and numerical computations
    Full-text · Article · Aug 2015 · Physics Letters A
  • J. Rossi · R. Carretero-Gonzalez · P. G. Kevrekidis
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    ABSTRACT: Recently, Galley [Phys. Rev. Lett. {\bf 110}, 174301 (2013)] proposed an initial value problem formulation of Hamilton's principle applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrodinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems; namely, the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein. We showcase the relevance of the NCVA method by exploring test case examples within the NLS setting including combinations of linear and density dependent loss and gain. We also present an example applied to exciton polariton condensates that intrinsically feature loss and a spatially dependent gain term.
    No preview · Article · Aug 2015

Publication Stats

10k Citations
1,122.25 Total Impact Points

Institutions

  • 2001-2016
    • University of Massachusetts Amherst
      • Department of Mathematics and Statistics
      Amherst Center, Massachusetts, United States
  • 2000-2015
    • Los Alamos National Laboratory
      • • Center for Nonlinear Studies
      • • Theoretical Division
      Лос-Аламос, California, United States
  • 2014
    • Aristotle University of Thessaloniki
      • School of Civil Engineering
      Saloníki, Central Macedonia, Greece
    • San Diego State University
      • Department of Mathematics and Statistics
      San Diego, California, United States
  • 2013
    • Tel Aviv University
      Tell Afif, Tel Aviv, Israel
  • 2011
    • Universität Heidelberg
      • Kirchhoff-Institute of Physics
      Heidelburg, Baden-Württemberg, Germany
  • 2007
    • University of Crete
      Retimo, Crete, Greece
    • University of Kansas
      • Department of Mathematics
      Lawrence, Kansas, United States
    • Australian National University
      • Nonlinear Physics Centre
      Canberra, Australian Capital Territory, Australia
  • 2006
    • Nankai University
      T’ien-ching-shih, Tianjin Shi, China
  • 2004
    • The University of Tokyo
      • Institute of Industrial Science
      Tokyo, Tokyo-to, Japan
  • 2002
    • University of Granada
      Granata, Andalusia, Spain
  • 2001-2002
    • Princeton University
      • • Department of Chemical and Biological Engineering
      • • Program in Applied and Computational Mathematics
      Princeton, New Jersey, United States
  • 1999-2001
    • Rutgers, The State University of New Jersey
      • Department Physics and Astronomy
      Newark, NJ, United States