P. G. Kevrekidis

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

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Publications (503)930.68 Total impact

  • [Show abstract] [Hide abstract]
    ABSTRACT: The dynamics of vortex ring pairs in the homogeneous nonlinear Schr\"odinger equation is studied. The generation of numerically-exact solutions of traveling vortex rings is described and their translational velocity compared to revised analytic approximations. The scattering behavior of co-axial vortex rings with opposite charge undergoing collision is numerically investigated for different scattering angles yielding a surprisingly simple result for its dependence as a function of the initial vortex ring parameters. We also study the leapfrogging behavior of co-axial rings with equal charge and compare it with the dynamics stemming from a modified version of the reduced equations of motion from a classical fluid model derived using the Biot-Savart law.
    Physics of Fluids 09/2014; 26:097101. · 1.94 Impact Factor
  • F. Li, C. Chong, J. Yang, P. G. Kevrekidis, C. Daraio
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    ABSTRACT: We present a dynamically tunable mechanism of wave transmission in 1D helicoidal phononic crystals in a shape similar to DNA structures. These helicoidal architectures allow slanted nonlinear contact among cylin- drical constituents, and the relative torsional movements can dynamically tune the contact stiffness between neighboring cylinders. This results in cross-talking between in-plane torsional and out-of-plane longitudinal waves. We numerically demonstrate their versatile wave mixing and controllable dispersion behavior in both wavenumber and frequency domains. Based on this principle, a suggestion towards an acoustic configuration bearing parallels to a transistor is further proposed, in which longitudinal waves can be switched on/off through torsional waves.
    08/2014;
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    J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena
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    ABSTRACT: In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross-Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-continuum limit of vanishing coupling). Numerous unexpected features are identified including a staggered solitary pattern emerging from a single site excitation, as well as two- and three-site excitations playing a role analogous to one- and two-site, respectively, excitations of the discrete nonlinear Schr\"odinger analogue of the model. Stability exchanges between the two- and three-site states are identified, as well as instabilities that appear to be persistent over the coupling strength $\epsilon$, for a subcritical value of the propagation constant $\Lambda$. Variations of the propagation constant, coupling parameter and nonlinearity exponent are all examined in terms of their existence and stability implications and long dynamical simulations are used to unravel the evolutionary phenomenology of the system (when unstable).
    08/2014;
  • Anna M. Barry, F. Hajir, P. G. Kevrekidis
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    ABSTRACT: In this work, we construct suitable generating functions for vortices of alternating signs in the realm of Bose-Einstein condensates. In addition to the vortex-vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend them to larger numbers of vortices. Moreover, particular moment conditions can be derived e.g. about the sum of the squared locations of the vortices for arbitrary vortex numbers. Furthermore, the relevant differential equation can be generalized appropriately in the two-dimensional complex plane and allows the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a vortex at the center that are again connected to earlier bibliography.
    07/2014;
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    ABSTRACT: The dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component rubidium Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analyzing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.
    07/2014;
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    ABSTRACT: We study a two-component nonlinear Schr\"odinger system with repulsive nonlinear interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in the one-component. Treating it as a "frozen" one, we explore the possibility of the formation of bright solitonic bound states in the other component. We identify bifurcation points of such states in the linear limit for the bright component, and explore their continuation in the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability not only of the single-peak ground state (the dark-bright solitary wave), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability manifestation.
    07/2014;
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    ABSTRACT: In this work, we consider the dynamics of vector rogue waves and dark-bright solitons in two-component nonlinear Schr\"odinger equations with various physically motivated time-dependent nonlinearity coefficients, as well as spatio-temporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeron-like soliton solutions of the latter are converted back into ones of the original non-autonomous model. Using direct numerical simulations we find that, in most cases, the rogue wave formation is rapidly followed by a modulational instability that leads to the emergence of an expanding soliton train. Scenarios different than this generic phenomenology are also reported.
    07/2014;
  • Jennie D'Ambroise, Stefano Lepri, Boris A. Malomed, Panayotis G. Kevrekidis
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    ABSTRACT: We consider a $\mathcal{PT}$-symmetric chain (ladder-shaped) system governed by the discrete nonlinear Schr\"{o}dinger equation where the cubic nonlinearity is carried solely by two central "rungs" of the ladder. Two branches of scattering solutions for incident plane waves are found. We systematically construct these solutions, analyze their stability, and discuss non-reciprocity of the transmission associated with them. To relate these results to finite-size wavepacket dynamics, we also perform direct simulations of the evolution of the wavepackets, which confirm that the transmission is indeed asymmetric in this nonlinear system with the mutually balanced gain and loss.
    07/2014;
  • N. Lu, J. Cuevas-Maraver, P. G. Kevrekidis
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    ABSTRACT: In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a $\mathcal{P T}$-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a $\mathcal{P T}$- symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that $\mathcal{P T}$-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink-antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their "delicate" existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.
    06/2014;
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    ABSTRACT: We study conditions under which vortices in a highly oblate harmonically trapped Bose-Einstein condensate (BEC) can be stabilized due to pinning by a blue-detuned Gaussian laser beam, with particular emphasis on the potentially destabilizing effects of laser beam positioning within the BEC. Our approach involves theoretical and numerical exploration of dynamically and energetically stable pinning of vortices with winding number up to S=6, in correspondence with experimental observations. Stable pinning is quantified theoretically via Bogoliubov-de Gennes excitation spectrum computations and confirmed via direct numerical simulations for a range of conditions similar to those of experimental observations. The theoretical and numerical results indicate that the pinned winding number, or equivalently the winding number of the superfluid current about the laser beam, decays as a laser beam of fixed intensity moves away from the BEC center. Our theoretical analysis helps explain previous experimental observations and helps define limits of stable vortex pinning for future experiments involving vortex manipulation by laser beams.
    Physical Review A 05/2014; 89:053606. · 3.04 Impact Factor
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    ABSTRACT: We consider the existence, stability and dynamics of the nodeless state and fundamental nonlinear excitations, such as vortices, for a quasi-two-dimensional polariton condensate in the presence of pumping and nonlinear damping. We find a series of interesting features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates (BECs). For sizeable parameter ranges, in line with earlier findings, the nodeless state becomes unstable towards the formation of stable nonlinear single or multi-vortex excitations. The potential instability of the single vortex is also examined and is found to possess similar characteristics to those of the nodeless cloud. We also report that, contrary to what is known, e.g., for the atomic BEC case, stable stationary gray ring solitons (that can be thought of as radial forms of Nozaki-Bekki holes) can be found for polariton condensates in suitable parametric regimes. In other regimes, however, these may also suffer symmetry-breaking instabilities. The dynamical, pattern-forming implications of the above instabilities are explored through direct numerical simulations and, in turn, give rise to waveforms with triangular or quadrupolar symmetry.
    Journal of Physics Condensed Matter 04/2014; 26(15):155801. · 2.22 Impact Factor
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    V Koukouloyannis, G Voyatzis, P G Kevrekidis
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    ABSTRACT: In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the energy (which is expressed through the Hamiltonian H) and the angular momentum L of the system. By using the integral of the angular momentum, we reduce the system to a 2-degrees-of-freedom one with L as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.
    Physical Review E 04/2014; 89(4-1):042905. · 2.31 Impact Factor
  • V. Koukouloyannis, G. Voyatzis, P. G. Kevrekidis
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    ABSTRACT: In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the energy (which is expressed through the Hamiltonian H) and the angular momentum L of the system. By using the integral of the angular momentum, we reduce the system to a 2-degrees-of-freedom one with L as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.
    03/2014; 89(4).
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    ABSTRACT: In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine-Gordon equation and the non-integrable $\phi^4$ model. We focus, in particular, on two of their prototypical solutions, namely the kink-like heteroclinic connections and the time-periodic, exponentially localized in space breather waveforms. Two limits of the discrete variants of these models are contrasted: on the one side, the analytically tractable original continuum limit, and on the opposite end, the highly discrete, so-called anti-continuum limit of vanishing coupling. Numerical computations are used to bridge these two limits, as regards the existence, stability and dynamical properties of the waves. Finally, a recent variant of this theme is presented in the form of $\mathcal{PT}$-symmetric Klein-Gordon field theories and a number of relevant results are touched upon.
    03/2014;
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    ABSTRACT: In the present work, we revisit the so-called regularized short pulse equation (RSPE) and, in particular, explore the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First, using a fixed point iteration scheme, we numerically integrate the equation to find solitary waves. It is found that these solutions are well approximated by a truncated series of hyperbolic secants. The dependence of the soliton's parameters (height, width, etc) to the parameters of the equation is also investigated. Second, by developing a multiple scale reduction of the RSPE to the nonlinear Schr\"odinger equation, we are able to construct (both standing and traveling) envelope wave breather type solutions of the former, based on the solitary wave structures of the latter. Both the regular and the breathing traveling wave solutions identified are found to be robust and should thus be amenable to observations in the form of few optical cycle pulses.
    Journal of Physics A Mathematical and Theoretical 03/2014; 47(31). · 1.77 Impact Factor
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    V. Achilleos, D. J. Frantzeskakis, P. G. Kevrekidis
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    ABSTRACT: We present families of beating dark-dark solitons in spin-orbit (SO) coupled Bose-Einstein condensates. These families consist of solitons residing simultaneously in the two bands of the energy spectrum. The soliton components are characterized by two different spatial and temporal scales, which are identified by a multiscale expansion method. The solitons are "beating" ones, as they perform density oscillations with a characteristic frequency, relevant to Zitterbewegung (ZB). We find that spin oscillations may occur, depending on the parity of each soliton branch, which consequently lead to ZB oscillations of the beating dark solitons. Analytical results are corroborated by numerical simulations, illustrating the robustness of the solitons.
    03/2014; 89(3).
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    ABSTRACT: By applying an out-of-phase actuation at the boundaries of a uniform chain of granular particles, we demonstrate experimentally that time-periodic and spatially localized structures with a nonzero background (so-called dark breathers) emerge for a wide range of parameter values and initial conditions. We demonstrate a remarkable control over the number of breathers within the multibreather pattern that can be "dialed in" by varying the frequency or amplitude of the actuation. The values of the frequency (or amplitude) where the transition between different multibreather states occurs are predicted accurately by the proposed theoretical model, which is numerically shown to support exact dark breather and multibreather solutions. Moreover, we visualize detailed temporal and spatial profiles of breathers and, especially, of multibreathers using a full-field probing technology and enable a systematic favorable comparison among theory, computation, and experiments. A detailed bifurcation analysis reveals that the dark and multibreather families are connected in a "snaking" pattern, providing a roadmap for the identification of such fundamental states and their bistability in the laboratory.
    Physical Review E 03/2014; 89(3-1):032924. · 2.31 Impact Factor
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    ABSTRACT: In the present work we consider the introduction of PT-symmetric terms in the context of classical Klein-Gordon field theories. We explore the implication of such terms on the spectral stability of coherent structures, namely kinks. We find that the conclusion critically depends on the location of the kink center relative to the center of the PT-symmetric term. The main result is that if these two points coincide, the kink's spectrum remains on the imaginary axis and the wave is spectrally stable. If the kink is centered on the "lossy side" of the medium, then it becomes stabilized. On the other hand, if it becomes centered on the "gain side" of the medium, then it is destabilized. The consequences of these two possibilities on the linearization (point and essential) spectrum are discussed in some detail.
    Studies in Applied Mathematics 02/2014; · 1.31 Impact Factor
  • D. Yan, F. Tsitoura, P. G. Kevrekidis, D. J. Frantzeskakis
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    ABSTRACT: In the present contribution, we explore a host of different stationary states, namely dark-bright solitons and their lattices, that arise in the context of multi-component atomic Bose-Einstein condensates. The latter, are modeled by systems of coupled Gross-Pitaevskii equations with general interaction (nonlinearity) coefficients $g_{ij}$. It is found that in some particular parameter ranges such solutions can be obtained in analytical form, however, numerically they are computed as existing in a far wider parametric range. Many features of the solutions under study, such as their analytical form without the trap or the stability/dynamical properties of one dark-bright soliton even in the presence of the trap are obtained analytically and corroborated numerically. Additional features, such as the stability of soliton lattice homogeneous states or their existence/stability in the presence of the trap, are examined numerically.
    02/2014;
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    ABSTRACT: Motivated by the recent successes of particle models in capturing the precession and interactions of vortex structures in quasi-two-dimensional Bose-Einstein condensates, we revisit the relevant systems of ordinary differential equations. We consider the number of vortices $N$ as a parameter and explore the prototypical configurations (``ground states'') that arise in the case of few or many vortices. In the case of few vortices, we modify the classical result of Havelock [Phil. Mag. {\bf 11}, 617 (1931)] illustrating that vortex polygons in the form of a ring are unstable for $N \geq7$. Additionally, we reconcile this modification with the recent identification of symmetry breaking bifurcations for the cases of $N=2,\dots,5$. We also briefly discuss the case of a ring of vortices surrounding a central vortex (so-called $N+1$ configuration). We finally examine the opposite limit of large $N$ and illustrate how a coarse-graining, continuum approach enables the accurate identification of the radial distribution of vortices in that limit.
    Proceedings. Mathematical, physical, and engineering sciences / the Royal Society. 01/2014; 470(2168).

Publication Stats

6k Citations
930.68 Total Impact Points

Institutions

  • 2001–2014
    • University of Massachusetts Amherst
      • Department of Mathematics and Statistics
      Amherst Center, Massachusetts, United States
  • 2013
    • University of Hartford
      • Department of Mathematics
      West Hartford, CT, United States
    • Technological Educational Institute of Lamia
      Lamia, Central Greece, Greece
    • University of Hamburg
      Hamburg, Hamburg, Germany
  • 2011–2013
    • Dickinson College
      • Department of Physics and Astronomy
      Carlisle, Pennsylvania, United States
    • National and Kapodistrian University of Athens
      Athínai, Attica, Greece
  • 2005–2013
    • San Diego State University
      • Department of Mathematics and Statistics
      San Diego, CA, United States
    • Columbia University
      • Department of Applied Physics and Applied Mathematics
      New York City, NY, United States
    • Georgia Institute of Technology
      • School of Physics
      Atlanta, GA, United States
    • University of New Mexico
      • Department of Mathematics & Statistics
      Albuquerque, NM, United States
    • University of Vermont
      • Department of Mathematics and Statistics
      Burlington, VT, United States
  • 2001–2013
    • Tel Aviv University
      • Faculty of Engineering
      Tell Afif, Tel Aviv, Israel
  • 2006–2012
    • California Institute of Technology
      • Department of Electrical Engineering
      Pasadena, California, United States
    • San Francisco State University
      • Department of Physics and Astronomy
      San Francisco, CA, United States
    • University of Bristol
      • Department of Engineering Mathematics
      Bristol, ENG, United Kingdom
  • 2001–2011
    • Universidad de Sevilla
      • Departamento de Física Aplicada I
      Sevilla, Andalusia, Spain
  • 2010
    • Morehouse College
      Atlanta, Georgia, United States
    • Newcastle University
      • School of Mathematics and Statistics
      Newcastle upon Tyne, ENG, United Kingdom
  • 2006–2010
    • Nankai University
      • Applied Physics School (APS)
      Tianjin, Tianjin Shi, China
  • 2009
    • National Technical University of Athens
      • Σχολή Εφαρμοσμένων Μαθηματικών & Φυσικών Επιστημών
      Athens, Attiki, Greece
    • University of Illinois, Urbana-Champaign
      • Department of Mathematics
      Urbana, IL, United States
    • Aristotle University of Thessaloniki
      • Department of Mathematical, Physical and Computational Sciences
      Thessaloníki, Kentriki Makedonia, Greece
  • 2008
    • Universität Heidelberg
      • Kirchhoff-Institute of Physics
      Heidelberg, Baden-Wuerttemberg, Germany
    • University of Oxford
      • Oxford Centre for Industrial and Applied Mathematics
      Oxford, ENG, United Kingdom
    • University of Tuebingen
      Tübingen, Baden-Württemberg, Germany
    • Heidelberg University
      Tiffin, Ohio, United States
    • São Paulo State University
      • Instituto de Física Teórica
      Bauru, Estado de Sao Paulo, Brazil
    • Western New England College
      Springfield, Missouri, United States
    • Calvin College
      • Department of Mathematics and Statistics
      Grand Rapids, MI, United States
    • University of Kansas
      • Department of Mathematics
      Lawrence, KS, United States
  • 2006–2008
    • Florida State University
      • Department of Mathematics
      Tallahassee, FL, United States
  • 2007
    • Amherst College
      • Department of Physics
      Amherst Center, Massachusetts, United States
    • University of Crete
      Retimo, Crete, Greece
    • Northwestern State University
      Louisiana, United States
  • 2004–2007
    • Escola Universitària Politècnica
      Hispalis, Andalusia, Spain
  • 2003–2007
    • The University of Tokyo
      • Institute of Industrial Science
      Tokyo, Tokyo-to, Japan
  • 2005–2006
    • McMaster University
      • Department of Mathematics and Statistics
      Hamilton, Ontario, Canada
  • 2001–2006
    • Princeton University
      • • Department of Chemical and Biological Engineering
      • • Program in Applied and Computational Mathematics
      Princeton, New Jersey, United States
  • 2002
    • Osaka Institute of Technology
      Ōsaka, Ōsaka, Japan
  • 2001–2002
    • Los Alamos National Laboratory
      • Center for Nonlinear Studies
      Los Alamos, NM, United States
  • 1999–2001
    • Rutgers, The State University of New Jersey
      • Department Physics and Astronomy
      New Brunswick, NJ, United States
  • 2000
    • Brown University
      • Department of Applied Mathematics
      Providence, Rhode Island, United States