P. G. Kevrekidis

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

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Publications (522)961.06 Total impact

  • [Show abstract] [Hide abstract]
    ABSTRACT: We consider a binary repulsive Bose-Einstein condensate in a harmonic trap in one spatial dimension and investigate particular solutions consisting of two dark-bright (DB) solitons. There are two different stationary solutions characterized by the phase difference in the bright component, in-phase and out-of-phase states. We show that above a critical particle number in the bright component, a symmetry breaking bifurcation of the pitchfork type occurs that leads to a new asymmetric solution whereas the parental branch, i.e., the out-of-phase state becomes unstable. These three different states support different small amplitude oscillations, characterized by an almost stationary density of the dark component and a tunneling of the bright component between the two dark solitons. Within a suitable effective double-well picture, these can be understood as the characteristic features of a Bosonic Josephson Junction (BJJ), and we show within a two-mode approach that all characteristic features of the BJJ phase space are recovered. For larger deviations from the stationary states, the simplifying double-well description breaks down due to the feedback of the bright component onto the dark one, causing the solitons to move. In this regime we observe intricate anharmonic and aperiodic dynamics, exhibiting remnants of the BJJ phase space.
    11/2014;
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    ABSTRACT: In the continuum a skyrmion is a topological nontrivial map between Riemannian manifolds, and a stationary point of a particular energy functional. This paper describes lattice analogues of the aforementioned skyrmions, namely a natural way of using the topological properties of the three-dimensional continuum Skyrme model to achieve topological stability on the lattice. In particular, using fixed point iterations, numerically exact lattice skyrmions are constructed; and their stability under small perturbations is verified by means of linear stability analysis. While stable branches of such solutions are identified, it is also shown that they possess a particularly delicate bifurcation structure, especially so in the vicinity of the continuum limit. The corresponding bifurcation diagram is elucidated and a prescription for selecting the branch asymptoting to the well-known continuum limit is given. Finally, the robustness of the solutions by virtue of direct numerical simulations is corroborated.
    11/2014;
  • Source
    E. G. Charalampidis, F. Li, C. Chong, J. Yang, P. G. Kevrekidis
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    ABSTRACT: In this work, we consider time-periodic structures of trimer granular crystals consisting of alternate chrome steel and tungsten carbide spherical particles yielding a spatial periodicity of three. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system become chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures.
    11/2014;
  • Roy H. Goodman, Panayotis G. Kevrekidis, Ricardo Carretero-Gonzalez
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    ABSTRACT: We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ordinary differential equations describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system we are able to construct complex periodic orbits in the original, partial differential equation, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.
    10/2014;
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    ABSTRACT: In this work we consider the dynamics of vector rogue waves and dark-bright solitons in two-component nonlinear Schrödinger equations with various physically motivated time-dependent nonlinearity coefficients, as well as spatiotemporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeronlike soliton solutions of the latter are converted back into ones of the original nonautonomous 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.
    Physical review. E, Statistical, nonlinear, and soft matter physics. 10/2014; 90(4-1):042912.
  • J. Cuevas-Maraver, A. Khare, P. G. Kevrekidis, H. Xu, A. Saxena
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    ABSTRACT: In the present work, we explore the case of a general PT-symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schr\"odinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations.
    09/2014;
  • Jennie D'Ambroise, Panayotis G. Kevrekidis, Boris A. Malomed
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    ABSTRACT: We introduce a ladder-shaped chain with each rung carrying a $\mathcal{PT}$ -symmetric gain-loss dipole. The polarity of the dipoles is staggered along the chain, meaning that a rung bearing gain-loss is followed by one bearing loss-gain. This renders the system $\mathcal{PT}$-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schr\"{o}dinger (DNLS) equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anti-continuum limit of uncoupled rungs and using the Newton's method for identifying solutions and parametric continuation in the inter-rung coupling for following the associated branches, we construct families of $\mathcal{PT}$-symmetric discrete solitons and identify their stability regions. Waveforms stemming from a single excited rung, as well as ones from multiple rungs are identified. Dynamics of unstable solitons is presented too.
    09/2014;
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    ABSTRACT: In the present work, we introduce a new $\mathcal{PT}$-symmetric variant of the Klein-Gordon field theoretic problem. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we obtain an explicit frequency condition, somewhat reminiscent of the classical Vakhitov-Kolokolov criterion, which sharply separates the regimes of spectral stability and instability. Our numerical computations corroborate the relevant theoretical result.
    09/2014;
  • Y. Xu, T. J. Alexander, H. Sidhu, P. G. Kevrekidis
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    ABSTRACT: Inspired by the experimental results of Cuevas et al. (Physical Review Letters 102, 224101 (2009)), we consider theoretically the behavior of a chain of planar rigid pendulums suspended in a uniform gravitational field and subjected to a horizontal periodic driving force applied to the pendulum pivots. We characterize the motion of a single pendulum, finding bistability near the fundamental resonance, and near the period-3 subharmonic resonance. We examine the development of modulational instability in a driven pendulum chain and find both a critical chain length and critical frequency for the appearance of the instability. We study the breather solutions and show their connection to the single pendulum dynamics, and extend our analysis to consider multi-frequency breathers connected to the period-3 periodic solution, showing also the possibility of stability in these breather states. Finally we examine the problem of breather generation and demonstrate a robust scheme for generation of on-site and off-site breathers.
    Physical review. E, Statistical, nonlinear, and soft matter physics. 09/2014; 90(4-1).
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    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;
  • Source
    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;
  • Y Shen, P G Kevrekidis, S Sen, A Hoffman
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    ABSTRACT: Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both one-soliton and two-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both copropagating and counterpropagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.
    Physical Review E 08/2014; 90(2-1):022905. · 2.31 Impact Factor
  • Source
    J. D'Ambroise, S. Lepri, B.A. Malomed, P.G. Kevrekidis
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    ABSTRACT: We consider a PTPT-symmetric chain (ladder-shaped) system governed by the discrete nonlinear Schrödinger 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.
    Physics Letters A 08/2014; · 1.63 Impact Factor
  • 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;
  • Source
    [Show abstract] [Hide abstract]
    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;
  • Source
    Jennie D'Ambroise, Stefano Lepri, Boris A. Malomed, Panayotis G. Kevrekidis
    [Show abstract] [Hide abstract]
    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;

Publication Stats

6k Citations
961.06 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
  • 2012
    • University of the Aegean
      • Department of Mathematics
      Kastro, North Aegean, Greece
  • 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