P. G. Kevrekidis

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

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Publications (557)1080.58 Total impact

  • [Show abstract] [Hide abstract]
    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.
    ASME - IMECE, San Antonio, TX; 11/2015
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    ABSTRACT: We study nonlinear wave propagation in one-dimensional granular dimer chains with no pre-compression. Here we define a dimer system in a more general sense, i.e., with alternating ‘heavy’ and ‘light’ masses, or alternating ‘stiff’ and ‘soft’ contacts. The system shows interesting nonlinear wave dynamics at different dimer mass and stiffness ratios. In particular, the resonance phenomenon leads to strong wave attenuation, whereas the anti-resonance of diatomic particles ensures the formation of solitary waves in the system. In this research, 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. Lastly, we closely investigate the resonance phenomenon, which shows a unique wave attenuation mechanism without any damping present in the system. In such scenario, we find that the primary pulse of a propagating wave transfers its energy to the near-field tailing pulse in the form of unique frequency oscillations, which eventually lead to chaotic oscillations in the far-field trailing pulse.
    ASME - McMat, Seattle, WA; 07/2015
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    ABSTRACT: Earlier work has shown that ring dark solitons in two-dimensional Bose-Einstein condensate are generically unstable. In this work, we propose a way of stabilizing the ring dark soliton via a radial Gaussian external potential. We investigate the existence and stability of the ring dark soliton upon variations of the chemical potential and also of the strength of the radial potential. Numerical results show that the ring dark soliton can be stabilized in a suitable interval of external potential strengths and chemical potentials. We also explore different proposed particle pictures considering the ring as a moving particle and find, where appropriate, results in very good qualitative and also reasonable quantitative agreement with the numerical findings.
  • Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein
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    ABSTRACT: We explore a recently proposed locally resonant granular system bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. In this system, we propose the existence of two types of configurations: (a) small-amplitude periodic traveling waves and (b) dark-breather solutions, i.e., exponentially localized, time periodic states mounted on top of a non-vanishing background. We also identify conditions under which the system admits long-lived bright breather solutions. Our results are obtained by means of an asymptotic reduction to a suitably modified version of the so-called discrete p-Schr{\"o}dinger (DpS) equation, which is established as controllably approximating the solutions of the original system for large but finite times (under suitable assumptions on the solution amplitude and the resonator mass). The findings are also corroborated by detailed numerical computations. A remarkable feature distinguishing our results from other settings where dark breathers are observed is the complete absence of precompression in the system, i.e., the absence of a linear spectral band.
  • Danial Saadatmand, Sergey V. Dmitriev, Panayotis G. Kevrekidis
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    ABSTRACT: Solitons are very effective in transporting energy over great distances and collisions between them can produce high energy density spots of relevance to phase transformations, energy localization and defect formation among others. It is then important to study how energy density accumulation scales in multi-soliton collisions. In this study, we demonstrate that the maximal energy density that can be achieved in collision of $N$ slowly moving kinks and antikinks in the integrable sine-Gordon field, remarkably, is proportional to $N^2$, while the total energy of the system is proportional to $N$. This maximal energy density can be achieved only if the difference between the number of colliding kinks and antikinks is minimal, i.e., is equal to 0 for even $N$ and 1 for odd $N$ and if the pattern involves an alternating array of kinks and anti-kinks. Interestingly, for odd (even) $N$ the maximal energy density appears in the form of potential (kinetic) energy, while kinetic (potential) energy is equal to zero. The results of the present study rely on the analysis of the exact multi-soliton solutions for $N=1,2,$ and 3 and on the numerical simulation results for $N=4,5,6,$ and 7. Based on these results one can speculate that the soliton collisions in the sine-Gordon field can, in principle, controllably produce very high energy density. This can have important consequences for many physical phenomena described by the Klein-Gordon equations.
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    ABSTRACT: We study wave propagation in strongly nonlinear 1D diatomic granular crystals under an impact load. Depending on the mass ratio of the `light' to `heavy' beads, this system exhibits rich wave dynamics from highly localized traveling waves to highly dispersive waves featuring strong attenuation. We experimentally demonstrate the nonlinear resonant and anti-resonant interactions of particles and verify that the nonlinear resonance results in strong wave attenuation, leading to highly efficient nonlinear energy cascading without relying on material damping. In this process, mechanical energy is transferred from low to high frequencies, while propagating waves emerge in both ordered and chaotic waveforms via a distinctive spatial cascading. This energy transfer mechanism from lower to higher frequencies and wavenumbers is of particular significance towards the design of novel nonlinear acoustic metamaterials with inherently passive energy redistribution properties.
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    ABSTRACT: We examine conditions for finite-time collapse of the solutions of the defocusing higher-order nonlinear Schr\"odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering, this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.
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    ABSTRACT: We investigate the nonlinear wave dynamics of origami-based metamaterials composed of Tachi-Miura polyhedron (TMP) unit cells. These cells exhibit strain softening behavior under compression, which can be tuned by modifying their geometrical configurations or initial folded conditions. We assemble these TMP cells into a cluster of origami-based metamaterials, and we theoretically model and numerically analyze their wave transmission mechanism under external impact. Numerical simulations show that origami-based metamaterials can provide a prototypical platform for the formation of nonlinear coherent structures in the form of rarefaction waves, which feature a tensile wavefront upon the application of compression to the system. We also demonstrate the existence of numerically exact traveling rarefaction waves. Origami-based metamaterials can be highly useful for mitigating shock waves, potentially enabling a wide variety of engineering applications.
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    Jesús Cuevas-Maraver, Panayotis G. Kevrekidis, Dmitry E. Pelinovsky
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    ABSTRACT: In the present work, we explore the possibility of excited breather states in a nonlinear Klein--Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stable. The mechanism for this fundamentally nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the Klein--Gordon lattice with a soft (Morse) and a hard ($\phi^4$) potential. Compared to the case of the nonlinear Schr{\"o}dinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the excited breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, excited breather states are observed to be nonlinearly stable.
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    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 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.
    Physical Review A 04/2015; 91(5):043637. DOI:10.1103/PhysRevA.91.043637
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    Panayotis Kevrekidis, Vakhtang Putkaradze, Zoi Rapti
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    ABSTRACT: We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the "proper" continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known "standard" discretizations, even in the case of highly discrete systems when the coupling between the adjacent nodes is weak. Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important.
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    B. Gertjerenken, P. G. Kevrekidis
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    ABSTRACT: We numerically investigate the influence of interactions on the generalized Hong-Ou-Mandel (HOM) effect for bosonic particles and show results for the cases of $N=2$, $N=3$ and $N=4$ bosons interacting with a beam splitter, whose role is played by a $\delta$-barrier. In particular, we focus on the effect of attractive interactions and compare the results with the repulsive case, as well as with the analytically available results for the non-interacting case (that we use as a benchmark). We observe a fermionization effect both for growing repulsive and attractive interactions, i.e., the dip in the HOM coincidence count is progressively smeared out, for increasing interaction strengths. The role of input asymmetries is also explored.
    Physics Letters A 03/2015; 379(30-31). DOI:10.1016/j.physleta.2015.04.001
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    ABSTRACT: We introduce a ladder-shaped chain with each rung carrying a parity-time- (PT-) symmetric gain-loss dimer. The polarity of the dimers is staggered along the chain, meaning alternation of gain-loss and loss-gain rungs. This structure, which can be implemented as an optical waveguide array, is the simplest one which renders the system PT-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schrödinger equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anticontinuum limit of uncoupled rungs and using the Newton's method for continuation of the solutions with the increase of the inter-rung coupling, we construct families of PT-symmetric discrete solitons and identify their stability regions. Waveforms stemming from a single excited rung and double ones are identified. Dynamics of unstable solitons is investigated too.
    Physical Review E 03/2015; 91(3-1):033207.
  • Wenlong Wang, P G Kevrekidis
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    ABSTRACT: We have performed a systematic study quantifying the variation of solitary wave behavior from that of an ordered cloud resembling a "crystalline" configuration to that of a disordered state that can be characterized as a soliton "gas." As our illustrative examples, we use both one-component, as well as two-component, one-dimensional atomic gases very close to zero temperature, where in the presence of repulsive interatomic interactions and of a parabolic trap, a cloud of dark (dark-bright) solitons can form in the one- (two-) component system. We corroborate our findings through three distinct types of approaches, namely a Gross-Pitaevskii type of partial differential equation, particle-based ordinary differential equations describing the soliton dynamical system, and Monte Carlo simulations for the particle system. We define an "empirical" order parameter to characterize the order of the soliton lattices and study how this changes as a function of the strength of the "thermally" (i.e., kinetically) induced perturbations. As may be anticipated by the one-dimensional nature of our system, the transition from order to disorder is gradual without, apparently, a genuine phase transition ensuing in the intermediate regime.
    Physical Review E 03/2015; 91(3-1):032905.
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    ABSTRACT: The aim of this introductory article is two-fold. First, we aim to offer a general introduction to the theme of Bose-Einstein condensates, and briefly discuss the evolution of a number of relevant research directions during the last two decades. Second, we introduce and present the articles that appear in this Special Volume of Romanian Reports in Physics celebrating the conclusion of the second decade since the experimental creation of Bose-Einstein condensation in ultracold gases of alkali-metal atoms.
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    ABSTRACT: We present a unified description of different types of matter-wave solitons that can emerge in quasi one-dimensional spin-orbit coupled (SOC) Bose-Einstein condensates (BECs). This description relies on the reduction of the original two-component Gross-Pitaevskii SOC-BEC model to a single nonlinear Schr\"{o}dinger equation, via a multiscale expansion method. This way, we find approximate bright and dark soliton solutions, for attractive and repulsive interatomic interactions respectively, for different regimes of the SOC interactions. Beyond this, our approach also reveals "negative mass" regimes, where corresponding "negative mass" bright or dark solitons can exist for repulsive or attractive interactions, respectively. Such a unique opportunity stems from the structure of the excitation spectrum of the SOC-BEC. Numerical results are found to be in excellent agreement with our analytical predictions.
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    ABSTRACT: We study scattering of quasi one-dimensional matter-waves at an interface of two spatial domains, one with repulsive and one with attractive interatomic interactions. It is shown that the incidence of a Gaussian wavepacket from the repulsive to the attractive region gives rise to generation of a soliton train. More specifically, the number of emergent solitons can be controlled e.g. by the variation of the amplitude or the width of the incoming wavepacket. Furthermore, we study the reflectivity of a soliton incident from the attractive region to the repulsive one. We find the reflection coefficient numerically and employ analytical methods, that treat the soliton as a particle (for moderate and large amplitudes) or a quasi-linear wavepacket (for small amplitudes), to determine the critical soliton momentum - as function of the soliton amplitude - for which total reflection is observed.
    Physical Review A 02/2015; 91(3). DOI:10.1103/PhysRevA.91.033633
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    Panayotis G. Kevrekidis, Atanas G. Stefanov, Haitao Xu
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    ABSTRACT: In the present work, we consider the mass in mass (or mass with mass) system of granular chains, namely a granular chain involving additionally an internal resonator. For these chains, we rigorously establish that under suitable "anti-resonance" conditions connecting the mass of the resonator and the speed of the wave, bell-shaped traveling wave solutions continue to exist in the system, in a way reminiscent of the results proven for the standard granular chain of elastic Hertzian contacts. We also numerically touch upon settings where the conditions do not hold, illustrating, in line also with recent experimental work, that non-monotonic waves bearing non-vanishing tails may exist in the latter case.
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    ABSTRACT: In the present paper we consider an optical system with a $\chi^{(2)}$-type nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions. Considering this as an inverse problem and positing a family of exact solutions in terms of cnoidal functions, we solve for the resulting potential functions in a way that ensures the potentials obey the requirements of $\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic solutions for which we present a stability analysis as a function of their amplitude parameters. Finally, we numerically explore the nonlinear dynamics of the associated waveforms to identify the outcome of the relevant dynamical instabilities of localized and extended states.
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    H. Xu, P. G. Kevrekidis, A. Stefanov
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    ABSTRACT: In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as Mass-in-Mass systems. We use 3 distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem. or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) of the problem in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, that will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of resonance conditions is identified for which solutions with genuinely monotonic decaying tails arise.
    Journal of Physics A Mathematical and Theoretical 12/2014; 48(19). DOI:10.1088/1751-8113/48/19/195204

Publication Stats

8k Citations
1,080.58 Total Impact Points


  • 2001–2015
    • University of Massachusetts Amherst
      • Department of Mathematics and Statistics
      Amherst Center, Massachusetts, United States
  • 2014
    • 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
  • 2001–2011
    • Princeton University
      • • Department of Mathematics
      • • Department of Chemical and Biological Engineering
      • • Program in Applied and Computational Mathematics
      Princeton, New Jersey, United States
  • 2007
    • University of Crete
      Retimo, Crete, Greece
    • University of Kansas
      • Department of Mathematics
      Lawrence, Kansas, United States
  • 2006
    • Nankai University
      T’ien-ching-shih, Tianjin Shi, China
  • 2005
    • University of New Mexico
      • Department of Mathematics & Statistics
      Albuquerque, New Mexico, United States
  • 2004
    • The University of Tokyo
      • Institute of Industrial Science
      Tokyo, Tokyo-to, Japan
  • 2000–2003
    • Los Alamos National Laboratory
      • • Center for Nonlinear Studies
      • • Theoretical Division
      Лос-Аламос, California, United States
  • 1999–2001
    • Rutgers, The State University of New Jersey
      • Department Physics and Astronomy
      Newark, NJ, United States