P. G. Kevrekidis

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

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Publications (578)1092.1 Total impact

  • Rajesh Chaunsali · E. Kim · H. Xu · J. Castillo · P. Kevrekidis · A. Vakakis · J. Yang
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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.
    ASME - IMECE, San Antonio, TX; 11/2015
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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.
    Physical Review E 10/2015; 92(4). DOI:10.1103/PhysRevE.92.042901 · 2.29 Impact Factor
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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.
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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.
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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.
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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
  • 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.
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    ABSTRACT: We numerically investigate an experimentally viable method, that we will refer to as the "chopsticks method", for generating and manipulating on-demand several vortices in a highly oblate atomic Bose-Einstein condensate (BEC) in order to initialize complex vortex distributions for studies of vortex dynamics. The method utilizes moving laser beams (the "chopsticks") to generate, capture and transport vortices inside and outside the BEC. We examine in detail this methodology and show a wide parameter range of applicability for the prototypical two-vortex case, and show case examples of producing and manipulating several vortices for which there is no net circulation, equal numbers of positive and negative circulation vortices, and for which there is one net quantum of circulation. We find that the presence of dissipation can help stabilize the pinning of the vortices on their respective laser beam pinning sites. Finally, we illustrate how to utilize laser beams as repositories that hold large numbers of vortices and how to deposit individual vortices in a sequential fashion in the repositories in order to construct superfluid flows about the repository beams with several quanta of circulation.
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    J. D'Ambroise · M. Salerno · P. G. Kevrekidis · F. Kh. Abdullaev
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    ABSTRACT: The existence of multidimensional lattice compactons in the discrete nonlinear Schr\"odinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, a new effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kind. These terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions: single site and vortex solutions are found to be always stable in the parametric regimes we examined. Other solutions such as double site in- and out-of-phase, four site symmetric and anti-symmetric, and a five site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged with those of the original dynamical equations without the averaging. Possible observation of compactons in the BEC loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is discussed.
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    ABSTRACT: In the present work, we consider a prototypical example of a PT-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PT-phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the PT-symmetric model of the solutions of the corresponding Hamiltonian model and find that the solutions can be continued robustly as stable ones all the way up to the PT-transition threshold. In the latter, they degenerate into linear waves. We also examine the dynamics of the model. Given the stability of the waveforms in the PT-exact phase we consider them as initial conditions for parameters outside of that phase. We find that both oscillatory dynamics and exponential growth may arise, depending on the size of the corresponding "quench". The former can be characterized by an interesting form of bi-frequency solutions that have been predicted on the basis of the SU(1,1) symmetry. Finally, we explore some special, analytically tractable, but not PT-symmetric solutions in the massless limit of the model.
    IEEE Journal of Selected Topics in Quantum Electronics 08/2015; DOI:10.1109/JSTQE.2015.2485607 · 2.83 Impact Factor
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    ABSTRACT: In the present work, we investigate how single- and multi-vortex-ring states can emerge from a planar dark soliton in three-dimensional (3D) Bose-Einstein condensates (confined in isotropic or anisotropic traps) through bifurcations. We characterize such bifurcations quantitatively using a Galerkin-type approach, and find good qualitative and quantitative agreement with our Bogoliubov-de Gennes (BdG) analysis. We also systematically characterize the BdG spectrum of the dark solitons, using perturbation theory, and obtain a quantitative match with our 3D BdG numerical calculations. We then turn our attention to the emergence of single- and multi-vortex-ring states. We systematically capture these as stationary states of the system and quantify their BdG spectra numerically. We find that although the vortex ring may be unstable when bifurcating, its instabilities weaken and may even eventually disappear, for sufficiently large chemical potentials and suitable trap settings. For instance, we demonstrate the stability of the vortex ring for an isotropic trap in the large chemical potential regime.
    Physical Review A 07/2015; 92(4). DOI:10.1103/PhysRevA.92.043601 · 2.81 Impact Factor
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    P. G. Kevrekidis · J. Cuevas-Maraver · A. Saxena · F. Cooper · A. Khare
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    ABSTRACT: In the present work, we combine the notion of $\mathcal{PT}$-symmetry with that of super-symmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the so-called P{\"o}schl-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 the corresponding solution of the regular nonlinear Schr{\"o}dinger equation with arbitrary power law nonlinearity does not possess. The spectral properties and dynamical implications of this instability are examined. We believe that these findings may pave the way towards initiating a fruitful interplay between the notions of $\mathcal{PT}$-symmetry, super-symmetric partner potentials and nonlinear interactions.
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    P. G. Kevrekidis · R. L. Horne · N. Whitaker · Q. E. Hoq · D. Kip
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    ABSTRACT: In the present work, we revisit the highly active research area of inhomogeneously nonlinear defocusing media and consider the existence, spectral stability and nonlinear dynamics of bright solitary waves in them. We use the anti-continuum limit of vanishing coupling as the starting point of our analysis, enabling in this way a systematic characterization of the branches of solutions. Our stability findings and bifurcation characteristics reveal the enhanced robustness and wider existence intervals of solutions with a broader support, culminating in the "extended" solution in which all sites are excited. Our eigenvalue predictions are corroborated by numerical linear stability analysis. Finally, the dynamics also reveal a tendency of the solution profiles to broaden, in line with the above findings. These results pave the way for further explorations of such states in discrete systems, including in higher dimensional settings.
    Journal of Physics A Mathematical and Theoretical 07/2015; 48(34). DOI:10.1088/1751-8113/48/34/345201 · 1.58 Impact Factor
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    ABSTRACT: We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude and frequency dependent boundary drive. Remarkably, despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period, and observe that they can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force ("antiresonances") in between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings, Neimark-Sacker bifurcations as well as multiple isolas (e.g. of period-3, -4 or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports an unprecedented example of a notion of a (purely) "nonlinear spectrum" in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). We rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.
  • Rajesh Chaunsali · E. Kim · H. Xu · J. Castillo · P. Kevrekidis · A. Vakakis · J. Yang
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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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    Q. E. Hoq · P. G. Kevrekidis · A. R. Bishop
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    ABSTRACT: In the present work, we consider the self-focusing discrete nonlinear Schrodinger equation on hexagonal and honeycomb lattice geometries. Our emphasis is on the study of the effects of anisotropy, motivated by the tunability afforded in recent optical and atomic physics experiments. We find that important classes of solutions, such as the so-called discrete vortices, undergo destabilizing bifurcations as the relevant anisotropy control parameter is varied. We quantify these bifurcations by means of explicit analytical calculations of the solutions, as well as of their spectral linearization eigenvalues. Finally, we corroborate the relevant stability picture through direct numerical computations. In the latter, we observe the prototypical manifestation of these instabilities to be the spontaneous rearrangement of the solution, for larger values of the coupling, into localized waveforms typically centered over fewer sites than the original unstable structure. For weak coupling, the instability appears to result in a robust breathing of the relevant waveforms.
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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.
    Physical Review A 06/2015; 92(3). DOI:10.1103/PhysRevA.92.033611 · 2.81 Impact Factor
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    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.
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    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.
    Physical Review D 06/2015; 92(5). DOI:10.1103/PhysRevD.92.056005 · 4.64 Impact Factor
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    E. Kim · R. Chaunsali · H. Xu · J. Jaworski · J. Yang · P. Kevrekidis · A. F. Vakakis
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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.

Publication Stats

8k Citations
1,092.10 Total Impact Points


  • 2001–2015
    • University of Massachusetts Amherst
      • Department of Mathematics and Statistics
      Amherst Center, Massachusetts, 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
  • 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
  • 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