P. G. Kevrekidis

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

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Publications (532)975.66 Total impact

  • [Show abstract] [Hide abstract]
    ABSTRACT: We consider a two-dimensional (2D) generalization of a recently proposed model [Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anti-continuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being what was considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (analytically, whenever possible). Typical scenarios of instability development are exhibited through direct simulations.
  • Zhi-Yuan Sun, Panayotis G. Kevrekidis, Peter Krüger
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    ABSTRACT: In the present work, we theoretically propose and numerically illustrate a mean-field analogue of the Hong-Ou-Mandel experiment with bright solitons. More specifically, we scatter two solitons off of each other (in our setup, the bright solitons play the role of a classical analogue to the quantum photons of the original experiment), while the role of the beam splitter is played by a repulsive Gaussian barrier. In our classical scenario, distinguishability of the particles yields, as expected, a $0.5$ split mass on either side. Nevertheless, for very slight deviations from the completely symmetric scenario a near-perfect transmission i.e., a $|2,0>e$ or a $|0,2 >$ state can be constructed instead, very similarly to the quantum mechanical output. We demonstrate this as a generic feature under slight variations of the relative soliton speed, or of the relative amplitude in a wide parametric regime. We also explore how variations of the properties of the "beam splitter" (i.e., the Gaussian barrier) affect this phenomenology.
  • P. G. Kevrekidis, D. E. Pelinovsky, A. Saxena
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    ABSTRACT: We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization's internal modes of negative energy with the wave continuum. In a broad class of nonlinear Schr{\"o}dinger (NLS) equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an anti-symmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.
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    ABSTRACT: In the present work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schr{\"o}dinger equation under the impact of external rotation. As in the well established Hamiltonian case, the rotation gives rise to the formation of vortices. We show, however, that the most unstable mode leading to this instability scales with an appropriate power of the chemical potential $\mu$ of the system, increasing proportionally to $\mu^{2/3}$. The precise form of the relevant formula, obtained through our asymptotic analysis, provides the most unstable mode as a function of the atomic density and the trap strength. We show how these unstable modes typically nucleate a large number of vortices in the periphery of the atomic cloud. However, through a pattern selection mechanism, prompted by symmetry-breaking, only few isolated vortices are pulled in sequentially from the periphery towards the bulk of the cloud resulting in highly symmetric stable vortex configurations with far fewer vortices than the original unstable mode. These results may be of relevance to the experimentally tractable realm of finite temperature atomic condensates.
  • E. Kim, F. Li, C. Chong, G. Theocharis, J. Yang, P. G. Kevrekidis
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    ABSTRACT: In the present work, we experimentally implement, numerically compute with and theoretically analyze a configuration in the form of a single column woodpile periodic structure. Our main finding is that a Hertzian, locally-resonant, woodpile lattice offers a test bed for the formation of genuinely traveling waves composed of a strongly-localized solitary wave on top of a small amplitude oscillatory tail. This type of wave, called a nanopteron, is not only motivated theoretically and numerically, but are also visualized experimentally by means of a laser Doppler vibrometer. This system can also be useful for manipulating stress waves at will, for example, to achieve strong attenuation and modulation of high-amplitude impacts without relying on damping in the system.
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    ABSTRACT: In this paper, we study the $\phi^4$ kink scattering from a spatially localized $\mathcal{PT}$-symmetric defect and the effect of the kink's internal mode (IM) is discussed. It is demonstrated that if a kink hits the defect from the gain side, a noticeable IM is excited, while for the kink coming from the opposite direction the mode excitation is much weaker. This asymmetry is a principal finding of the present work. Similar to the case of the sine-Gordon kink studied earlier, it is found that the $\phi^4$ kink approaching the defect from the gain side always passes through the defect, while in the opposite case it must have sufficiently large initial velocity, otherwise it is trapped by the loss region. It is found that for the kink with IM the critical velocity is smaller, meaning that the kink bearing IM can pass more easily through the loss region. This feature, namely the "increased transparency" of the defect as regards the motion of the kink in the presence of IM is the second key finding of the present work. A two degree of freedom collective variable model offered recently by one of the co-authors is shown to be capable of reproducing both principal findings of the present work. A simpler, analytically tractable single degree of freedom collective variable method is used to calculate analytically the kink phase shift and the kink critical velocity sufficient to pass through the defect. Comparison with the numerical results suggests that the collective variable method is able to predict these parameters with a high accuracy.
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    ABSTRACT: We study spreading processes of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to be fundamentally different from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder: an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements). For all three types of disorder, the long-time asymptotic behavior of the second moment $\tilde{m}_2$ and the inverse participation ratio $P^{-1}$ satisfy the following scaling relations: $\tilde{m}_2\sim t^{\gamma}$ and $P^{-1}\sim t^{-\eta}$. For the Anderson-like uncorrelated disorder, we find a transition from subdiffusive ($\gamma<1$) to superdiffusive ($\gamma>1$) dynamics that depends on the amount of precompression in the chain. By contrast, for the correlated disorders, we find that the dynamics is superdiffusive for all precompression levels that we consider. Additionally, for strong precompression, the inverse participation ratio decreases slowly (with $\eta<0.1$) for all three types of disorder, and the dynamics leads to a partial localization around the leading edge of the wave. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly $\gamma\approx 1.7$ and $\eta\approx 1$) for all three types of disorder.
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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 (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.
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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.
  • 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.
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    ABSTRACT: The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized parity-time-symmetric perturbation (defect) with a balanced gain and loss is investigated numerically. It is demonstrated that if a kink passes the defect, it always restores its initial momentum and energy, and the only effect of the interaction with the defect is a phase shift of the kink. A kink approaching the defect from the gain side always passes, while in the opposite case it must have sufficiently large initial momentum to pass through the defect instead of being trapped in the loss region. The kink phase shift and critical velocity are calculated by means of the collective variable method. Kink-kink (kink-antikink) collisions at the defect are also briefly considered, showing how their pairwise repulsive (respectively, attractive) interaction can modify the collisional outcome of a single kink within the pair with the defect. For the breather, the result of its interaction with the defect depends strongly on the breather parameters (velocity, frequency, and initial phase) and on the defect parameters. The breather can gain some energy from the defect and as a result potentially even split into a kink-antikink pair, or it can lose a part of its energy. Interestingly, the breather translational mode is very weakly affected by the dissipative perturbation, so that a breather penetrates more easily through the defect when it comes from the lossy side, than a kink. In all studied soliton-defect interactions, the energy loss to radiation of small-amplitude extended waves is negligible.
    Physical review. E, Statistical, nonlinear, and soft matter physics. 11/2014; 90(5-1):052902.
  • 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.
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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.
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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.
    International Journal of Theoretical Physics 09/2014; · 1.09 Impact Factor
  • 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.
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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.
  • 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
  • Nan Lu, Jes\'us Cuevas-Maraver, Panayotis\ 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 {\it 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.
    Journal of Physics A Mathematical and Theoretical 09/2014; · 1.77 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.

Publication Stats

6k Citations
975.66 Total Impact Points


  • 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