Mathematics Department Faculty Selected Works pagesCopyright (c) 2018 University of Massachusetts Amherst All rights reserved.
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Recent documents in Mathematics Department Faculty Selected Works pagesen-usSun, 14 Jan 2018 13:23:42 PST3600Wave mixing in coupled phononic crystals via a variable stiffness mechanism
http://works.bepress.com/panos_kevrekidis/419
http://works.bepress.com/panos_kevrekidis/419Wed, 25 Jan 2017 21:23:24 PST
We investigate wave mixing eects in a phononic crystal that couples the wavedynamics of two channels { primary and control ones { via a variable stinessmechanism. We demonstrate analytically and numerically that the wave transmissionin the primary channel can be manipulated by the control channel'ssignal. We show that the application of control waves allows the selection ofa specic mode through the primary channel. We also demonstrate that themixing of two wave modes is possible whereby a modulation eect is observed.A detailed study of the design parameters is also carried out to optimize theswitching capabilities of the proposed system. Finally, we verify that the systemcan fulll both switching and amplication functionalities, potentially enablingthe realization of an acoustic transistor.
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Panos Kevrekidis et al.Vector Dark-Antidark Solitary Waves in Multi-Component Bose-Einstein condensates
http://works.bepress.com/panos_kevrekidis/418
http://works.bepress.com/panos_kevrekidis/418Wed, 25 Jan 2017 21:19:25 PST
Multi-component Bose-Einstein condensates exhibit an intriguing variety of nonlinear structures.In recent theoretical work, the notion of magnetic solitons has been introduced. Here we generalizethis concept to vector dark-antidark solitary waves in multi-component Bose-Einstein condensates.We rst provide concrete experimental evidence for such states in an atomic BEC and subsequentlyillustrate the broader concept of these states, which are based on the interplay between miscibilityand inter-component repulsion. Armed with this more general conceptual framework, we expandthe notion of such states to higher dimensions presenting the possibility of both vortex-antidarkstates and ring-antidark-ring (dark soliton) states. We perform numerical continuation studies,investigate the existence of these states and examine their stability using the method of BogolyubovdeGennes analysis. Dark-antidark and vortex-antidark states are found to be stable for broadparametric regimes. In the case of ring dark solitons, where the single-component ring state isknown to be unstable, the vector entity appears to bear a progressively more and more stabilizingrole as the inter-component coupling is increased.
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Panos Kevrekidis et al.To Infinity and Beyond: Some ODE and PDE Case Studies
http://works.bepress.com/panos_kevrekidis/417
http://works.bepress.com/panos_kevrekidis/417Wed, 25 Jan 2017 21:10:13 PST
When mathematical/computational problems “reach” infinity, extending analysisand/or numerical computation beyond it becomes a notorious challenge. We suggestthat, upon suitable singular transformations (that can in principle be computationallydetected on the fly) it becomes possible to “go beyond infinity” to the other side, withthe solution becoming again well behaved and the computations continuing normally.In our lumped, Ordinary Differential Equation (ODE) examples this “infinity crossing”can happen instantaneously; at the spatially distributed, Partial Differential Equation(PDE) level the crossing of infinity may even persist for finite time, necessitating theintroduction of conceptual (and computational) buffer zones in which an appropriatesingular transformation is continuously (locally) detected and performed. Theseobservations (and associated tools) could set the stage for a systematic approach tobypassing infinity (and thus going beyond it) in a broader range of evolutionequations; they also hold the promise of meaningfully and seamlessly performing therelevant computations. Along the path of our analysis, we present a regularizationprocess via complexification and explore its impact on the dynamics; we also discuss aset of compactification transformations and their intuitive implications..
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Panos Kevrekidis et al.The emergence and analysis of Kuramoto-Sakaguchi-like models as an effective description for the dynamics of coupled Wien-bridge oscillators
http://works.bepress.com/panos_kevrekidis/416
http://works.bepress.com/panos_kevrekidis/416Wed, 25 Jan 2017 21:06:38 PST
We derive the Kuramoto-Sakaguchi model from the basic circuit equations governing two coupledWien-bridge oscillators. A Wien-bridge oscillator is a particular realization of a tunable autonomousoscillator that makes use of frequency filtering (via a RC band-pass filter) and positive feedback (viaan Op-Amp). In the last few years, such oscillators have started to be utilized in synchronizationstudies. We first show that the Wien-bridge circuit equations can be cast in the form of a coupledpair of Duffing - Van der Pol equations. Subsequently, by applying the method of multiple timescales, we derive the differential equations that govern the slow evolution of the oscillator phasesand amplitudes. These equations are directly reminiscent of the Kuramoto-Sakaguchi type modelsfor the study of synchronization. We analyze the resulting system in terms of existence and stabilityof various coupled oscillator solutions and explain on that basis how their synchronization emerges.The phase-amplitude equations are also compared numerically to the original circuit equations,and good agreement is found. Finally, we report on experimental measurements on two coupledWien-bridge oscillators and relate the results back to the theoretical predictions.
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Panos Kevrekidis et al.SO(2)-induced breathing patterns in multi-component Bose-Einstein condensates
http://works.bepress.com/panos_kevrekidis/415
http://works.bepress.com/panos_kevrekidis/415Wed, 25 Jan 2017 21:03:46 PST
In this work, we employ the SO(2)-rotations of a two-component, one-, two- and three-dimensionalnonlinear Schrodinger system at and near the Manakov limit, to construct vector solitons and vortexstructures. This way, stable stationary dark-bright solitons and their higher-dimensional siblingsare transformed into robust oscillatory dark-dark solitons (and generalizations thereof), with andwithout a harmonic connement. By analogy to the one-dimensional case, vector higher-dimensionalstructures take the form of vortex-vortex states in two dimensions and, e.g., vortex ring-vortex ringones in three dimensions. We consider the eects of unequal (self- and cross-) interaction strengths,where the SO(2) symmetry is only approximately satised, showing the dark-dark soliton oscillationis generally robust. Similar features are found in higher dimensions too, although our case examplessuggest that phenomena such as phase separation may contribute to the associated dynamics. Theseresults, in connection with the experimental realization of one-dimensional variants of such statesin optics and Bose-Einstein condensates (BECs), suggest the potential observation of the higherdimensionalbound states proposed herein.
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Panos Kevrekidis et al.Shock and Rarefaction Waves in Generalized Hertzian Contact Models
http://works.bepress.com/panos_kevrekidis/414
http://works.bepress.com/panos_kevrekidis/414Wed, 25 Jan 2017 20:59:52 PST
In the present work motivated by generalized forms of the Hertzian dynamics associated with granular crystals,we consider the possibility of such models to give rise to both shock and rarefaction waves. Dependingon the value p of the nonlinearity exponent, we find that both of these possibilities are realizable. We use aquasi-continuum approximation of a generalized inviscid Burgers model in order to predict the solution profileup to times near the shock formation, as well as to estimate when it will occur. Beyond that time threshold,oscillations associated with the discrete nature of the underlying model emerge that cannot be captured bythe quasi-continuum approximation. Our analytical characterization of the above features is complemented bysystematic numerical computations.
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Panos Kevrekidis et al.Scattering of Waves by Impurities in Precompressed Granular Chains
http://works.bepress.com/panos_kevrekidis/413
http://works.bepress.com/panos_kevrekidis/413Wed, 25 Jan 2017 20:54:29 PST
We study scattering of waves by impurities in strongly precompressed granular chains. We ex-plore the linear scattering of plane waves and identify a closed-form expression for the reectionand transmission coecients for the scattering of the waves from both a single impurity and a dou-ble impurity. For single-impurity chains, we show that, within the transmission band of the hostgranular chain, high-frequency waves are strongly attenuated (such that the transmission coecientvanishes as the wavenumber k ! ), whereas low-frequency waves are well-transmitted throughthe impurity. For double-impurity chains, we identify a resonance | enabling full transmission at aparticular frequency | in a manner that is analogous to the Ramsauer{Townsend (RT) resonancefrom quantum physics. We also demonstrate that one can tune the frequency of the RT resonanceto any value in the pass band of the host chain. We corroborate our theoretical predictions bothnumerically and experimentally, and we directly observe complete transmission for frequencies closeto the RT resonance frequency. Finally, we show how this RT resonance can lead to the existenceof reectionless modes even in granular chains (including disordered ones) with multiple doubleimpurities.
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Panos Kevrekidis et al.Resonant interaction of Phi^4 kink with spatially periodic PT-symmetric perturbation
http://works.bepress.com/panos_kevrekidis/411
http://works.bepress.com/panos_kevrekidis/411Wed, 25 Jan 2017 20:40:07 PST
The resonant interaction of the φ4 kink with a periodic PT -symmetric perturbation is observedin the frame of the continuum model and with the help of a two degree of freedom collectivevariable model derived in PRA 89, 010102(R). When the kink interacts with the perturbation, thekink’s internal mode is excited with the amplitude varying in time quasiperiodically. The maximalvalue of the amplitude was found to grow when the kink velocity is such that it travels one periodof perturbation in nearly one period of the kink’s internal mode. It is also found that the kink’stranslational and vibrational modes are coupled in a way that an increase in the kink’s internal modeamplitude results in a decrease in kink velocity. The results obtained with the collective variablemethod are in a good qualitative agreement with the numerical simulations for the continuum model.The results of the present study suggest that kink dynamics in open systems with balanced gainand loss can have new features in comparison with the case of conservative systems.
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Panos Kevrekidis et al.Performing Hong-Ou-Mandel-type Numerical Experiments with Repulsive Condensates: The case of Dark and Dark-bright Solitons
http://works.bepress.com/panos_kevrekidis/410
http://works.bepress.com/panos_kevrekidis/410Wed, 25 Jan 2017 20:36:09 PST
The Hong-Ou-Mandel experiment leads indistinguishable photons simultaneously reach-ing a 50:50 beam splitter to emerge on the same port through two-photon interference.Motivated by this phenomenon, we consider numerical experiments of the same flavor forclassical, wave objects in the setting of repulsive condensates. We examine dark solitonsinteracting with a repulsive barrier, a case in which we find no significant asymmetries inthe emerging waves after the collision, presumably due to their topological nature. We alsoconsider case examples of two-component systems, where the dark solitons trap a brightstructure in the second-component (dark-bright solitary waves). For these, pronouncedasymmetries upon collision are possible for the non-topological bright component. Wealso show an example of a similar phenomenology for ring dark-bright structures in twodimensions.
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Panos Kevrekidis et al.On Two-Component Dark-Bright Solitons in Three-dimensional Atomic Bose-Einstein Condensates
http://works.bepress.com/panos_kevrekidis/409
http://works.bepress.com/panos_kevrekidis/409Wed, 25 Jan 2017 20:33:24 PST
In the present work, we revisit two-component Bose-Einstein condensates in their fully threedimensional(3d) form. Motivated by earlier studies of dark-bright solitons in the 1d case, weexplore the stability of these structures in their fully 3d form in two variants. In one the darksoliton is planar and trapping a planar bright (disk) soliton. In the other case, a dark sphericalshell soliton creates an eective potential in which a bright spherical shell of atoms is trapped inthe second component. We identify these solutions as numerically exact states (up to a prescribedaccuracy) and perform a Bogolyubov-de Gennes linearization analysis that illustrates that bothstructures can be dynamically stable in suitable intervals of suciently low chemical potentials. Wecorroborate this nding theoretically by analyzing the stability via degenerate perturbation theorynear the linear limit of the system. When the solitary waves are found to be unstable, we exploretheir dynamical evolution via direct numerical simulations which, in turn, reveal novel waveformsthat are more robust. Finally, using the SO(2) symmetry of the model, we produce multi-dark-brightplanar or shell solitons involved in pairwise oscillatory motion.
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Panos Kevrekidis et al.