arXiv subjects

P. G. Kevrekidis

Publications and source records attributed to P. G. Kevrekidis.

At least 289 records · Page 16Linked to original sources

arXiv2014

Generalized Dimers and their Stokes-variable Dynamics

In the present work, we generalize the setting of dimers with potential gain and loss which have been extensively considered recently in $\mathcal{P T}$-symmetric contexts. We consider a pair of waveguides which are evanescently coupled but may also be actively coupled and may possess onsite gain and loss, as well as (possibly non-uniform) nonlinearity. We identify (and where appropriate review from earlier work) a plethora of interesting dynamical scenaria ranging from the existence of stable and unstable fixed points and integrable dynamics, to the emergence of pitchfork or Hopf bifurcations and the generation of additional fixed points and limit cycles, respectively, as well as the potential deviation of trajectories to infinity. Thus, a catalogue of a large number of possible cases is given and their respective settings physically justified (where appropriate).

nlin.PS↗

arXiv2014

sine-Gordon Equation: From Discrete to Continuum

In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine-Gordon equation and the non-integrable $ϕ^4$ model. We focus, in particular, on two of their prototypical solutions, namely the kink-like heteroclinic connections and the time-periodic, exponentially localized in space breather waveforms. Two limits of the discrete variants of these models are contrasted: on the one side, the analytically tractable original continuum limit, and on the opposite end, the highly discrete, so-called anti-continuum limit of vanishing coupling. Numerical computations are used to bridge these two limits, as regards the existence, stability and dynamical properties of the waves. Finally, a recent variant of this theme is presented in the form of $\mathcal{PT}$-symmetric Klein-Gordon field theories and a number of relevant results are touched upon.

nlin.PS↗

arXiv2014

Traveling waves of the regularized short pulse equation

In the present work, we revisit the so-called regularized short pulse equation (RSPE) and, in particular, explore the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First, using a fixed point iteration scheme, we numerically integrate the equation to find solitary waves. It is found that these solutions are well approximated by a truncated series of hyperbolic secants. The dependence of the soliton's parameters (height, width, etc) to the parameters of the equation is also investigated. Second, by developing a multiple scale reduction of the RSPE to the nonlinear Schrödinger equation, we are able to construct (both standing and traveling) envelope wave breather type solutions of the former, based on the solitary wave structures of the latter. Both the regular and the breathing traveling wave solutions identified are found to be robust and should thus be amenable to observations in the form of few optical cycle pulses.

nlin.PS↗

arXiv2014

Beating dark-dark solitons and Zitterbewegung in spin-orbit coupled Bose-Einstein condensates

We present families of beating dark-dark solitons in spin-orbit (SO) coupled Bose-Einstein condensates. These families consist of solitons residing simultaneously in the two bands of the energy spectrum. The soliton components are characterized by two different spatial and temporal scales, which are identified by a multiscale expansion method. The solitons are "beating" ones, as they perform density oscillations with a characteristic frequency, relevant to Zitterbewegung (ZB). We find that spin oscillations may occur, depending on the parity of each soliton branch, which consequently lead to ZB oscillations of the beating dark solitons. Analytical results are corroborated by numerical simulations, illustrating the robustness of the solitons.

cond-mat.quant-gas↗

arXiv2014

Dynamic and Energetic Stabilization of Persistent Currents in Bose-Einstein Condensates

We study conditions under which vortices in a highly oblate harmonically trapped Bose-Einstein condensate (BEC) can be stabilized due to pinning by a blue-detuned Gaussian laser beam, with particular emphasis on the potentially destabilizing effects of laser beam positioning within the BEC. Our approach involves theoretical and numerical exploration of dynamically and energetically stable pinning of vortices with winding number up to $S=6$, in correspondence with experimental observations. Stable pinning is quantified theoretically via Bogoliubov-de Gennes excitation spectrum computations and confirmed via direct numerical simulations for a range of conditions similar to those of experimental observations. The theoretical and numerical results indicate that the pinned winding number, or equivalently the winding number of the superfluid current about the laser beam, decays as a laser beam of fixed intensity moves away from the BEC center. Our theoretical analysis helps explain previous experimental observations, and helps define limits of stable vortex pinning for future experiments involving vortex manipulation by laser beams.

cond-mat.quant-gas↗

arXiv2014

On the spectral stability of kinks in some PT-symmetric variants of the classical Klein-Gordon Field Theories

In the present work we consider the introduction of PT-symmetric terms in the context of classical Klein-Gordon field theories. We explore the implication of such terms on the spectral stability of coherent structures, namely kinks. We find that the conclusion critically depends on the location of the kink center relative to the center of the PT-symmetric term. The main result is that if these two points coincide, the kink's spectrum remains on the imaginary axis and the wave is spectrally stable. If the kink is centered on the "lossy side" of the medium, then it becomes stabilized. On the other hand, if it becomes centered on the "gain side" of the medium, then it is destabilized. The consequences of these two possibilities on the linearization (point and essential) spectrum are discussed in some detail.

nlin.PS↗

arXiv2014

Dark-bright Solitons and their Lattices in Atomic Bose-Einstein Condensates

In the present contribution, we explore a host of different stationary states, namely dark-bright solitons and their lattices, that arise in the context of multi-component atomic Bose-Einstein condensates. The latter, are modeled by systems of coupled Gross-Pitaevskii equations with general interaction (nonlinearity) coefficients $g_{ij}$. It is found that in some particular parameter ranges such solutions can be obtained in analytical form, however, numerically they are computed as existing in a far wider parametric range. Many features of the solutions under study, such as their analytical form without the trap or the stability/dynamical properties of one dark-bright soliton even in the presence of the trap are obtained analytically and corroborated numerically. Additional features, such as the stability of soliton lattice homogeneous states or their existence/stability in the presence of the trap, are examined numerically.

cond-mat.quant-gas↗

arXiv2013

Nonlinear modes and symmetries in linearly-coupled pairs of PT-invariant dimers

The subject of the work are pairs of linearly coupled PT-symmetric dimers. Two different settings are introduced, namely, straight-coupled dimers, where each gain site is linearly coupled to one gain and one loss site, and cross-coupled dimers, with each gain site coupled to two lossy ones. The latter pair with equal coupling coefficients represents a "PT-hypersymmetric" quadrimer. We find symmetric and antisymmetric solutions in these systems, chiefly in an analytical form, and explore the existence, stability and dynamical behavior of such solutions by means of numerical methods. We thus identify bifurcations occurring in the systems, including spontaneous symmetry breaking and saddle-center bifurcations. Simulations demonstrate that evolution of unstable branches typically leads to blowup. However, in some cases unstable modes rearrange into stable ones.

nlin.PS↗

arXiv2013

${\cal PT}$-symmetric coupler with $χ^{(2)}$ nonlinearity

We introduce the notion of a ${\cal PT}$-symmetric dimer with a $χ^{(2)}$ nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain/loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first or the second harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical time-evolution simulations exploring the evolution of the different configurations, when unstable.

physics.optics↗

arXiv2013

Justification of Leading Order Quasicontinuum Approximations of Strongly Nonlinear Lattices

We consider the leading order quasicontinuum limits of a one-dimensional granular medium governed by the Hertz contact law under precompression. The approximate model which is derived in this limit is justified by establishing asymptotic bounds for the error with the help of energy estimates. The continuum model predicts the development of shock waves, which are also studied in the full system with the aid of numerical simulations. We also show that existing results concerning the Nonlinear Schrodinger (NLS) and Korteweg de-Vries (KdV) approximation of FPU models apply directly to a precompressed granular medium in the weakly nonlinear regime.

nlin.PS↗

arXiv2013

Nonlinear PT-symmetric models bearing exact solutions

We study the nonlinear Schr$\ddot{o}$dinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.

nlin.PS↗

arXiv2013

Directed Ratchet Transport in Granular Crystals

Directed-ratchet transport (DRT) in a one-dimensional lattice of spherical beads, which serves as a prototype for granular crystals, is investigated. We consider a system where the trajectory of the central bead is prescribed by a biharmonic forcing function with broken time-reversal symmetry. By comparing the mean integrated force of beads equidistant from the forcing bead, two distinct types of directed transport can be observed ---spatial and temporal DRT. Based on the value of the frequency of the forcing function relative to the cutoff frequency, the system can be categorized by the presence and magnitude of each type of DRT. Furthermore, we investigate and quantify how varying additional parameters such as the biharmonic weight affects DRT velocity and magnitude. Finally, friction is introduced into the system and is found to significantly inhibit spatial DRT. In fact, for sufficiently low forcing frequencies, the friction may even induce a switching of the DRT direction.

nlin.PS↗

arXiv2013

Exploring Rigidly Rotating Vortex Configurations and their Bifurcations in Atomic Bose-Einstein Condensates

In the present work, we consider the problem of a system of few vortices $N \leq 5$ as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. On the one hand, we use a Monte-Carlo method parametrizing the vortex "particles" by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for $N=2, ..., 5$ and different vortex particle angular momenta. We then complement this picture with a dynamical systems analysis of the possible rigidly rotating states. The latter reveals all the supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise exposing the full wealth of the problem even at such low dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte-Carlo approach selects for different values of the angular momentum which is used as a bifurcation parameter.

nlin.PS↗

arXiv2013

From Nodeless Clouds and Vortices to Gray Rings and Symmetry-Broken States in Two-Dimensional Polariton Condensates

We consider the existence, stability and dynamics of the nodeless state and fundamental nonlinear excitations, such as vortices, for a quasi-two-dimensional polariton condensate in the presence of pumping and nonlinear damping. We find a series of interesting features that can be directly contrastedto the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates (BECs). For sizeable parameter ranges, in line with earlier findings, the nodeless state becomes unstable towards the formation of {\em stable} nonlinear single or multi-vortex excitations. The potential instability of the single vortex is also examined and is found to possess similar characteristics to those of the nodeless cloud. We also report that, contrary to what is known, e.g., for the atomic BEC case, {\it stable} stationary gray rings (that can be thought of as radial forms of a Nozaki-Bekki hole) can be found for polariton condensates in suitable parametric regimes. In other regimes, however, these may also suffer symmetry breaking instabilities. The dynamical, pattern-forming implications of the above instabilities are explored through direct numerical simulations and, in turn, give rise to waveforms with triangular or quadrupolar symmetry.

nlin.PS↗

arXiv2013

PT-symmetry Management in Oligomer Systems

We study the effects of management of the PT-symmetric part of the potential within the setting of Schrödinger dimer and trimer oligomer systems. This is done by rapidly modulating in time the gain/loss profile. This gives rise to a number of interesting properties of the system, which are explored at the level of an averaged equation approach. Remarkably, this rapid modulation provides for a controllable expansion of the region of exact PT-symmetry, depending on the strength and frequency of the imposed modulation. The resulting averaged models are analyzed theoretically and their exact stationary solutions are translated into time-periodic solutions through the averaging reduction. These are, in turn, compared with the exact periodic solutions of the full non-autonomous PT-symmetry managed problem and very good agreement is found between the two.

nlin.PS↗

arXiv2013

Inelastic Collisions of Solitary Waves in Anisotropic Bose-Einstein Condensates: Sling-Shot Events and Expanding Collision Bubbles

We study experimentally and theoretically the dynamics of apparent dark soliton stripes in an elongated Bose-Einstein condensate. We show that for the trapping strengths corresponding to our experimental setup, the transverse confinement along one of the tight directions is not strong enough to arrest the formation of solitonic vortices or vortex rings. These solitonic vortices and vortex rings, when integrated along the transverse direction, appear as dark soliton stripes along the longitudinal direction thereby hiding their true character. The latter significantly modifies the interaction dynamics during collision events and can lead to apparent examples of inelasticity and what may appear experimentally even as a merger of two dark soliton stripes. We explain this feature by means of the interaction of two solitonic vortices leading to a sling shot event with one of the solitonic vortices being ejected at a relatively large speed. Furthermore we observe expanding collision bubbles which consist of repeated inelastic collisions of a dark soliton stripe pair with an {\it increasing} time interval between collisions.

cond-mat.quant-gas↗

arXiv2013

Matter-wave dark solitons and their excitation spectra in spin-orbit coupled Bose-Einstein condensates

We present three types of dark solitons in quasi-one-dimensional spin-orbit coupled repulsive Bose-Einstein condensates. Among these families, two are always stable, while the third one is only stable sufficiently close to the linear regime. The solitons' excitation spectra reveal the potential existence of a second anomalous mode. While the first such mode describes the soliton oscillatory motion in a parabolic trap, the second, when present, reflects the double well structure of the underlying single-particle spectrum. This novel mode results in moving density stripes in the vicinity of the soliton core, or in an out-of-phase oscillation of the constituent components, with little effect on the nearly stationary striped total density of the composite soliton.

cond-mat.quant-gas↗

arXiv2013

PT-Symmetric Dimer of Coupled Nonlinear Oscillators

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schrödinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

nlin.PS↗