arXiv subjects

Panayotis G. Kevrekidis

Publications and source records attributed to Panayotis G. Kevrekidis.

At least 109 records · Page 6Linked to original sources

arXiv2015

Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates

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.

cond-mat.quant-gas↗

arXiv2015

Staggered $\mathcal{PT}$-symmetric ladders with cubic nonlinearity

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 Schrödinger (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.

nlin.PS↗

arXiv2015

Traveling waves for the Mass in Mass model of granular chains

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.

nlin.PS↗

arXiv2014

Asymmetric Wave Propagation Through Saturable Nonlinear Oligomers

In the present paper we consider nonlinear dimers and trimers (more generally, oligomers) embedded within a linear Schr{ö}dinger lattice where the nonlinear sites are of saturable type. We examine the stationary states of such chains in the form of plane waves, and analytically compute their reflection and transmission coefficients through the nonlinear oligomer, as well as the corresponding rectification factors which clearly illustrate the asymmetry between left and right propagation in such systems. We examine not only the existence but also the dynamical stability of the plane wave states. Lastly, we generalize our numerical considerations to the more physically relevant case of Gaussian initial wavepackets and confirm that the asymmetry in the transmission properties also persists in the case of such wavepackets.

nlin.PS↗

arXiv2014

A Mean-Field Analogue of the Hong-Ou-Mandel Experiment With Bright Solitons

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.

nlin.PS↗

arXiv2014

Kink scattering from a parity-time-symmetric defect in the $ϕ^4$ model

In this paper, we study the $ϕ^4$ 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 $ϕ^4$ 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.

nlin.PS↗

arXiv2014

Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation

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.

nlin.PS↗

arXiv2014

Self trapping transition for a nonlinear impurity within a linear chain

In the present work we revisit the issue of the self-trapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schrödinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied in the fully dynamical problem in the case of a 3D-lattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the self-trapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations.

nlin.PS↗

arXiv2014

$\mathcal{PT}$-symmetric ladders with a scattering core

We consider a $\mathcal{PT}$-symmetric chain (ladder-shaped) system governed by the discrete nonlinear Schrödinger equation where the cubic nonlinearity is carried solely by two central "rungs" of the ladder. Two branches of scattering solutions for incident plane waves are found. We systematically construct these solutions, analyze their stability, and discuss non-reciprocity of the transmission associated with them. To relate these results to finite-size wavepacket dynamics, we also perform direct simulations of the evolution of the wavepackets, which confirm that the transmission is indeed asymmetric in this nonlinear system with the mutually balanced gain and loss.

nlin.PS↗

arXiv2014

Dynamics of Three Non-co-rotating Vortices in Bose-Einstein Condensates

In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the {\it energy} (which is expressed through the Hamiltonian $H$) and the {\it angular momentum} $L$ of the system. By using the integral of the angular momentum, we reduce the system to a two degree-of-freedom one with $L$ as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.

nlin.CD↗

arXiv2013

Solitary Matter Waves in Combined Linear and Nonlinear Potentials: Detection, Stability, and Dynamics

We study statically homogeneous Bose-Einstein condensates with spatially inhomogeneous interactions and outline an experimental realization of compensating linear and nonlinear potentials that can yield constant-density solutions. We illustrate how the presence of a step in the nonlinearity coefficient can only be revealed dynamically and consider, in particular, how to reveal it by exploiting the inhomogeneity of the sound speed with a defect-dragging experiment. We conduct computational experiments and observe the spontaneous emergence of dark solitary waves. We use effective-potential theory to perform a detailed analytical investigation of the existence and stability of solitary waves in this setting, and we corroborate these results computationally using a Bogoliubov-de Gennes linear stability analysis. We find that dark solitary waves are unstable for all step widths, whereas bright solitary waves can become stable through a symmetry-breaking bifurcation as one varies the step width. Using phase-plane analysis, we illustrate the scenarios that permit this bifurcation and explore the dynamical outcomes of the interaction between the solitary wave and the step.

nlin.PS↗

arXiv2013

Nonlinear dynamics in PT-symmetric lattices

We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schr{ö}dinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We prove that the solutions of the dNLS equation do not blow up in a finite time and the trajectories starting with small initial data remain bounded for all times. Nevertheless, for arbitrary values of the gain and loss parameter, there exist trajectories starting with large initial data that grow exponentially fast for larger times with a rate that is rigorously identified. Numerical computations illustrate these analytical results for dimers and quadrimers.

nlin.SI↗

arXiv2013

Nonlinear stationary states in PT-symmetric lattices

In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem at an analogue of the anti-continuum limit for the dNLS equation. Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation. Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.

nlin.PS↗

arXiv2012

Dark lattice solitons in one-dimensional waveguide arrays with defocusing saturable nonlinearities and alternating couplings

In the present work, we examine "binary" waveguide arrays, where the coupling between adjacent sites alternates between two distinct values $C_1$ and $C_2$ and a saturable nonlinearity is present on each site. Motivated by experimental investigations of this type of system in fabricated LiNbO$_3$ arrays, we proceed to analyze the nonlinear wave excitations arising in the self-defocusing nonlinear regime, examining, in particular, dark solitons and bubbles. We find that such solutions may, in fact, possess a reasonably wide, experimentally relevant parametric interval of stability, while they may also feature both prototypical types of instabilities, namely exponential and oscillatory ones, for the same configuration. The spectral properties of the linearization around the solutions and the presence of two bands and associated mini-gaps are discussed and the nontrivial differences from the standard cubic discrete nonlinear Schr{ö}dinger model are highlighted.

nlin.PS↗

arXiv2011

Dynamics of bright solitons and soliton arrays in the nonlinear Schrodinger equation with a combination of random and harmonic potentials

We report results of systematic simulations of the dynamics of solitons in the framework of the one-dimensional nonlinear Schrödinger equation (NLSE), which includes the harmonic-oscillator (HO) potential and a random potential. The equation models experimentally relevant spatially disordered settings in Bose-Einstein condensates (BECs) and nonlinear optics. First, the generation of soliton arrays from a broad initial quasi-uniform state by the modulational instability (MI) is considered, following a sudden switch of the nonlinearity from repulsive to attractive. Then, we study oscillations of a single soliton in this setting, which models a recently conducted experiment in BEC. Basic characteristics of the MI-generated array, such as the number of solitons and their mobility, are reported as functions of the strength and correlation length of the disorder, and of the total norm. For the single oscillating soliton, its survival rate is found. Main features of these dependences are explained qualitatively.

nlin.PS↗

arXiv2011

Dark-bright gap solitons in coupled-mode one-dimensional saturable waveguide arrays

In the present work, we consider the dynamics of dark solitons as one mode of a defocusing photorefractive lattice coupled with bright solitons as a second mode of the lattice. Our investigation is motivated by an experiment which illustrates that such coupled states can exist with both components in the first gap of the linear band spectrum. This finding is further extended by the examination of different possibilities from a theoretical perspective, such as symbiotic ones where the bright component is supported by states of the dark component in the first or second gap, or non-symbiotic ones where the bright soliton is also a first-gap state coupled to a first or second gap state of the dark component. While the obtained states are generally unstable, these instabilities typically bear fairly small growth rates which enable their observation for experimentally relevant propagation distances.

nlin.PS↗

arXiv2009

On the stability of multibreathers in Klein-Gordon chains

In the present paper, a theorem, which determines the linear stability of multibreathers in Klein-Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive inter-site coupling, only structures with adjacent sites excited out-of-phase may be stable, while only in-phase ones may be stable for negative coupling. The situation is reversed for hard nonlinearities. This theorem can be applied in $n$-site breathers, where $n$ is any finite number and provides an $\cal{O}(\sqrtε)$ estimation of the characteristic exponents of the solution. To complement the analysis, we perform numerical simulations and establish that the results are in excellent agreement with the theoretical predictions, at least for small values of the coupling constant $ε$.

nlin.PS↗

arXiv2008

Radiationless energy exchange in three-soliton collisions

We revisit the problem of the three-soliton collisions in the weakly perturbed sine-Gordon equation and develop an effective three-particle model allowing to explain many interesting features observed in numerical simulations of the soliton collisions. In particular, we explain why collisions between two kinks and one antikink are observed to be practically elastic or strongly inelastic depending on relative initial positions of the kinks. The fact that the three-soliton collisions become more elastic with an increase in the collision velocity also becomes clear in the framework of the three-particle model. The three-particle model does not involve internal modes of the kinks, but it gives a qualitative description to all the effects observed in the three-soliton collisions, including the fractal scattering and the existence of short-lived three-soliton bound states. The radiationless energy exchange between the colliding solitons in weakly perturbed integrable systems takes place in the vicinity of the separatrix multi-soliton solutions of the corresponding integrable equations, where even small perturbations can result in a considerable change in the collision outcome. This conclusion is illustrated through the use of the reduced three-particle model.

nlin.PS↗