arXiv subjects

P. G. Kevrekidis

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

At least 253 records · Page 14Linked to original sources

arXiv2015

On the Bifurcation and Stability of Single and Multiple Vortex Rings in Three-Dimensional Bose-Einstein Condensates

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.

cond-mat.quant-gas↗

arXiv2015

Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schrödinger equation

We study various properties of solutions of an extended nonlinear Schrödinger (ENLS) equation, which arises in the context of geometric evolution problems -- including vortex filament dynamics -- and governs propagation of short pulses in optical fibers and nonlinear metamaterials. For the periodic initial-boundary value problem, we derive conservation laws satisfied by local in time, weak $H^2$ (distributional) solutions, and establish global existence of such weak solutions. The derivation is obtained by a regularization scheme under a balance condition on the coefficients of the linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS model. Next, we investigate conditions for the existence of traveling wave solutions, focusing on the case of bright and dark solitons. The balance condition on the coefficients is found to be essential for the existence of exact analytical soliton solutions; furthermore, we obtain conditions which define parameter regimes for the existence of traveling solitons for various linear dispersion strengths. Finally, we study the modulational instability of plane waves of the ENLS equation, and identify important differences between the ENLS case and the corresponding NLS counterpart. The analytical results are corroborated by numerical simulations, which reveal notable differences between the bright and the dark soliton propagation dynamics, and are in excellent agreement with the analytical predictions of the modulation instability analysis.

nlin.PS↗

arXiv2015

PT Meets Supersymmetry and Nonlinearity: An Analytically Tractable Case Example

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{ö}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{ö}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.

nlin.PS↗

arXiv2015

Bright Discrete Solitons in Spatially Modulated DNLS Systems

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.

nlin.PS↗

arXiv2015

Nonlinear Resonances and Antiresonances of a Forced Sonic Vacuum

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.

nlin.PS↗

arXiv2015

Discrete Solitons and Vortices in Anisotropic Hexagonal and Honeycomb Lattices

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.

nlin.PS↗

arXiv2015

Stabilization of ring dark solitons in Bose-Einstein condensates

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.

cond-mat.quant-gas↗

arXiv2015

Effects of interactions on the generalized Hong-Ou-Mandel effect

We numerically investigate the influence of interactions on the generalized Hong-Ou-Mandel (HOM) effect for bosonic particles and show results for the cases of $N=2$, $N=3$ and $N=4$ bosons interacting with a beam splitter, whose role is played by a $δ$-barrier. In particular, we focus on the effect of attractive interactions and compare the results with the repulsive case, as well as with the analytically available results for the non-interacting case (that we use as a benchmark). We observe a fermionization effect both for growing repulsive and attractive interactions, i.e., the dip in the HOM coincidence count is progressively smeared out, for increasing interaction strengths. The role of input asymmetries is also explored.

cond-mat.quant-gas↗

arXiv2015

Bose-Einstein condensation: Twenty years after

The aim of this introductory article is two-fold. First, we aim to offer a general introduction to the theme of Bose-Einstein condensates, and briefly discuss the evolution of a number of relevant research directions during the last two decades. Second, we introduce and present the articles that appear in this Special Volume of Romanian Reports in Physics celebrating the conclusion of the second decade since the experimental creation of Bose-Einstein condensation in ultracold gases of alkali-metal atoms.

cond-mat.quant-gas↗

arXiv2015

Positive and negative mass solitons in spin-orbit coupled Bose-Einstein condensates

We present a unified description of different types of matter-wave solitons that can emerge in quasi one-dimensional spin-orbit coupled (SOC) Bose-Einstein condensates (BECs). This description relies on the reduction of the original two-component Gross-Pitaevskii SOC-BEC model to a single nonlinear Schrödinger equation, via a multiscale expansion method. This way, we find approximate bright and dark soliton solutions, for attractive and repulsive interatomic interactions respectively, for different regimes of the SOC interactions. Beyond this, our approach also reveals "negative mass" regimes, where corresponding "negative mass" bright or dark solitons can exist for repulsive or attractive interactions, respectively. Such a unique opportunity stems from the structure of the excitation spectrum of the SOC-BEC. Numerical results are found to be in excellent agreement with our analytical predictions.

cond-mat.quant-gas↗

arXiv2015

Scattering of matter-waves in spatially inhomogeneous environments

We study scattering of quasi one-dimensional matter-waves at an interface of two spatial domains, one with repulsive and one with attractive interatomic interactions. It is shown that the incidence of a Gaussian wavepacket from the repulsive to the attractive region gives rise to generation of a soliton train. More specifically, the number of emergent solitons can be controlled e.g. by the variation of the amplitude or the width of the incoming wavepacket. Furthermore, we study the reflectivity of a soliton incident from the attractive region to the repulsive one. We find the reflection coefficient numerically and employ analytical methods, that treat the soliton as a particle (for moderate and large amplitudes) or a quasi-linear wavepacket (for small amplitudes), to determine the critical soliton momentum - as function of the soliton amplitude - for which total reflection is observed.

cond-mat.quant-gas↗

arXiv2015

Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

We study a two-component nonlinear Schr{ö}dinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright-solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability, not only of the single-peak ground state (the dark-bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although our principal focus is on the homogeneous setting, we also briefly touch upon the counterintuitive impact of the potential presence of a parabolic trap on the states of interest.

nlin.PS↗

arXiv2015

Some case example exact solutions for quadratically nonlinear optical media with $\mathcal{PT}$-symmetric potentials

In the present paper we consider an optical system with a $χ^{(2)}$-type nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions. Considering this as an inverse problem and positing a family of exact solutions in terms of cnoidal functions, we solve for the resulting potential functions in a way that ensures the potentials obey the requirements of $\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic solutions for which we present a stability analysis as a function of their amplitude parameters. Finally, we numerically explore the nonlinear dynamics of the associated waveforms to identify the outcome of the relevant dynamical instabilities of localized and extended states.

nlin.PS↗

arXiv2014

Lattice Three-Dimensional Skyrmions Revisited

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 explored 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 spectrally stable solutions is corroborated by virtue of direct numerical simulations .

nlin.PS↗

arXiv2014

Traveling Waves and their Tails in Locally Resonant Granular Systems

In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as Mass-in-Mass systems. We use 3 distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem. or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) of the problem in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, that will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of resonance conditions is identified for which solutions with genuinely monotonic decaying tails arise.

nlin.PS↗

arXiv2014

Solitons and Vortices in Two-dimensional Discrete Nonlinear Schrodinger Systems with Spatially Modulated Nonlinearity

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.

nlin.PS↗

arXiv2014

Transitions From Order to Disorder in Multi-Dark and Multi-Dark-Bright Soliton Atomic Clouds

We have performed a systematic study quantifying the variation of solitary wave behavior from that of an ordered cloud resembling a "crystalline" configuration to that of a disordered state that can be characterized as a soliton "gas". As our illustrative examples, we use both one-component, as well as two-component, one dimensional atomic gases very close to zero temperature, where in the presence of repulsive inter-atomic interactions and of a parabolic trap, a cloud, respectively of dark (dark-bright) solitons can form in the one- (two-) component system. We corroborate our findings through three distinct types of approaches, namely a Gross-Pitaevskii type of partial differential equation, particle-based ordinary differential equations describing the soliton dynamical system and Monte-Carlo simulations for the particle system. We define an "empirical" order parameter to characterize the order of the soliton lattices and study how this changes as a function of the strength of the "thermally" (i.e., kinetically) induced perturbations. As may be anticipated by the one-dimensional nature of our system, the transition from order to disorder is gradual without, apparently, a genuine phase transition ensuing in the intermediate regime.

nlin.PS↗

arXiv2014

When Does Linear Stability Not Exclude Nonlinear Instability ?

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{ö}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.

nlin.PS↗