arXiv subjects

P. G. Kevrekidis

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

At least 271 records · Page 15Linked to original sources

arXiv2014

Vortex Nucleation in a Dissipative Variant of the Nonlinear Schrödinger Equation under Rotation

In the present work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schr{ö}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 $μ$ of the system, increasing proportionally to $μ^{2/3}$. 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.

cond-mat.quant-gas↗

arXiv2014

Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures

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.

cond-mat.other↗

arXiv2014

Stability and Tunneling Dynamics of a Dark-Bright Soliton Pair in a Harmonic Trap

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.

cond-mat.quant-gas↗

arXiv2014

Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals

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.

nlin.PS↗

arXiv2014

Rogue Waves In Nonlinear Schrödinger Models with Variable Coefficients: Application to Bose-Einstein Condensates

We explore the form of rogue wave solutions in a select set of case examples of nonlinear Schrödinger equations with variable coefficients. We focus on systems with constant dispersion, and present three different models that describe atomic Bose-Einstein condensates in different experimentally relevant settings. For these models, we identify exact rogue wave solutions. Our analytical findings are corroborated by direct numerical integration of the original equations, performed by two different schemes. Very good agreement between numerical results and analytical predictions for the emergence of the rogue waves is identified. Additionally, the nontrivial fate of small numerically induced perturbations to the exact rogue wave solutions is also discussed.

nlin.PS↗

arXiv2014

Vector rogue waves and dark-bright boomeronic solitons in autonomous and non-autonomous settings

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 spatio-temporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeron-like soliton solutions of the latter are converted back into ones of the original non-autonomous 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.

nlin.PS↗

arXiv2014

PT-Symmetric dimer in a generalized model of coupled nonlinear oscillators

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 Schrödinger 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.

nlin.PS↗

arXiv2014

Instability dynamics and breather formation in a horizontally shaken pendulum chain

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.

nlin.PS↗

arXiv2014

PT-symmetric sine-Gordon breathers

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 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.

nlin.PS↗

arXiv2014

Time- and space-variant wave transmission in helicoidal phononic crystals

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.

nlin.PS↗

arXiv2014

Solitary Waves in a Discrete Nonlinear Dirac equation

In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross-Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-continuum limit of vanishing coupling). Numerous unexpected features are identified including a staggered solitary pattern emerging from a single site excitation, as well as two- and three-site excitations playing a role analogous to one- and two-site, respectively, excitations of the discrete nonlinear Schrödinger analogue of the model. Stability exchanges between the two- and three-site states are identified, as well as instabilities that appear to be persistent over the coupling strength $ε$, for a subcritical value of the propagation constant $Λ$. Variations of the propagation constant, coupling parameter and nonlinearity exponent are all examined in terms of their existence and stability implications and long dynamical simulations are used to unravel the evolutionary phenomenology of the system (when unstable).

nlin.PS↗

arXiv2014

Interaction of sine-Gordon kinks and breathers with a $\mathcal{PT}$-symmetric defect

The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized $\mathcal{PT}$-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 with the use 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 strongly depends 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.

nlin.PS↗

arXiv2014

Generating Functions, Polynomials and Vortices with Alternating Signs in Bose-Einstein Condensates

In this work, we construct suitable generating functions for vortices of alternating signs in the realm of Bose-Einstein condensates. In addition to the vortex-vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend them to larger numbers of vortices. Moreover, particular moment conditions can be derived e.g. about the sum of the squared locations of the vortices for arbitrary vortex numbers. Furthermore, the relevant differential equation can be generalized appropriately in the two-dimensional complex plane and allows the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a vortex at the center that are again connected to earlier bibliography.

nlin.PS↗

arXiv2014

Nonlinear dressed states at the miscibility-immiscibility threshold

The dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component rubidium Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analyzing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.

cond-mat.quant-gas↗

arXiv2014

Quasi-energies, parametric resonances, and stability limits in ac-driven $\mathcal{PT}$-symmetric systems

We introduce a simple model for implementing the concepts of quasi-energy and parametric resonances (PRs) in systems with the $\mathcal{PT}$ symmetry, i.e., a pair of coupled and mutually balanced gain and loss elements. The parametric (ac) forcing is applied through periodic modulation of the coefficient accounting for the coupling of the two degrees of freedom. The system may be realized in optics as a dual-core waveguide with the gain and loss applied to different cores, and the thickness of the gap between them subject to a periodic modulation. The onset and development of the parametric instability for a small forcing amplitude ($V_{1}$) is studied in an analytical form. The full dynamical chart of the system is generated by systematic simulations. At sufficiently large values of the forcing frequency, $ω$, tongues of the parametric instability originate, with the increase of $V_{1}$, as predicted by the analysis. However, the tongues following further increase of $V_{1}$ feature a pattern drastically different from that in usual (non-$\mathcal{PT}$)\ parametrically driven systems: instead of bending down to larger values of the dc coupling constant, $V_{0}$, they maintain a direction parallel to the $V_{1}$ axis. The system of the parallel tongues gets dense with the decrease of $ω$, merging into a complex small-scale structure of alternating regions of stability and instability. The cases of $ω\rightarrow 0$ and $ω\rightarrow \infty $ are studied analytically by means of the adiabatic and averaging approximation, respectively. The cubic nonlinearity, if added to the system, alters the picture, destabilizing many originally robust dynamical regimes, and stabilizing some which were unstable.

nlin.PS↗

arXiv2014

Characterizing Traveling Wave Collisions in Granular Chains Starting from Integrable Limits: the case of the KdV and the Toda Lattice

Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both 1-soliton and 2-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both co-propagating and counter-propagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.

nlin.PS↗

arXiv2014

A Tale of Two Distributions: From Few To Many Vortices In Quasi-Two-Dimensional Bose-Einstein Condensates

Motivated by the recent successes of particle models in capturing the precession and interactions of vortex structures in quasi-two-dimensional Bose-Einstein condensates, we revisit the relevant systems of ordinary differential equations. We consider the number of vortices $N$ as a parameter and explore the prototypical configurations ("ground states") that arise in the case of few or many vortices. In the case of few vortices, we modify the classical result of Havelock [Phil. Mag. ${\bf 11}$, 617 (1931)] illustrating that vortex polygons in the form of a ring are unstable for $N \geq7$. Additionally, we reconcile this modification with the recent identification of symmetry breaking bifurcations for the cases of $N=2,\dots,5$. We also briefly discuss the case of a ring of vortices surrounding a central vortex (so-called $N+1$ configuration). We finally examine the opposite limit of large $N$ and illustrate how a coarse-graining, continuum approach enables the accurate identification of the radial distribution of vortices in that limit.

cond-mat.quant-gas↗

arXiv2014

Scattering and leapfrogging of vortex rings in a superfluid

The dynamics of vortex ring pairs in the homogeneous nonlinear Schrödinger 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.

cond-mat.other↗