arXiv subjects

P. G. Kevrekidis

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

At least 163 records · Page 9Linked to original sources

arXiv2018

Speed-of-light pulses in a nonlinear Weyl equation

We introduce a prototypical nonlinear Weyl equation, motivated by recent developments in massless Dirac fermions, topological semimetals and photonics. We study the dynamics of its pulse solutions and find that a localized one-hump initial condition splits into a localized two-hump pulse, while an associated phase structure emerges in suitable components of the spinor field. For times larger than a transient time $t_s$ this pulse moves with the speed of light (or Fermi velocity in Weyl semimetals), effectively featuring linear wave dynamics and maintaining its shape (both in two and three dimensions). We show that for the considered nonlinearity, this pulse represents an exact solution of the nonlinear Weyl (NLW) equation. Finally, we comment on the generalization of the results to a broader class of nonlinearities and on their emerging potential for observation in different areas of application.

nlin.PS↗

arXiv2018

Dynamical transitions between equilibria in a dissipative Klein-Gordon lattice

We consider the energy landscape of a dissipative Klein-Gordon lattice with a $ϕ^4$ on-site potential. Our analysis is based on suitable energy arguments, combined with a discrete version of the Łojasiewicz inequality, in order to justify the convergence to a single, nontrivial equilibrium for all initial configurations of the lattice. Then, global bifurcation theory is explored, to illustrate that in the discrete regime all linear states lead to nonlinear generalizations of equilibrium states. Direct numerical simulations reveal the rich structure of the equilibrium set, consisting of non-trivial topological (kink-shaped) interpolations between the adjacent minima of the on-site potential, and the wealth of dynamical convergence possibilities. These dynamical evolution results also provide insight on the potential stability of the equilibrium branches, and glimpses of the emerging global bifurcation structure, elucidating the role of the interplay between discreteness, nonlinearity and dissipation.

nlin.PS↗

arXiv2018

Planar and Radial Kinks in Nonlinear Klein-Gordon Models: Existence, Stability and Dynamics

We consider effectively one-dimensional planar and radial kinks in two-dimensional nonlinear Klein-Gordon models and focus on the sine-Gordon model and the $ϕ^4$ variants thereof. We adapt an adiabatic invariant formulation recently developed for nonlinear Schr{ö}dinger equations, and we study the transverse stability of these kinks. This enables us to characterize one-dimensional planar kinks as solitonic filaments, whose stationary states and corresponding spectral stability can be characterized not only in the homogeneous case, but also in the presence of external potentials. Beyond that, the full nonlinear (transverse) dynamics of such filaments are described using the reduced, one-dimensional, adiabatic invariant formulation. For radial kinks, this approach confirms their azimuthal stability. It also predicts the possibility of creating stationary and stable ring-like kinks. In all cases we corroborate the results of our methodology with full numerics on the original sine-Gordon and $ϕ^4$ models.

nlin.PS↗

arXiv2018

Experimental and numerical observation of dark and bright discrete solitons in the band-gap of a diatomic-like electrical lattice

We observe dark and bright intrinsic localized modes (ILMs) or discrete breathers (DB) experimentally and numerically in a diatomic-like electrical lattice. The generation of dark ILMs by driving a dissipative lattice with spatially-homogenous amplitude is, to our knowledge, unprecedented. In addition, the experimental manifestation of bright breathers within the bandgap is also novel in this system. In experimental measurements the dark modes appear just below the bottom of the top branch in frequency. As the frequency is then lowered further into the band-gap, the dark DBs persist, until the nonlinear localization pattern reverses and bright DBs appear on top of the finite background. Deep into the bandgap, only a single bright structure survives in a lattice of 32 nodes. The vicinity of the bottom band also features bright and dark self-localized excitations. These results pave the way for a more systematic study of dark breathers and their bifurcations in diatomic-like chains.

nlin.PS↗

arXiv2018

2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

We derive stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schrödinger (HDNLS) equation by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling parameter. We use pseudo-arclength continuation to capture the relevant branches of solutions and explore their corresponding stability and dynamical properties (i.e., their fate when unstable). We focus on nine primary types of solutions: single site, double site in- and out-of-phase, squares with four sites in-phase and out-of phase in each of the vertical and horizontal directions, four sites out-of-phase arranged in a line horizontally, and two additional solutions having respectively six and eight nonzero sites. The chosen configurations are found to merge into four distinct bifurcation events. We unveil the nature of the bifurcation phenomena and identify the critical points associated with these states and also explore the consequences of the termination of the branches on the dynamical phenomenology of the model.

nlin.PS↗

arXiv2018

Dynamics of Dirac solitons in networks

We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, allowing reflectionless transmission over a Y-junction are derived. The analytical results are confirmed by direct numerical simulations.

nlin.PS↗

arXiv2018

N-break states in a chain of nonlinear oscillators

In the present work we explore a pre-stretched oscillator chain where the nodes interact via a pairwise Lennard-Jones potential. In addition to a homogeneous solution, we identify solutions with one or more (so-called) `breaks', i.e., jumps. As a function of the canonical parameter of the system, namely the precompression strain $d$, we find that the most fundamental one break solution changes stability when the monotonicity of the Hamiltonian changes with $d$. We provide a proof for this (motivated by numerical computations) observation. This critical point separates stable and unstable segments of the one break branch of solutions. We find similar branches for 2 through 5 break branches of solutions. Each of these higher `excited state' solutions possesses an additional unstable pair of eigenvalues. We thus conjecture that $k$ break solutions will possess at least $k-1$ (and at most $k$) pairs of unstable eigenvalues. Our stability analysis is corroborated by direct numerical computations of the evolutionary dynamics.

nlin.PS↗

arXiv2018

Spectral and dynamical analysis of a single vortex ring in anisotropic harmonically trapped three-dimensional Bose-Einstein condensates

In the present work, motivated by numerous recent experimental developments we revisit the dynamics of a single vortex ring in anisotropic harmonic traps. At the theoretical level, we start from a general Lagrangian dynamically capturing the evolution of a vortex ring and not only consider its spectrum of linearized excitations, but also explore the full nonlinear dynamical evolution of the ring as a vortical filament. The theory predicts that the ring is stable for $1 \leq λ\leq 2$, where $λ=ω_z/ω_r$ is the ratio of the trapping frequencies along the $z$ and $r$ axes, i.e., for spherical to slightly oblate condensates. We compare this prediction with direct numerical simulations of the full 3D Gross-Pitaevskii equation (GPE) capturing the linearization spectrum of the ring for different values of the chemical potential and as a function of the anisotropy parameter $λ$. We identify this result as being only asymptotically valid as the chemical potential $μ\rightarrow \infty$, revealing how the stability interval narrows and, in particular, its upper bound decreases for finite $μ$. Finally, we compare at the dynamical level the results of the GPE with the ones effectively capturing the ring dynamics, revealing the unstable evolution for different values of $λ$, as well as the good agreement between the two.

nlin.PS↗

arXiv2018

Interactions and scattering of quantum vortices in a polariton fluid

Quantum vortices, the quantized version of classical vortices, play a prominent role in superfluid and superconductor phase transitions. However, their exploration at a particle level in open quantum systems has gained considerable attention only recently. Here we study vortex pair interactions in a resonant polariton fluid created in a solid-state microcavity. By tracking the vortices on picosecond time scales, we reveal the role of nonlinearity, as well as of density and phase gradients, in driving their rotational dynamics. Such effects are also responsible for the split of composite spin-vortex molecules into elementary half-vortices, when seeding opposite vorticity between the two spinorial components. Remarkably, we also observe that vortices placed in close proximity experience a pull-push scenario leading to unusual scattering-like events that can be described by a tunable effective potential. Understanding vortex interactions can be useful in quantum hydrodynamics and in the development of vortex-based lattices, gyroscopes, and logic devices.

cond-mat.quant-gas↗

arXiv2018

Many-Body Dissipative Flow of a Confined Scalar Bose-Einstein Condensate Driven by a Gaussian Impurity

The many-body dissipative flow induced by a mobile aussian impurity harmonically oscillating within a cigar-shaped Bose-Einstein condensate is investigated. For very {small and large driving frequencies} the superfluid phase is preserved. Dissipation is identified, for intermediate driving frequencies, by the non-zero value of the drag force whose abrupt increase {signals the spontaneous downstream emission of an array of gray solitons. After each emission event, typically each of the solitary waves formed decays and splits into two daughter gray solitary waves that are found to be robust propagating in the bosonic background for large evolution times.} In particular, a smooth transition towards dissipation is observed, with the {\it critical} velocity for solitary wave formation depending on both the characteristics of the obstacle, namely its driving frequency and width as well as on the interaction strength. The variance of a sample of single-shot simulations indicates the fragmented nature of the system; here it is found to increase during evolution for driving frequencies where the coherent structure formation becomes significant. Finally, we demonstrate that for fairly large particle numbers in-situ single-shot images directly capture the gray soliton's decay and splitting.

cond-mat.quant-gas↗

arXiv2018

Resonant Localized Modes in Electrical Lattices with Second Neighbor Coupling

We demonstrate experimentally and corroborate numerically that an electrical lattice with nearest-neighbor and second-neighbor coupling can simultaneously support long-lived coherent structures in the form of both standard intrinsic localized modes (ILMs), as well as resonant ILMs. In the latter case, the wings of the ILM exhibit oscillations due to resonance with a degenerate plane-wave mode. This kind of localized mode has also been termed nanopteron. Here we show experimentally and using realistic simulations of the system that the nanopteron can be stabilized via both direct and subharmonic driving. In the case of excitations at the zone center (i.e., at wavenumber $k=0$), we observed stable ILMs, as well as a periodic localization pattern in certain driving regimes. In the zone boundary case (of wavenumber $k=π/a$, where $a$ is the lattice spacing), the ILMs are always resonant with a plane-wave mode, but can nevertheless be stabilized by direct (staggered) and subharmonic driving.

nlin.PS↗

arXiv2018

Solitary Waves of the Two-Dimensional Camassa-Holm--Nonlinear Schrödinger Equation

In this work, we study solitary waves in a (2+1)-dimensional variant of the defocusing nonlinear Schrödinger (NLS) equation, the so-called Camassa-Holm NLS (CH-NLS) equation. We use asymptotic multiscale expansion methods to reduce this model to a Kadomtsev--Petviashvili (KP) equation. The KP model includes both the KP-I and KP-II versions, which possess line and lump soliton solutions. Using KP solitons, we construct approximate solitary wave solutions on top of the stable continuous-wave solution of the original CH-NLS model, which are found to be of both the dark and anti-dark type. We also use direct numerical simulations to investigate the validity of the approximate solutions, study their evolution, as well as their head-on collisions.

nlin.PS↗

arXiv2018

ϕ^4 Solitary Waves in a Parabolic Potential: Existence, Stability, and Collisional Dynamics

We explore a ϕ^4 model with an added external parabolic potential term. This term dramatically alters the spectral properties of the system. We identify single and multiple kink solutions and examine their stability features; importantly, all of the stationary structures turn out to be unstable. We complement these with a dynamical study of the evolution of a single kink in the trap, as well as of the scattering of kink and anti-kink solutions of the model. We see that some of the key characteristics of kink-antikink collisions, such as the critical velocity and the multi-bounce windows, are sensitively dependent on the trap strength parameter, as well as the initial displacement of the kink and antikink.

math-ph↗

arXiv2018

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids

We study a two-dimensional incoherently pumped exciton-polariton condensate described by an open-dissipative Gross-Pitaevskii equation for the polariton dynamics coupled to a rate equation for the exciton density. Adopting a hydrodynamic approach, we use multiscale expansion methods to derive several models appearing in the context of shallow water waves with viscosity. In particular, we derive a Boussinesq/Benney-Luke type equation and its far-field expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and left-going waves. From the KP-I model, we predict the existence of vorticity-free, weakly (algebraically) localized two-dimensional dark-lump solitons. We find that, in the presence of dissipation, dark lumps exhibit a lifetime three times larger than that of planar dark solitons. Direct numerical simulations show that dark lumps do exist, and their dissipative dynamics is well captured by our analytical approximation. It is also shown that lump-like and vortex-like structures can spontaneously be formed as a result of the transverse "snaking" instability of dark soliton stripes.

cond-mat.quant-gas↗

arXiv2018

Lattices With Internal Resonator Defects

We consider a variety of settings involving chains with one or more defects stemming from the introduction of nodes bearing internal resonators. Motivated by experimental results in woodpile elastic lattices with one or two defects, we consider a variety of different theoretical scenarios. These include multi-defect chains and their ability to transmit, reflect, and especially trap energy; they also include settings with linear vs. nonlinear defects of variable interaction exponent. Moreover, they involve defects which are spatially separated and either statically, or more effectively dynamically, enable the confinement of energy between the separated defects. Wherever possible, comparisons of the experiments with numerical simulations, as well as with theoretical intuition are also offered, to provide a justification for the observed findings.

nlin.PS↗

arXiv2018

Adiabatic invariant analysis of dark and dark-bright soliton stripes in two-dimensional Bose-Einstein condensates

In the present work, we develop an adiabatic invariant approach for the evolution of quasi-one-dimensional (stripe) solitons embedded in a two-dimensional Bose-Einstein condensate. The results of the theory are obtained both for the one-component case of dark soliton stripes, as well as for the considerably more involved case of the two-component dark-bright (alias "filled dark") soliton stripes. In both cases, analytical predictions regarding the stability and dynamics of these structures are obtained. One of our main findings is the determination of the instability modes of the waves as a function of the parameters of the system (such as the trap strength and the chemical potential). Our analytical predictions are favorably compared with results of direct numerical simulations.

cond-mat.quant-gas↗

arXiv2018

Dark-bright soliton pairs: bifurcations and collisions

The statics, stability and dynamical properties of dark-bright soliton pairs are investigated motivated by applications in a homogeneous system of two-component repulsively interacting Bose-Einstein condensate. One of the intra-species interaction coefficients is used as the relevant parameter controlling the deviation from the integrable Manakov limit. Two different families of stationary states are identified consisting of dark-bright solitons that are either antisymmetric (out-of-phase) or asymmetric (mass imbalanced) with respect to their bright soliton. Both of the above dark-bright configurations coexist at the integrable limit of equal intra- and inter-species repulsions and are degenerate in that limit. However, they are found to bifurcate from it in a transcritical bifurcation. The latter interchanges the stability properties of the bound dark-bright pairs rendering the antisymmetric states unstable and the asymmetric ones stable past the associated critical point (and vice versa before it). Finally, on the dynamical side, it is found that large kinetic energies and thus rapid soliton collisions are essentially unaffected by the intra-species variation, while cases involving near equilibrium states or breathing dynamics are significantly modified under such a variation.

nlin.PS↗

arXiv2018

Correlation Effects in the Quench-Induced Phase Separation Dynamics of a Two-Species Ultracold Quantum Gas

We explore the quench dynamics of a binary Bose-Einstein condensate crossing the miscibility-immiscibility threshold and vice versa, both within and in particular beyond the mean-field approximation. Increasing the interspecies repulsion leads to the filamentation of the density of each species, involving shorter wavenumbers and longer spatial scales in the many-body approach. These filaments appear to be strongly correlated and exhibit domain-wall structures. Following the reverse quench process multiple dark-antidark solitary waves are spontaneously generated and subsequently found to decay in the many-body scenario. We simulate single-shot images to connect our findings to possible experimental realizations. Finally, the growth rate of the variance of a sample of single-shots probes the degree of entanglement inherent in the system.

cond-mat.quant-gas↗