arXiv subjects

P. G. Kevrekidis

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

At least 361 records · Page 20Linked to original sources

arXiv2011

Defect Modes in One-Dimensional Granular Crystals

We study the vibrational spectra of one-dimensional statically compressed granular crystals (arrays of elastic particles in contact) containing defects. We focus on the prototypical settings of one or two spherical defects (particles of smaller radii) interspersed in a chain of larger uniform spherical particles. We measure the near-linear frequency spectrum within the spatial vicinity of the defects, and identify the frequencies of the localized defect modes. We compare the experimentally determined frequencies with those obtained by numerical eigen-analysis and by analytical expressions based on few-site considerations. We also present a brief numerical and experimental example of the nonlinear generalization of a single-defect localized mode.

cond-mat.other↗

arXiv2011

Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

In this note we propose a new set of coordinates to study the hyperbolic or non-elliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.

nlin.PS↗

arXiv2011

Nonlinear Excitations, Stability Inversions and Dissipative Dynamics in Quasi-one-dimensional Polariton Condensates

We consider the existence, stability and dynamics of the ground state and nonlinear excitations, in the form of dark solitons, for a quasi-one-dimensional polariton condensate in the presence of pumping and nonlinear damping. We find a series of remarkable features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates. For some sizeable parameter ranges, the nodeless ("ground") state becomes {\it unstable} towards the formation of {\em stable} nonlinear single or {\em multi} dark-soliton excitations. It is also observed that for suitable parametric choices, the instability of single dark solitons can nucleate multi-dark-soliton states. Also, for other parametric regions, {\em stable asymmetric} sawtooth-like solutions exist. Finally, we consider the dragging of a defect through the condensate and the interference of two initially separated condensates, both of which are capable of nucleating dark multi-soliton dynamical states.

cond-mat.quant-gas↗

arXiv2011

Control of the symmetry breaking in double-well potentials by the resonant nonlinearity management

We introduce a one-dimensional model of Bose-Einstein condensates (BECs), combining the double-well potential, which is a well-known setting for the onset of spontaneous-symmetry-breaking (SSB) effects, and time-periodic modulation of the nonlinearity, which may be implemented by means of the Feshbach-resonance-management (FRM) technique. Both cases of the nonlinearity which is repulsive or attractive on the average are considered. In the former case, the main effect produced by the application of the FRM is spontaneous self-trapping of the condensate in either of the two potential wells in parameter regimes where it would remain untrapped in the absence of the management. In the weakly nonlinear regime, the frequency of intrinsic oscillations in the FRM-induced trapped state is very close to half the FRM frequency, suggesting that the effect is accounted for by a parametric resonance. In the case of the attractive nonlinearity, the FRM-induced effect is the opposite, i.e., enforced detrapping of a state which is self-trapped in its unmanaged form. In the latter case, the frequency of oscillations of the untrapped mode is close to a quarter of the driving frequency, suggesting that a higher-order parametric resonance may account for this effect.

cond-mat.quant-gas↗

arXiv2011

PT-Symmetric Oligomers: Analytical Solutions, Linear Stability and Nonlinear Dynamics

In the present work we focus on the case of (few-site) configurations respecting the PT-symmetry. We examine the case of such "oligomers" with not only 2-sites, as in earlier works, but also the cases of 3- and 4-sites. While in the former case of recent experimental interest, the picture of existing stationary solutions and their stability is fairly straightforward, the latter cases reveal a considerable additional complexity of solutions, including ones that exist past the linear PT-breaking point in the case of the trimer, and more complex, even asymmetric solutions in the case of the quadrimer with nontrivial spectral and dynamical properties. Both the linear stability and the nonlinear dynamical properties of the obtained solutions are discussed.

nlin.PS↗

arXiv2011

Variational approximations in discrete nonlinear Schrödinger equations with next-nearest-neighbor couplings

Solitons of a discrete nonlinear Schrödinger equation which includes the next-nearest-neighbor interactions are studied by means of a variational approximation and numerical computations. A large family of multi-humped solutions, including those with a nontrivial phase structure which are a feature particular to the next-nearest-neighbor interaction model, are accurately predicted by the variational approximation. Bifurcations linking solutions with the trivial and nontrivial phase structures are also captured remarkably well, including a prediction of critical parameter values.

nlin.PS↗

arXiv2010

Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter. We employ a novel technique combining concentration--compactness and spectral properties of linearized Schrödinger type operators to show that the symmetric ground states can either be uniquely continued for the entire interval of the eigenvalue parameter or they undergo a symmetry--breaking pitchfork bifurcation due to the second eigenvalue of the linearized operator crossing zero. In addition we prove the appropriate scaling for the stationary states in the limit of large values of the eigenvalue parameter. The scaling and our novel technique imply that all ground states at large eigenvalues must be localized near a critical point of the potential and bifurcate from the soliton of the focusing NLS equation without potential localized at the same point. The theoretical results are illustrated numerically for a double-well potential obtained after the splitting of a single-well potential. We compare the cases before and after the splitting, and numerically investigate bifurcation and stability properties of the ground states which are beyond the reach of our theoretical tools.

math-ph↗

arXiv2010

Dark-Bright Discrete Solitons: A Numerical Study of Existence, Stability and Dynamics

In the present work, we numerically explore the existence and stability properties of different types of configurations of dark-bright solitons, dark-bright soliton pairs and pairs of dark-bright and dark solitons in discrete settings, starting from the anti-continuum limit. We find that while single discrete dark-bright solitons have similar stability properties to discrete dark solitons, their pairs may only be stable if the bright components are in phase and are always unstable if the bright components are out of phase. Pairs of dark-bright solitons with dark ones have similar stability properties as individual dark or dark-bright ones. Lastly, we consider collisions between dark-bright solitons and between a dark-bright one and a dark one. Especially in the latter and in the regime where the underlying lattice structure matters, we find a wide range of potential dynamical outcomes depending on the initial soliton speed.

nlin.PS↗

arXiv2010

Discrete breathers at the interface between a diatomic and monoatomic granular chain

In the present work, we develop a systematic examination of the existence, stability and dynamical properties of a discrete breather at the interface between a diatomic and a monoatomic granular chain. We remarkably find that such an "interface breather" is more robust than its bulk diatomic counterpart throughout the gap of the linear spectrum. The latter linear spectral gap needs to exist for the breather state to arise and the relevant spectral conditions are discussed. We illustrate the minimal excitation conditions under which such an interface breather can be "nucleated" and analyze its apparently weak interaction with regular highly nonlinear solitary waveforms.

nlin.PS↗

arXiv2010

Emergence and Stability of Vortex Clusters in Bose-Einstein Condensates: a Bifurcation Approach near the Linear Limit

We study the existence and stability properties of clusters of alternating charge vortices in Bose-Einstein condensates. It is illustrated that such states emerge from cascades of symmetry-breaking bifurcations that can be analytically tracked near the linear limit of the system via weakly nonlinear few-mode expansions. We present the resulting states that emerge near the first few eigenvalues of the linear limit, and illustrate how the nature of the bifurcations can be used to understand their stability. Rectilinear, polygonal and diagonal vortex clusters are only some of the obtained states while mixed states, consisting of dark solitons and vortex clusters, are identified as well.

cond-mat.quant-gas↗

arXiv2010

Impact of anisotropy on vortex clusters and their dynamics

We investigate the effects of anisotropy on the stability and dynamics of vortex cluster states which arise in Bose-Einstein condensates. Sufficiently strong anisotropies are shown to stabilize states with arbitrary numbers of vortices that are highly unstable in the isotropic limit. Conversely, anisotropy can be used to destabilize states which are stable in the isotropic limit. Near the linear limit, we identify the bifurcations of vortex states including their emergence from linear eigenstates, while in the strongly nonlinear limit, a particle-like description of the dynamics of the vortices in the anisotropic trap is developed. Both are in very good agreement with numerical results. Collective modes of stabilized many vortex cluster states are demonstrated.

cond-mat.quant-gas↗

arXiv2010

Symmetry breaking, coupling management, and localized modes in dual-core discrete nonlinear-Schrödinger lattices

We introduce a system of two linearly coupled discrete nonlinear Schrödinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC). Using an averaging procedure based on the multiscale method, we derive a system of averaged (autonomous) equations, which take the form of coupled DNLSEs with additional nonlinear coupling terms of the four-wave-mixing type. We identify stability regions for fundamental onsite discrete symmetric solitons (single-site modes with equal norms in both components), as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The symmetry-breaking bifurcation, which destabilizes the fundamental symmetric solitons and gives rise to their asymmetric counterparts, is investigated too. It is demonstrated that the averaged equations provide a good approximation in all the cases. In particular, the symmetry-breaking bifurcation, which is of the pitchfork type in the framework of the averaged equations, corresponds to a Hopf bifurcation in terms of the original system.

nlin.PS↗

arXiv2010

Compactons in Nonlinear Schrödinger Lattices with Strong Nonlinearity Management

The existence of compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. In the averaged DNLS equation the resulting effective inter-well tunneling depends on modulation parameters {\it and} on the field amplitude. This introduces nonlinear dispersion in the system and can lead to a prototypical realization of single- or multi-site stable discrete compactons in nonlinear optical waveguide and BEC arrays. These structures can dynamically arise out of Gaussian or compactly supported initial data.

nlin.PS↗

arXiv2010

Tunable Vibrational Band Gaps in One-Dimensional Diatomic Granular Crystals with Three-Particle Unit Cells

We investigate the tunable vibration filtering properties of one-dimensional diatomic granular crystals composed of arrays of stainless steel spheres and cylinders interacting via Hertzian contact. The arrays consist of periodically repeated three-particle unit cells (steel-cylinder-sphere) in which the length of the cylinder is varied systematically. We apply static compression to linearize the dynamic response of the crystals and characterize their linear frequency spectrum. We find good agreement between theoretical dispersion relation analysis (for infinite systems), state-space analysis (for finite systems), and experiments. We report the observation of up to three distinct pass bands and two finite band gaps and show their tunability for variations in cylinder length and static compression.

cond-mat.other↗

arXiv2010

Nonlinear Waves in Lattices: Past, Present, Future

In the present work, we attempt a brief summary of various areas where nonlinear waves have been emerging in the phenomenology of lattice dynamical systems. These areas include nonlinear optics, atomic physics, mechanical systems, electrical lattices, nonlinear metamaterials, plasma dynamics and granular crystals. We give some of the recent developments in each one of these areas and speculate on some of the potentially interesting directions for future study.

nlin.PS↗

arXiv2010

Intrinsic Energy Localization through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals

We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy profile. We investigate their existence, structure, and stability throughout the band gap of the linear spectrum and classify them into four regimes: a regime near the lower optical band edge of the linear spectrum, a moderately discrete regime, a strongly discrete regime that lies deep within the band gap of the linearized version of the system, and a regime near the upper acoustic band edge. We contrast discrete breathers in anharmonic FPU-type diatomic chains with those in diatomic granular crystals, which have a tensionless interaction potential between adjacent particles, and highlight in that the asymmetric nature of the latter interaction potential may lead to a form of hybrid bulk-surface localized solutions.

cond-mat.other↗

arXiv2010

Stable Vortex-Bright Soliton Structures in Two-Component Bose Einstein Condensates

We report the numerical realization and demonstration of robustness of certain 2-component structures in Bose-Einstein Condensates in 2 and 3 spatial dimensions with non-trivial topological charge in one of the components. In particular, we identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, as a result of the effective potential created by a stable vortex in the other component. The resulting vortex-bright solitary waves, which naturally generalize the recently experimentally observed dark-bright solitons, are examined both in the homogeneous medium and in the presence of parabolic and periodic external confinement and are found to be very robust.

nlin.PS↗

arXiv2010

Stable structures with high topological charge in nonlinear photonic quasicrystals

Stable vortices with topological charge of 3 and 4 are examined numerically and analytically in photonic quasicrystals created by interference of 5 as well as 8 beams, in the cases of cubic as well as saturable nonlinearities. These structures are experimentally realizable, including a prototypical example of a stable charge 4 vortex. Direct numerical simulations corroborate the analytical and numerical linear stability analysis predictions.

nlin.PS↗