Vassilis Koukouloyannis

On the stability of multibreathers in Klein-Gordon chains

In this paper, a theorem, which determines the linear stability of multibreathers excited over adjacent coupled oscillators in Klein–Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive nearest–neighbour 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 method can be applied to n-site breathers, where n is any finite number and provides a detailed count of the number of real and imaginary characteristic exponents of the breather, based on its configuration. In addition, an estimation of these exponents can be extracted through this procedure. 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 e.

Dynamics of three noncorotating 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 energy (which is expressed through the Hamiltonian H) and the angular momentum L of the system. By using the integral of the angular momentum, we reduce the system to a 2-degrees-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.

Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate

Motivated by recent experimental works, we investigate a system of vortex dynamics in an atomic Bose-Einstein condensate (BEC), consisting of three vortices, two of which have the same charge. These vortices are modeled as a system of point particles which possesses a Hamiltonian structure. This tripole system constitutes a prototypical model of vortices in BECs exhibiting chaos. By using the angular momentum integral of motion, we reduce the study of the system to the investigation of a two degree of freedom Hamiltonian model and acquire quantitative results about its chaotic behavior. Our investigation tool is the construction of scan maps by using the Smaller ALignment Index as a chaos indicator. Applying this approach to a large number of initial conditions, we manage to accurately and efficiently measure the extent of chaos in the model and its dependence on physically important parameters like the energy and the angular momentum of the system.

Existence and stability of multisite breathers in honeycomb and hexagonal lattices

We study the existence and stability of multisite discrete breathers in two prototypical non-square Klein–Gordon lattices, namely a honeycomb and a hexagonal one. In the honeycomb case we consider six-site configurations and find that for soft potential and positive coupling the out-of-phase breather configuration and the charge-two vortex breather are linearly stable, while the in-phase and charge-one vortex states are unstable. In the hexagonal lattice, we first consider three-site configurations. In the case of soft potential and positive coupling, the in-phase configuration is unstable and the charge-one vortex is linearly stable. The out-of-phase configuration here is found to always be linearly unstable. We then turn to six-site configurations in the hexagonal lattice. The stability results in this case are the same as in the six-site configurations in the honeycomb lattice. For all configurations in both lattices, the stability results are reversed in the setting of either hard potential or negative coupling. The study is complemented by numerical simulations which are in very good agreement with the theoretical predictions. Since neither the form of the on-site potential nor the sign of the coupling parameter involved have been prescribed, this description can accommodate inverse-dispersive systems (e.g. supporting backward waves) such as transverse dust-lattice oscillations in dusty plasma (Debye) crystals or analogous modes in molecular chains.

Discrete breathers and vortices in hexagonal dust crystals

We consider the existence and stability of multipole, vortex and soliton type localized modes in hexagonal and honeycomb lattices, with an aim to model strongly nonlinear transverse oscillations of charged dust grains in dusty plasma lattices (DPLs). The theory is essentially formulated in terms of the coupling (discreteness) parameter epsiv = omegaT 2/omega0 2 (here omegaT is the eigenfrequency of the tranverse DPL mode, and omega0 is related to electrostatic inter-grain interactions), thus allowing for a direct interpretation in terms of experimentally measurable plasma quantities. Two approaches, relying on the discrete nonlinear Schrodinger and the Klein-Gordon paradigms, are employed and found to agree to a satisfactory extent. The former approach has been associated to beam propagation in optical waveguide arrays. In contours involving three sites, the vortex configuration of topological charge S=1 is the most stable one. For 6-site configurations, we obtain the surprising feature that vortices of lower topological charge (S=1) may be unstable, while those of higher topological charge (e.g., S=2) may be stable. The analysis is complemented by numerical simulations and bifurcation studies. Extending previous work on breather excitations in 1D crystals, the anticontinuum-limit method is employed, to investigate single- and multi-site excitations. Instability occurs beyond a threshold value of epsiv, which is lower for multi-breathers than for single-site ones. These results complement earlier ones regarding quasi-continuum 2D envelope dust-lattice modes.

Localized excitations in dusty plasma crystals: A survey of theoretical results

The nonlinear aspects of dust motion in one- (1D) and two-dimensional (2D) dust lattices are reviewed. Horizontal (longitudinal, acoustic) as well as vertical (transverse, optic-like) dust grain motion in 1D monolayer is studied. Dust crystals are shown to support nonlinear kink-shaped solitary excitations (density solitons), related to longitudinal (in-plane) dust grain displacement, as well as modulated envelope localized modes associated with either longitudinal (in-plane, acoustic) or transverse (off-plane, inverse-optic) oscillations. Highly localized excitations (Discrete Breathers), associated with transverse dust-grain motion in 1D dust crystals, may also exist, as recently shown from first principles. Hexagonal (2D) dust lattices sustain modulated envelope structures, formed via modulational instability of in-plane vibrations. Discrete analysis of hexagonal crystals also suggests the occurrence of ultra-localized modes and vortices. With the exception of longitudinal density solitons, the above theoretical predictions have not yet been tested in the laboratory. This provides a challenging test-bed for experimental investigations, which will hopefully confirm these results.

Localized excitations in dusty plasma crystals: on the interface among plasma physics and nonlinear lattice theories

Introduction. Dusty plasma crystals (DPCs) are strongly-coupled charged particle configurations, which occur in dysty plasmas (DP) [1] when the average electrostatic potential energy substantially exceeds the mean kinetic energy. In laboratory, DPCs are formed in low-temperature plasma discharges, wherein the charged dust particles are suspended under the combined action of gravity and electric forces [2]. DPC configurations typically consist of two-dimensional (2D) – hexagonal in general – layers, but also one(1D) chains, when appropriate trapping potentials are employed for lateral confinement [3]. Our aim here is to revisit the nonlinear aspects of dust grain motion in 1D and 2D DPCs, from first principles, by reviewing earlier analytical results [4] and presenting more recent ones [5, 6].

Existence of localized periodic motions in systems of coupled integrable symplectic maps

Abstract We study a one-dimensional chain of discrete non-linear maps with a weak coupling. We consider solutions of the integrable anticontinuous limit, where one or more “central” oscillators move in resonant non-isolated periodic orbits while the other oscillators are at rest. We prove the continuation of these motions for weak non-zero coupling and determine their initial conditions and stability. We apply the above results and perform numerical investigation in the case where the uncoupled motion of each oscillator is described by the integrable Suris map.