26 research outputs found
Interface solitons in one-dimensional locally-coupled lattice systems
Fundamental solitons pinned to the interface between two discrete lattices
coupled at a single site are investigated. Serially and parallel-coupled
identical chains (\textit{System 1} and \textit{System 2}), with the
self-attractive on-site cubic nonlinearity, are considered in one dimension. In
these two systems, which can be readily implemented as arrays of nonlinear
optical waveguides, symmetric, antisymmetric and asymmetric solitons are
investigated by means of the variational approximation (VA) and numerical
methods. The VA demonstrates that the antisymmetric solitons exist in the
entire parameter space, while the symmetric and asymmetric modes can be found
below some critical value of the coupling parameter. Numerical results confirm
these predictions for the symmetric and asymmetric fundamental modes. The
existence region of numerically found antisymmetric solitons is also limited by
a certain value of the coupling parameter. The symmetric solitons are
destabilized via a supercritical symmetry-breaking pitchfork bifurcation, which
gives rise to stable asymmetric solitons, in both systems. The antisymmetric
fundamental solitons, which may be stable or not, do not undergo any
bifurcation. In bistability regions stable antisymmetric solitons coexist with
either symmetric or asymmetric ones.Comment: 9 figure
Wave interactions in localizing media - a coin with many faces
A variety of heterogeneous potentials are capable of localizing linear
non-interacting waves. In this work, we review different examples of
heterogeneous localizing potentials which were realized in experiments. We then
discuss the impact of nonlinearity induced by wave interactions, in particular
its destructive effect on the localizing properties of the heterogeneous
potentials.Comment: Review submitted to Intl. Journal of Bifurcation and Chaos Special
Issue edited by G. Nicolis, M. Robnik, V. Rothos and Ch. Skokos 21 Pages, 8
Figure
Nonlinear Lattice Waves in Random Potentials
Localization of waves by disorder is a fundamental physical problem
encompassing a diverse spectrum of theoretical, experimental and numerical
studies in the context of metal-insulator transition, quantum Hall effect,
light propagation in photonic crystals, and dynamics of ultra-cold atoms in
optical arrays. Large intensity light can induce nonlinear response, ultracold
atomic gases can be tuned into an interacting regime, which leads again to
nonlinear wave equations on a mean field level. The interplay between disorder
and nonlinearity, their localizing and delocalizing effects is currently an
intriguing and challenging issue in the field. We will discuss recent advances
in the dynamics of nonlinear lattice waves in random potentials. In the absence
of nonlinear terms in the wave equations, Anderson localization is leading to a
halt of wave packet spreading.
Nonlinearity couples localized eigenstates and, potentially, enables
spreading and destruction of Anderson localization due to nonintegrability,
chaos and decoherence. The spreading process is characterized by universal
subdiffusive laws due to nonlinear diffusion. We review extensive computational
studies for one- and two-dimensional systems with tunable nonlinearity power.
We also briefly discuss extensions to other cases where the linear wave
equation features localization: Aubry-Andre localization with quasiperiodic
potentials, Wannier-Stark localization with dc fields, and dynamical
localization in momentum space with kicked rotors.Comment: 45 pages, 19 figure
Kolmogorov turbulence, Anderson localization and KAM integrability
The conditions for emergence of Kolmogorov turbulence, and related weak wave
turbulence, in finite size systems are analyzed by analytical methods and
numerical simulations of simple models. The analogy between Kolmogorov energy
flow from large to small spacial scales and conductivity in disordered solid
state systems is proposed. It is argued that the Anderson localization can stop
such an energy flow. The effects of nonlinear wave interactions on such a
localization are analyzed. The results obtained for finite size system models
show the existence of an effective chaos border between the
Kolmogorov-Arnold-Moser (KAM) integrability at weak nonlinearity, when energy
does not flow to small scales, and developed chaos regime emerging above this
border with the Kolmogorov turbulent energy flow from large to small scales.Comment: 8 pages, 6 figs, EPJB style
Self-consistent field theory of polarized BEC: dispersion of collective excitation
We suggest the construction of a set of the quantum hydrodynamics equations
for the Bose-Einstein condensate (BEC), where atoms have the electric dipole
moment. The contribution of the dipole-dipole interactions (DDI) to the Euler
equation is obtained. Quantum equations for the evolution of medium
polarization are derived. Developing mathematical method allows to study effect
of interactions on the evolution of polarization. The developing method can be
applied to various physical systems in which dynamics is affected by the DDI.
Derivation of Gross-Pitaevskii equation for polarized particles from the
quantum hydrodynamics is described. We showed that the Gross-Pitaevskii
equation appears at condition when all dipoles have the same direction which
does not change in time. Comparison of the equation of the electric dipole
evolution with the equation of the magnetization evolution is described.
Dispersion of the collective excitations in the dipolar BEC, either affected or
not affected by the uniform external electric field, is considered using our
method. We show that the evolution of polarization in the BEC leads to the
formation of a novel type of the collective excitations. Detailed description
of the dispersion of collective excitations is presented. We also consider the
process of wave generation in the polarized BEC by means of a monoenergetic
beam of neutral polarized particles. We compute the possibilities of the
generation of Bogoliubov and polarization modes by the dipole beam.Comment: 16 pages, 15 figures. arXiv admin note: substantial text overlap with
arXiv:1106.082
Nonlinear localized flat-band modes with spin-orbit coupling
We report the coexistence and properties of stable compact localized states (CLSs) and discrete solitons (DSs) for nonlinear spinor waves on a flat-band network with spin-orbit coupling (SOC). The system can be implemented by means of a binary Bose-Einstein condensate loaded in the corresponding optical lattice. In the linear limit, the SOC opens a minigap between flat and dispersive bands in the system’s band-gap structure, and preserves the existence of CLSs at the flat-band frequency, simultaneously lowering their symmetry. Adding on-site cubic nonlinearity, the CLSs persist and remain available in an exact analytical form, with frequencies that are smoothly tuned into the minigap. Inside of the minigap, the CLS and DS families are stable in narrow areas adjacent to the FB. Deep inside the semi-infinite gap, both the CLSs and DSs are stable too. ©2016 American Physical Society1991sciescopu
Two routes to the one-dimensional discrete nonpolynomial Schrodinger equation
The Bose\u2013Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schr\uf6dinger equation (NPSE). Both models are derived from the three-dimensional Gross\u2013Pitaevskii equation (3D GPE). To produce \u201cmodel 1\u201d (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. \u201cModel 2,\u201d which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2\u2014in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC
Localized gap modes in nonlinear dimerized Lieb lattices
Compact localized modes of ring type exist in many two-dimensional lattices with a flat linear band, such as the Lieb lattice. The uniform Lieb lattice is gapless, but gaps surrounding the flat band can be induced by various types of bond alternations (dimerizations) without destroying the compact linear eigenmodes. Here, we investigate the conditions under which such diffractionless modes can be formed and propagated also in the presence of a cubic on-site (Kerr) nonlinearity. For the simplest type of dimerization with a three-site unit cell, nonlinearity destroys the exact compactness, but strongly localized modes with frequencies inside the gap are still found to propagate stably for certain regimes of system parameters. By contrast, introducing a dimerization with a 12-site unit cell, compact (diffractionless) gap modes are found to exist as exact nonlinear solutions in continuation of flat band linear eigenmodes. These modes appear to be generally weakly unstable, but dynamical simulations show parameter regimes where localization would persist for propagation lengths much larger than the size of typical experimental waveguide array configurations. Our findings represent an attempt to realize conditions for full control of light propagation in photonic environments.Funding Agencies|Swedish Research Council [348-2013-6752]; Ministry of Education, Science and Technological Development of the Republic of Serbia [III 45010]</p
WAVE INTERACTIONS IN LOCALIZING MEDIA - A COIN WITH MANY FACES
A variety of heterogeneous potentials are capable of localizing linear noninteracting waves. In this work, we review different examples of heterogeneous localizing potentials which were realized in experiments. We then discuss the impact of nonlinearity induced by wave interactions, in particular, its destructive effect on the localizing properties of the heterogeneous potentials