Nonlinear excitations in arrays of Bose-Einstein condensates

The dynamics of localized excitations in array of Bose-Einstein condensates is investigated in the framework of the nonlinear lattice theory. The existence of temporarily stable ground states displaying an atomic population distributions localized on very few lattice sites (intrinsic localized modes), as well as, of atomic population distributions involving many lattice sites (envelope solitons), is studied both numerically and analytically. The origin and properties of these modes are shown to be inherently connected with the interplay between macroscopic quantum tunnelling and nonlinearity induced self-trapping of atoms in coupled BECs. The phenomenon of Bloch oscillations of these excitations is studied both for zero and non zero backgrounds. We find that in a definite range of parameters, homogeneous distributions can become modulationally unstable. We also show that bright solitons and excitations of shock wave type can exist in BEC arrays even in the case of positive scattering length. Finally, we argue that BEC array with negative scattering length in presence of linear potentials can display collapse.Comment: Submitted to Phys. Rev.

Three-wave interaction in two-component quadratic nonlinear lattices

We investigate a two-component lattice with a quadratic nonlinearity and find with the multiple scale technique that integrable three-wave interaction takes place between plane wave solutions when these fulfill resonance conditions. We demonstrate that energy conversion and pulse propagation known from three-wave interaction is reproduced in the lattice and that exact phase matching of parametric processes can be obtained in non-phase-matched lattices by tilting the interacting plane waves with respect to each other.info:eu-repo/semantics/publishedVersio

Spatial optical solitons in nonlinear photonic crystals

We study spatial optical solitons in a one-dimensional nonlinear photonic crystal created by an array of thin-film nonlinear waveguides, the so-called Dirac-comb nonlinear lattice. We analyze modulational instability of the extended Bloch-wave modes and also investigate the existence and stability of bright, dark, and ``twisted'' spatially localized modes in such periodic structures. Additionally, we discuss both similarities and differences of our general results with the simplified models of nonlinear periodic media described by the discrete nonlinear Schrodinger equation, derived in the tight-binding approximation, and the coupled-mode theory, valid for shallow periodic modulations of the optical refractive index.Comment: 15 pages, 21 figure

Dark-in-Bright Solitons in Bose-Einstein Condensates with Attractive Interactions

We demonstrate a possibility to generate localized states in effectively one-dimensional Bose-Einstein condensates with a negative scattering length in the form of a dark soliton in the presence of an optical lattice (OL) and/or a parabolic magnetic trap. We connect such structures with twisted localized modes (TLMs) that were previously found in the discrete nonlinear Schr{\"o}dinger equation. Families of these structures are found as functions of the OL strength, tightness of the magnetic trap, and chemical potential, and their stability regions are identified. Stable bound states of two TLMs are also found. In the case when the TLMs are unstable, their evolution is investigated by means of direct simulations, demonstrating that they transform into large-amplitude fundamental solitons. An analytical approach is also developed, showing that two or several fundamental solitons, with the phase shift $\pi$ between adjacent ones, may form stable bound states, with parameters quite close to those of the TLMs revealed by simulations. TLM structures are found numerically and explained analytically also in the case when the OL is absent, the condensate being confined only by the magnetic trap.Comment: 13 pages, 7 figures, New Journal of Physics (in press

Parametric localized modes in quadratic nonlinear photonic structures

We analyze two-color spatially localized modes formed by parametrically coupled fundamental and second-harmonic fields excited at quadratic (or chi-2) nonlinear interfaces embedded into a linear layered structure --- a quasi-one-dimensional quadratic nonlinear photonic crystal. For a periodic lattice of nonlinear interfaces, we derive an effective discrete model for the amplitudes of the fundamental and second-harmonic waves at the interfaces (the so-called discrete chi-2 equations), and find, numerically and analytically, the spatially localized solutions --- discrete gap solitons. For a single nonlinear interface in a linear superlattice, we study the properties of two-color localized modes, and describe both similarities and differences with quadratic solitons in homogeneous media.Comment: 9 pages, 8 figure

Solitons in Triangular and Honeycomb Dynamical Lattices with the Cubic Nonlinearity

We study the existence and stability of localized states in the discrete nonlinear Schr{\"o}dinger equation (DNLS) on two-dimensional non-square lattices. The model includes both the nearest-neighbor and long-range interactions. For the fundamental strongly localized soliton, the results depend on the coordination number, i.e., on the particular type of the lattice. The long-range interactions additionally destabilize the discrete soliton, or make it more stable, if the sign of the interaction is, respectively, the same as or opposite to the sign of the short-range interaction. We also explore more complicated solutions, such as twisted localized modes (TLM's) and solutions carrying multiple topological charge (vortices) that are specific to the triangular and honeycomb lattices. In the cases when such vortices are unstable, direct simulations demonstrate that they turn into zero-vorticity fundamental solitons.Comment: 17 pages, 13 figures, Phys. Rev.

Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition

Oscillatory instabilities in Hamiltonian anharmonic lattices are known to appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions of multibreather type. Here, we analyze the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear: the three-site discrete nonlinear Schr\"odinger model with periodic boundary conditions. The stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes. We numerically analyze the nature of the two bifurcations leading to this instability and find them to be of two different types. Close to the lower-amplitude threshold stable two-frequency quasiperiodic solutions exist surrounding the unstable stationary solution, and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small. By contrast, close to the higher-amplitude threshold all two-frequency quasiperiodic solutions are detached from the unstable stationary solution, and the resulting dynamics is of 'population-inversion' type involving also the originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen. Revised and shortened version with few clarifying remarks adde

The use of descriptive statistics for the characteristics of the empirical samples macroeconomic indicators

In the article, the statistical tests of the empirical distributions of economic indicators of the Russian Federation in 2011 (average monthly nominal accrued salary, the unemployment rate), the unemployment rate of the countries of the European Union in 2011, on conformity to their normal distribution law (distribution fitting) are made. On the basis of the calculation of the statistical criterion of the Kolmogorov-Smirnov, (K-S test) the distributions being not normal distributions is proved. Therefore, for their characteristics it is inappropriate to use the arithmetic mean as indicators of economic status and development. For the selected empirical distributions of the calculations of such indicators descriptive statistics, as the median, the lower and upper quartiles, interquartile range are conducted, the advantages of graphical representation of the data in the form of a chart «box and whisker» are shown. On the example of the two Russian districts of the Southern Federal district and the Volga Federal district, the comparison of the average monthly salary in 2010 with the help of nonparametric test Mann-Whitney U test is carried out, the difference in the interpretation of data compared with the use of simple averages is shown. The results of the performed calculations can be used for the analysis of economic indicators average monthly salary and the unemployment rate in 83 subjects of the Russian Federation in 2011, the unemployment rate in the EU countries in 2011, as well as the methodology of statistical analysis of the different empirical distribution economic indicators