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V V Konotop

Publications and source records attributed to V V Konotop.

12 recordsLinked to original sources
PubMed2006

Nature of the intrinsic relation between Bloch-band tunneling and modulational instability.

In an example of Bose-Einstein condensates embedded in two-dimensional optical lattices, we show that in nonlinear periodic systems modulational instability and interband tunneling are intrinsically related phenomena. By direct numerical simulations we find that tunneling results in attenuation or enhancement of instability. On the other hand, instability results in asymmetric nonlinear tunneling. The effect strongly depends on the band gap structure and it is especially significant in the case of the resonant tunneling. The symmetry of the coherent structures emerging from the instability reflects the symmetry of both the stable and the unstable states between which the tunneling occurs. Our results provide evidence of the profound effect of the band structure on the superfluid-insulator transition.

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PubMed2005

Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations: application to two-component Bose-Einstein condensates.

The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schrödinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.

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PubMed2004

Two-component Bose-Einstein condensates in periodic potential.

Coupled nonlinear Schrödinger (CNLS) equations with an external elliptic function potential model with high accuracy a quasi-one-dimensional interacting two-component Bose-Einstein condensate (BEC) trapped in a standing wave generated by a few laser beams. The construction of stationary solutions of the two-component CNLS equation with a periodic potential is detailed and their stability properties are studied by direct numerical simulations. Some of these solutions allow reduction to the Manakov system. From a physical point of view the trivial phase solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against weak modulations of long wavelength. By numerical simulations it is shown that the modulationally unstable solutions lead to the formation of localized ground states of the coupled BEC system.

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PubMed2004

Nonlinearity from geometric interactions: a case example.

We propose a ladder model wherein dynamical nonlinearity arises from geometry. It includes two strings of particles which are set along rigid rails of a "railroad" and coupled by linear springs. Physical realizations of the model include dust-particle strings in plasma sheaths and chains of microparticles trapped in a strong optical lattice. The transverse couplings between the strings, along with the motion constraint imposed by the rails, generate nonlinearity. It gives rise to robust solitary waves, which are found analytically in the long-wavelength limit, and are obtained in simulations of the full system.

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PubMed2003

Bohr-Sommerfeld quantization condition for the Gross-Pitaevskii equation.

The discrete spectrum of the nonlinear eigenvalue problem associated to the one-dimensional Gross-Pitaevskii equation with a smooth potential is studied in the quasiclassical limit. We particularly focus on the corrections to the Bohr-Sommerfeld quantization rule for the excited energy levels due to the nonlinearity. Explicit predictions are obtained analytically for these corrections and are supported by numerical computations.

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PubMed2002

Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential.

In the present paper we use the Wannier function basis to construct lattice approximations of the nonlinear Schrödinger equation with a periodic potential. We show that the nonlinear Schrödinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding approximation, i.e., the approximation when nearest neighbor interactions are dominant. The results are relevant to the Bose-Einstein condensate theory as well as to other physical systems, such as, for example, electromagnetic wave propagation in nonlinear photonic crystals.

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PubMed2002

Bright compact breathers.

In this communication we will consider the potential of some general classes of nonlinear lattice models to support bright discrete compact breather solutions (compactlets). We analyze the conditions for which such solutions are possible and classify the models as belonging in three general categories: a class with no compact breather solutions, one with one-parameter families of solutions, and a class with "isolated" solutions (i.e., no free parameters). In the latter two cases we construct the solutions and analyze their linear stability. The drastically different stability features of these solutions in comparison with their smoothly decaying counterparts are discussed. Stable breather solutions with compact support are identified in the one-parameter families of solutions, while the corresponding solutions found in the zero-parameter families are always found to be unstable.

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PubMed2001

Nonlinear modulation of multidimensional lattice waves.

The equations governing weakly nonlinear modulations of N-dimensional lattices are considered using a quasidiscrete multiple-scale approach. It is found that the evolution of a short wave packet for a lattice system with cubic and quartic interatomic potentials is governed by the generalized Davey-Stewartson (GDS) equations, which include mean motion induced by the oscillatory wave packet through cubic interatomic interaction. The GDS equations derived here are more general than those known in the theory of water waves because of the anisotropy inherent in lattices. The generalized Kadomtsev-Petviashvili equations describing the evolution of long-wavelength acoustic modes in two- and three-dimensional lattices are also presented. Then the modulational instability of an N-dimensional Stokes lattice wave is discussed based on the N-dimensional GDS equations obtained. Finally, the one- and two-soliton solutions of two-dimensional GDS equations are provided by means of Hirota's bilinear transformation method.

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PubMed2001

Quantization of weakly nonlinear lattices: envelope solitons.

A method of quantizing weakly nonlinear lattices is proposed. It is based on introducing "pseudofield" operators. In this formalism quantum envelope solitons together with phonons are regarded as elementary quasiparticles making up a boson gas. In the classical limit the excitations corresponding to frequencies above a linear cutoff frequency are reduced to conventional envelope solitons. The approach allows one to identify a quantum soliton that is localized in space and to understand the existence of a narrow soliton frequency band.

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PubMed1999

Stationary dark localized modes: discrete nonlinear Schrödinger equations.

Various kinds of stationary dark localized modes in discrete nonlinear Schrödinger equations are considered. A criterion for the existence of such excitations is introduced and an estimation of a localization region is provided. The results are illustrated in examples of the deformable discrete nonlinear Schrödinger equation, of the model of Frenkel excitons in a chain of two-level atoms, and of the model of a one-dimensional Heisenberg ferromagnetic in the stationary phase approximation. The three models display essentially different properties. It is shown that at an arbitrary amplitude of the background it is impossible to reach strong localization of dark modes. In the meantime, in the model of Frenkel excitons, exact dark compacton solutions are found.

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PubMed1999

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.

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