V. V. Konotop

Nonlinear waves in PT -symmetric systems

The concept of parity-time symmetric systems is rooted in non-Hermitian quantum mechanics where complex potentials obeying this symmetry could exhibit real spectra. The concept has applications in many fields of physics, e.g., in optics, metamaterials, acoustics, Bose-Einstein condensation, electronic circuitry, etc. The inclusion of nonlinearity has led to a number of new phenomena for which no counterparts exist in traditional dissipative systems. Several examples of nonlinear parity-time symmetric systems in different physical disciplines are presented and their implications discussed.

THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

Nonlinear Random Waves

Contents: Introduction Linear Random Waves. Some Basic Results Exactly Solvable Models Direct Perturbation Methods From Inverse Scattering to Perturbative Approach Dynamical Solitons under Random Perturbations Sine-Gordon Kinks under Random Perturbations Random Wave Packets in Non-linear Media Dynamics of Randomly Modulated Solitons Waves in Non-linear Stationary Inhomogeneous Media Numerical Study of the Single-Paftiele Motion Numerical Studies: A Panoramic View Non-linear Klein-Gordon Models Simulations with Dynamical and Envelope Solitons.

Modulational instability in Bose-Einstein condensates in optical lattices

A self-consistent theory of a cylindrically shaped Bose-Einstein condensate (BEC) periodically modulated by a laser beam is presented. We show, both analytically and numerically, that modulational instability/stability isthe mechanism by which wave functions of soliton type can be generated in a cylindrically shaped BEC subject to a one-dimensional optical lattice. The theory explains why bright solitons can exist in a BEC with positive scattering length and why condensates with negative scattering length can be stable and give rise to dark solitary pulses.

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.

Nonlinear modes in finite-dimensional PT-symmetric systems.

By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schrödinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra does not imply similarity of the structure or stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. In addition, the stability is not directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use a graph representation of PT-symmetric networks allowing for a simple illustration of linearly equivalent networks and indicating their possible experimental design.

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.

Dynamics of coupled dark and bright optical solitons.

We theoretically examine the interaction between two solitons in a double-mode optical fiber. The bound state between two solitons of different modes is investigated, including both dark and bright solitons. It is shown that interaction between dark solitons as well as between bright solitons is always attractive, but the interaction between bright and dark solitons may be repulsive. Analytical results are in agreement with numerical ones.

Three-dimensional rogue waves in nonstationary parabolic potentials

Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1) -dimensional inhomogeneous nonlinear Schrödinger (NLS) equation with variable coefficients and parabolic potential to the (1+1) -dimensional NLS equation with constant coefficients. This transformation allows us to relate certain class of localized exact solutions of the (3+1) -dimensional case to the variety of solutions of integrable NLS equation of the (1+1) -dimensional case. As an example, we illustrated our technique using two lowest-order rational solutions of the NLS equation as seeding functions to obtain rogue wavelike solutions localized in three dimensions that have complicated evolution in time including interactions between two time-dependent rogue wave solutions. The obtained three-dimensional rogue wavelike solutions may raise the possibility of relative experiments and potential applications in nonlinear optics and Bose-Einstein condensates.

Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length.

Evolution of periodic matter waves in one-dimensional Bose-Einstein condensates with time-dependent scattering length is described. It is shown that variation of the effective nonlinearity is a powerful tool for controlled generation of bright and dark solitons starting with periodic waves.