Dmitry A. Zezyulin

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.

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.

Stability of solitons in -symmetric nonlinear potentials

We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias ) symmetric potential. We are particularly focusing on the case where the spatially dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and tanh-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, which suggests that the relation between width of the modes and spatial size of the complex potential defines the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.

Discrete solitons in PT-symmetric lattices

We prove the existence of discrete solitons in infinite parity-time () symmetric lattices by means of analytical continuation from the anticontinuum limit. The energy balance between dissipation and gain implies that in the anticontinuum limit the solitons are constructed from elementary -symmetric blocks such as dimers, quadrimers, or more general oligomers. We consider in detail a chain of coupled dimers, analyze bifurcations of discrete solitons from the anticontinuum limit and show that the solitons are stable in a sufficiently large region of the lattice parameters. The generalization of the approach is illustrated on two examples of networks of quadrimers, for which stable discrete solitons are also found.

Discrete solitons in \chem {\cal PT} -symmetric lattices

We prove the existence of discrete solitons in infinite parity-time () symmetric lattices by means of analytical continuation from the anticontinuum limit. The energy balance between dissipation and gain implies that in the anticontinuum limit the solitons are constructed from elementary -symmetric blocks such as dimers, quadrimers, or more general oligomers. We consider in detail a chain of coupled dimers, analyze bifurcations of discrete solitons from the anticontinuum limit and show that the solitons are stable in a sufficiently large region of the lattice parameters. The generalization of the approach is illustrated on two examples of networks of quadrimers, for which stable discrete solitons are also found.

Stability of localized modes in PT-symmetric nonlinear potentials

We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias ) symmetric potential. We are particularly focusing on the case where the spatially dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and tanh-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, which suggests that the relation between width of the modes and spatial size of the complex potential defines the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.

Solitons in a medium with linear dissipation and localized gain

We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system possesses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, existence of modes in focusing media at negative propagation constant values, simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions.

Nonlinear modes for the Gross–Pitaevskii equation—a demonstrative computation approach

A method for the study of steady-state nonlinear modes for Gross-Pitaevskii equation (GPE) is described. It is based on exact statement about coding of the steady-state solutions of GPE which vanish as

Nonlinear modes in a generalized -symmetric discrete nonlinear Schrödinger equation

x\to+\infty

Giant amplification of modes in parity-time symmetric waveguides

by reals. This allows to fulfill {\it demonstrative computation} of nonlinear modes of GPE i.e. the computation which allows to guarantee that {\it all} nonlinear modes within a given range of parameters have been found. The method has been applied to GPE with quadratic and double-well potential, for both, repulsive and attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these cases are represented. The stability of these modes has been discussed.A method for the study of steady-state nonlinear modes for the Gross–Pitaevskii equation (GPE) is described. It is based on the exact statement about the coding of the steady-state solutions of GPE which vanish as x → +∞ by reals. This allows us to fulfil the demonstrative computation of nonlinear modes of GPE, i.e. the computation which allows us to guarantee that all nonlinear modes within a given range of parameters have been found. The method has been applied to GPE with quadratic and double-well potentials, for both repulsive and attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these cases are represented. The stability of these modes has been discussed.