I. V. Barashenkov

Optical solitons in PT-symmetric nonlinear couplers with gain and loss

We study spatial and temporal solitons in the

Breathers in PT-symmetric optical couplers

\mathcal{PT}

Stability and evolution of the quiescent and travelling solitonic bubbles

symmetric coupler with gain in one waveguide and loss in the other. Stability properties of the high- and low-frequency solitons are found to be completely determined by a single combination of the solitons amplitude and the gain/loss coefficient of the waveguides. The unstable perturbations of the high-frequency soliton break the symmetry between its active and lossy components which results in a blowup of the soliton or a formation of a long-lived breather state. The unstable perturbations of the low-frequency soliton separate its two components in space blocking the power drainage of the active component and cutting the power supply to the lossy one. Eventually this also leads to the blowup or breathing.

Soliton-like “bubbles” in a system of interacting bosons

We show that parity-time- (PT-) symmetric coupled optical waveguides with gain and loss support localized oscillatory structures similar to the breathers of the classical φ 4 model. The power carried by the PT breather oscillates periodically, switching back and forth between the waveguides, so that the gain and loss are compensated on the average. The breathers are found to coexist with solitons and to be prevalent in the products of the soliton collisions. We demonstrate that the evolution of a small-amplitude breather’s envelope is governed by a system of two coupled nonlinear Schr¨ odinger equations and employ this Hamiltonian system to show that small-amplitude PT breathers are stable.

Topography of attractors of the parametrically driven nonlinear Schro¨dinger equation

Abstract We simulate numerically the dynamics of the bubble-like solitons of the cubic-quintic nonlinear Schrodinger equation. In agreement with earlier predictions, slow moving bubbles have been observed to be unstable, in contrast to rapid bubbles which stabilize above a certain critical velocity. The empirical formula for the critical velocity has been confirmed to a high degree of accuracy. Based on the choice of some critical perturbation, we derive an integral criterion for stability of kinks and bubbles of the general nonlinear Schrodinger equation which provides an explanation of the appearance of the critical velocity. Finally, we follow numerically the nonlinear evolution of the unstable bubbles in one, two and three dimensions.

Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability

Abstract We study the ψ3–ψ5 nonlinear Schrodinger equation describing the boson gas with 2- and 3-body interactions. This equation is shown to possess, in 1, 2, and 3 dimensions, a new type of solitons which may be interpreted as bubbles in Bose condensate. The static “bubbles” turn out to be essentially unstable solutions.

Stability of the soliton-like “bubbles”

Abstract The parametrically driven, damped NLS equation is numerically simulated in the neighbourhood of its exact soliton solution. We obtain the attractor chart on the control parameter plane in the domain of the soliton instability. Regions of the period-doubling and quasiperiodic transitions to chaos are found, and the existence of a critical point where the two scenaria meet, is demonstrated.

Two- and three-dimensional oscillons in nonlinear faraday resonance.

Solitary waves in conservative and near-conservative systems may become unstable due to a resonance of two internal oscillation modes. We study the parametrically driven, damped nonlinear Schrequation, a prototype system exhibiting this oscillatory instability. An asymptotic multi-scale expansion is used to derive a reduced amplitude equation describing the nonlinear stage of the instability and supercritical dynamics of the soliton in the weakly dissipative case. We also derive the amplitude equation in the strongly dissipative case, when the bifurcation is of the Hopf type. The analysis of the reduced equations shows that in the undamped case the temporally periodic spatially localized structures are suppressed by the nonlinearity-induced radiation. In this case the unstable stationary soliton evolves either into a slowly decaying long- lived breather, or into a radiating soliton whose amplitude grows without bound. However, adding a small damping is sufficient to bring about a stably oscillating soliton of finite amplitude. PACS numbers: 0340K, 0545, 7530D

Nonrelativistic Cherns-Simons theory for the repulsive Bose gas

Abstract We show that the recently found bubble-like soliton solutions of D-dimensional nonlinear Schrodinger (NLS) equation are unstable for any D and that this fact does not depend on the choice of nonlinearity. For the particular case of the ψ3−αψ5 NLS equation which arises in a variety of physical contexts the bubbles growth rates are numerically calculated.

Dimer with gain and loss: Integrability and

We study 2D and 3D localized oscillating patterns in a simple model system exhibiting nonlinear Faraday resonance. The corresponding amplitude equation is shown to have exact soliton solutions which are found to be always unstable in 3D. On the contrary, the 2D solitons are shown to be stable in a certain parameter range; hence the damping and parametric driving are capable of suppressing the nonlinear blowup and dispersive decay of solitons in two dimensions. The negative feedback loop occurs via the enslaving of the solitons phase, coupled to the driver, to its amplitude and width.