Multi-channel pulse dynamics in a stabilized Ginzburg-Landau system

We study the stability and interactions of chirped solitary pulses in a system of nonlinearly coupled cubic Ginzburg-Landau (CGL) equations with a group-velocity mismatch between them, where each CGL equation is stabilized by linearly coupling it to an additional linear dissipative equation. In the context of nonlinear fiber optics, the model describes transmission and collisions of pulses at different wavelengths in a dual-core fiber, in which the active core is furnished with bandwidth-limited gain, while the other, passive (lossy) one is necessary for stabilization of the solitary pulses. Complete and incomplete collisions of pulses in two channels in the cases of anomalous and normal dispersion in the active core are analyzed by means of perturbation theory and direct numerical simulations. It is demonstrated that the model may readily support fully stable pulses whose collisions are quasi-elastic, provided that the group-velocity difference between the two channels exceeds a critical value. In the case of quasi-elastic collisions, the temporal shift of pulses, predicted by the analytical approach, is in semi-quantitative agrement with direct numerical results in the case of anomalous dispersion (in the opposite case, the perturbation theory does not apply). We also consider a simultaneous collision between pulses in three channels, concluding that this collision remains quasi-elastic, and the pulses remain completely stable. Thus, the model may be a starting point for the design of a stabilized wavelength-division-multiplexed (WDM) transmission system.Comment: a text file in the revtex4 format, and 16 pdf files with figures. Physical Review E, in pres

Static and rotating domain-wall crosses in Bose-Einstein condensates

For a Bose-Einstein condensate (BEC) in a two-dimensional (2D) trap, we introduce cross patterns, which are generated by intersection of two domain walls (DWs) separating immiscible species, with opposite signs of the wave functions in each pair of sectors filled by the same species. The cross pattern remains stable up to the zero value of the immiscibility parameter $|\Delta |$, while simpler rectilinear (quasi-1D) DWs exist only for values of $|\Delta |$ essentially exceeding those in BEC mixtures (two spin states of the same isotope) currently available to the experiment. Both symmetric and asymmetric cross configurations are investigated, with equal or different numbers $N_{1,2}$ of atoms in the two species. In rotating traps, ``propellers'' (stable revolving crosses) are found too. A full stability region for of the crosses and propellers in the system's parameter space is identified, unstable crosses evolving into arrays of vortex-antivortex pairs. Stable rotating rectilinear DWs are found too, at larger vlues of $|\Delta |$. All the patterns produced by the intersection of three or more DWs are unstable, rearranging themselves into ones with two DWs. Optical propellers are also predicted in a twisted nonlinear photonic-crystal fiber carrying two different wavelengths or circular polarizations, which can be used for applications to switching and routing.Comment: 9 pages, 10 figures, Phys. Rev. A (in press

Soliton Dynamics in Linearly Coupled Discrete Nonlinear Schr\"odinger Equations

We study soliton dynamics in a system of two linearly coupled discrete nonlinear Schr\"odinger equations, which describe the dynamics of a two-component Bose gas, coupled by an electromagnetic field, and confined in a strong optical lattice. When the nonlinear coupling strengths are equal, we use a unitary transformation to remove the linear coupling terms, and show that the existing soliton solutions oscillate from one species to the other. When the nonlinear coupling strengths are different, the soliton dynamics is numerically investigated and the findings are compared to the results of an effective two-mode model. The case of two linearly coupled Ablowitz-Ladik equations is also investigated.Comment: to be published in Mathematics and Computers in Simulation, proceedings of the fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory (Athens, Georgia - April 2007

Dynamical oscillations in nonlinear optical media

The spatial dynamics of pulses in Kerr media with parabolic index profile are examined. It is found that when diffraction and graded-index have opposite signs propagating pulses exhibit an oscillatory pattern, similar to a breathing behavior. Furthermore, if the pulse and the index profile are not aligned the pulse oscillates around the index origin with frequency that depends on the values of the diffraction and index of refraction. These oscillations are not observed when diffraction and graded-index share the same sign

Linearly Coupled Bose-Einstein Condensates: From Rabi Oscillations and Quasi-Periodic Solutions to Oscillating Domain Walls and Spiral Waves

In this paper, an exact unitary transformation is examined that allows for the construction of solutions of coupled nonlinear Schr{\"o}dinger equations with additional linear field coupling, from solutions of the problem where this linear coupling is absent. The most general case where the transformation is applicable is identified. We then focus on the most important special case, namely the well-known Manakov system, which is known to be relevant for applications in Bose-Einstein condensates consisting of different hyperfine states of $^{87}$Rb. In essence, the transformation constitutes a distributed, nonlinear as well as multi-component generalization of the Rabi oscillations between two-level atomic systems. It is used here to derive a host of periodic and quasi-periodic solutions including temporally oscillating domain walls and spiral waves.Comment: 6 pages, 4 figures, Phys. Rev. A (in press

Families of Matter-Waves for Two-Component Bose-Einstein Condensates

We produce several families of solutions for two-component nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in the one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to Bose-Einstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs).Comment: 6 pages, 4 figures, EPJD in pres