Modulational instability and pattern formation in discrete dissipative systems

Dissipative solitons in the discrete Ginzburg-Landau equation with saturable nonlinearity

The modulational instability of nonlinear plane waves and the existence of periodic and localized dissipative solitons and waves of the discrete Ginzburg-Landau equation with saturable nonlinearity are investigated. Explicit analytic expressions for periodic solutions with a zero and a finite background are derived and their stability properties investigated by means of direct numerical simulations. We find that while discrete periodic waves and solitons on a zero background are stable under time evolution, they may become modulationally unstable on finite backgrounds. The effects of a linear ramp potential on stable localized dissipative solitons are also briefly discussed.

Effects of nonlinear gradient terms on the defect turbulence regime in weakly dissipative systems

We investigate the behavior of traveling waves in a defect turbulence regime with the periodic boundary conditions by using the lowest-order complex Ginzburg-Landau equation (CGLE), and we show the effect of the nonlinear gradient terms in the system. It is found that the nonlinear gradient terms which appear at the same order as the quintic term can change the behavior of the wave patterns. The presence of the nonlinear gradient terms can cause major changes in the behavior of the solution. They can be considered like the stabilizing terms. The system which was initially unstable or chaotic can become stable by including the nonlinear gradient terms.

Modulational instability of a trapped Bose-Einstein condensate with two- and three-body interactions

We investigate analytically and numerically the modulational instability of a Bose-Einstein condensate with both two- and three-body interatomic interactions and trapped in an external parabolic potential. Analytical investigations performed lead us to establish an explicit time-dependent criterion for the modulational instability of the condensate. The effects of the potential as well as of the quintic nonlinear interaction are studied. Direct numerical simulations of the Gross-Pitaevskii equation with two- and three-body interactions describing the dynamics of the condensate agree with the analytical predictions.

How does an inclined holding beam affect discrete modulational instability and solitons in nonlinear cavities?

We study light propagation in arrays of weakly coupled nonlinear cavities driven by an inclined holding beam.We show analytically that both discreteness and inclination of the driving field can dramatically change the conditions for modulational instability in discrete nonlinear systems. We find numerically the families of resting and moving dissipative solitons for an arbitrary inclination angle of the driving field, both in the discrete and a quasi-continuous limits. We analyze a crossover between resting and moving cavity solitons, and also observe novel features in the soliton collision.