Thresholds between modulational stability, rogue waves and soliton regimes in saturable nonlinear media

We study a third-order nonlinear Schrodinger equation for saturable nonlinear media, which can represent either optical pulse propagation in 1D nonlinear waveguide arrays or electron dynamics in one-dimensional lattices. Here, we describe how stable uniform solutions can evolve into regimes in which they are localized and the influence of saturation on this transition. We observed regular and chaotic-like breathing dynamics as intermediate regimes, which have their domain of existence significantly altered by the saturation parameter. Critical nonlinear strengths separating the existing regimes are shown in phase diagrams, evidencing the role played by saturation. Numerical data and analytical approach show the nonlinear strength above which uniform solutions become breather solutions increasing with the saturation parameter. In the regular breathing regime, we reveal the breathing frequency for media with saturable nonlinearity displaying faster growth as it moves away from the critical point of uniform solutions. Furthermore, critical nonlinear strength separating the regimes of regular and chaotic-like breather solutions exhibits a decreasing behavior as the saturation parameter increases. The latter ones present clear signatures of emerging rogue waves, such as peaks showing long-tailed statistics. Thresholds of this regime are increased by the saturable nonlinearity. Thus, the regime of localized solutions, which we have shown to be well-described by bright soliton-like structures, becomes less accessible with increasing saturation.