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

Abstract We study a third-order nonlinear Schr ö dinger equation for a 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. We analyze the existence conditions for such dynamical regimes, as well as the intermediate (breather and chaotic-like solutions). Since regimes are significantly changed by the saturable parameter, we show phase diagrams with critical nonlinear strengths separating such regimes. Numerical data and analytical approach show the nonlinear strength above which uniform solutions become breather solutions increasing with the saturation parameter. We also reveal chaotic-like solutions exhibiting clear signatures of emerging rogue waves, such as peaks showing long-tailed statistics. The thresholds of this regime are increased by the saturable nonlinearity. On the other hand, the regime of localized solutions, which are well-described by bright soliton-like structures, becomes less accessible with increase of saturation.