We introduce a vector form of the cubic complex Ginzburg-Landau equation describing the dynamics of dissipative solitons in the two-component helicoidal spin-orbit coupled open Bose-Einstein condensates (BECs), where the addition of dissipative interactions is done through coupled rate equations. Furthermore, the standard linear stability analysis is used to investigate theoretically the stability of continuous-wave (cw) solutions and to obtain an expression for the modulational instability gain spectrum. Using direct simulations of the Fourier space, we numerically investigate the dynamics of the modulational instability in the presence of helicoidal spin-orbit coupling. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold for amplitude perturbations.

We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schrödinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

The modulational instability (MI) phenomenon is addressed in a nonlocal medium under controllable saturation. The linear stability analysis of a plane-wave solution is used to derive an expression for the growth rate of MI that is exploited to parametrically discuss the possibility for the plane wave to disintegrate into nonlinear localized light patterns. The influence of the nonlocal parameter, the saturation coefficient, and the saturation index are mainly explored in the context of a Gaussian nonlocal response. It is pointed out that the instability spectrum, which tends to be quenched by the high nonlocality parameter, gets amplified under the right choices of the saturation parameters, especially the saturation index. Via direct numerical simulations, confirmations of analytical predictions are given, where competing nonlocal and saturable nonlinearities enable the emergence of trains of patterns as manifestations of MI. The comprehensive parametric analysis carried out throughout the numerical experiment reveals the robustness of the obtained rogue waves of A- and B-type Akhmediev breathers, as the nonlinear signature of MI, providing the saturation index as a suitable tool to manipulate nonlinear waves in nonlocal media.