Impact of spectral filtering on the stability of a stationary dissipative soliton in the complex cubic–quintic Ginzburg–Landau equation in the presence of higher-order effects

Abstract

In this work, we investigate the influence of linear and nonlinear effects on the stability of stationary solitons in the framework of the Complex Cubic–Quintic Ginzburg–Landau Equation (CCQGLE). In this regard, analytical and numerical results of the stationary soliton revealed the existence of two types of pulses in the stationary state for the given parameters. These pulses in the parameter space propagate with different amplitudes and frequencies. When the amplitude of a pulse increases (resp. decreases), its frequency decreases (resp. increases). In the first pulse (first fixed point), we obtained several dynamics (source, circle and sink) and we showed that there is a boundary between the source and sink dynamics. This boundary is the dynamics of an unstable limit cycle (subcritical bifurcation) representing the bifurcation in the sense of Poincaré–Andronov–Hopf (P. A. H). The second pulse (second fixed point), showed us a single dynamic called saddle. When we take a coordinate in the parameter space corresponding to a dynamics, we find that the structure of the soliton is preserved in the case of a sink, and is destroyed at a certain distance in the other cases. This proves that the soliton envelope is stable in the sink dynamics and unstable in the other dynamics (source, saddle and circle). We also plotted the bifurcation parameters as a function of the gain dispersion. This allowed us to highlight the areas corresponding to each dynamic. Direct numerical simulations of the perturbed CCQGLE confirmed our theoretical predictions. We observed the robustness of the stable soliton by superimposing the noise on the initial pulse and found that the soliton structure is preserved up to a certain limit. These results can allow us to transmit information over long distances.