Impact of higher-order effects on the stationary and pulsating localized soliton in a nonlinear and dissipative medium

Abstract

In this paper, we have investigated by numerical solution of the basic equation as well as by the dynamical model the influence of higher-order effects on the dissipative soliton mapping in the complex cubic-quintic Ginzburg–Landau equation (CCQGLE). We generated stationary and pulsating solutions, localized and delocalized from the equilibrium point. This was done using a mathematical formula developed on the basis of our finite degree of freedom dynamical model. This formula linking all the parameters of disturbance, allowed us to explain certain phenomena. It presented several aspects under the influence of the fourth-order dispersion. It was shown that long-lived pulsed solutions with relatively small fluctuations can be considered as nonlinear solutions. We found that the limit cycle that appeared was faster in the case where the initial frequency was zero and the initial chirp was non-zero. The higher-order effect that moved the trajectory away from the limit cycle was compensated by another effect (the nonlinearity compensated the dispersion). The oscillations were maintained (oscillation of the energy around a fixed value) thanks to the stable limit cycles and we obtained preserved structures over long distances. In the presence of noise, we found that the structure was more robust in the self-frequency shift case compared to the fourth-order dispersion case. The resulting pulsating solutions are distorted and frequency shifted. The self-frequency shift allowed us to modify the initial pulse by changing its amplitude, frequency and bandwidth. Intrapulse Raman scattering allowed us to have a large stability domain. All higher-order effects can influence the dynamics of the initial pulse. A comparison between the results obtained by the basic equation and the dynamical model was presented to verify its applicability. These results allow us to transmit information over long distances.