Long-living periodic solutions of complex cubic-quintic Ginzburg-Landau equation in the presence of intrapulse Raman scattering: A bifurcation and numerical study

We have found long-living periodic solutions of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) perturbed with intrapulse Raman scattering. To achieve this we have applied a model system of ordinary differential equations (SODE). A set of the fixed points of the system has been described. A complete phase portrait as well as phase portraits near the fixed points have been built for a proper choice of parameters. The behavior of the model system near the fixed points has been determined. We have presented a detailed description of the subcritical Poincaré-Andronov-Hopf bifurcation due to the intrapulse Raman scattering that appears at one of the fixed points. We have established that there appears an unstable limit cycle in the SODE. To check the validity of the obtained results from the model system we have compared them with the results of the numerical solution of the CCQGLE perturbed with intrapulse Raman scattering. There has been found a remarkable correspondence between the obtained numerical results for the amplitude and frequency of the soliton pulses and the results for these parameters of the bifurcation theory. We have observed that the numerical characteristics of the propagating solitonlike pulses-amplitude, frequency, width, and position-periodically change if we change the distance with a period determined by the bifurcation analysis.