Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation.

Finite-dimensional dynamical models for solitons of the cubic-quintic complex Ginzburg-Landau equation (CGLE) are derived. The models describe the evolution of the pulse parameters, such as the maximum amplitude, pulse width, and chirp. A clear correspondence between attractors of the finite-dimensional dynamical systems and localized waves of the continuous dissipative system is demonstrated. It is shown that stationary solitons of the CGLE correspond to fixed points, while pulsating solitons are associated with stable limit cycles. The models show that a transformation from a stationary soliton to a pulsating soliton is the result of a Hopf bifurcation in the reduced dynamical system. The appearance of moving fronts (kinks) in the CGLE is related to the loss of stability of the limit cycles. Bifurcation boundaries and pulse behavior in the regions between the boundaries, for a wide range of system parameters, are found from analysis of the reduced dynamical models. We also provide a comparison between various models and their correspondence to the exact results.