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

Abstract

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.

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

“…More specifically, we should point out that the thirdorder cubic CGL model that we consider herein is essentially different from the second-order cubic-quintic model discussed in [10][11][12], not only from a mathematical but also from a physical point of view: indeed, in the context of optics, the model considered in the latter works refers to propagation of short pulses, in the picosecond regime, in media featuring saturation of the nonlinear refractive index, while the model we consider here is relevant to propagation of ultrashort pulses in the sub-picosecond or femtosecond regimes [3]. For this reason, our model includes third-order dispersion and higher-order nonlinear effects, that appear naturally as higher-order corrections of the usual NLS model in the framework of the reductive perturbation method.…”

Low-Dimensional Description of Pulses under the Action of Global Feedback Control

“…There exist several methods of a reduction of an original equation to a finite set of ordinary differential equations (ODE). One of these methods utilizes the integrals of energy, momentum and a finite number of higher-order generalized moments, and it has been applied for the analysis of the dynamics of localized waves of a cubic-quintic complex Ginzburg-Landau equation (CGLE) [25]. Another method is based on a modified variational technique.…”

Semi-Analytic Theory of Similariton Amplifiers and Laser Oscillators Using a Shape-Adaptive Model Pulse

“…Furthermore, A, T , β , α, and ϕ denote the amplitude, pulse duration, linear and third order chirp parameter, and phase, respectively. Applying the method of moments [4] to the governing propagation equation, evolution equations can be formulated for the pulse parameters, which are taken to be functions of the position z. Specifically, an evolution equation is set up for n (z), allowing to monitor the change of the pulse shape during propagation.…”

“…Thus, semi-analytic approaches have been employed, yielding evolution equations for the characteristic pulse parameters, such as the pulse energy and duration. However, such approaches typically rely on fixed pulse shapes such as Gaussian or sech pulses [4][5][6], apart from very few exceptions [7].…”