Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation

Abstract

Stationary to pulsating soliton bifurcation analysis of the complex Ginzburg-Landau equation (CGLE) is presented. The analysis is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Stationary solitons, with constant amplitude and width, are associated with fixed points in the model. For the first time, pulsating solitons are shown to be stable limit cycles in the finite-dimensional dynamical system. The boundaries between the two types of solutions are obtained approximately from the reduced model. These boundaries are reasonably close to those predicted by direct numerical simulations of the CGLE.

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.…”

“…Remarkable phenomena are also exhibited by its higher-order variants, emerging in a diverse spectrum of applications, such as nonlinear optics [3], nonlinear metamaterials [4], and water waves in finite depth [5][6][7]. On the other hand, dissipative variants of NLS models incorporating gain and loss have also been used in optics [8], e.g., in the physics of mode-locked lasers [9,10] (see also the relevant works [11,12]) and polariton superfluids [13] -see, e.g., Ref. [14] for various applications.…”

“…Note that higherorder versions of the CGL equation, have only recently started attracting attention [27], while extended secondorder CGL models have been extensively studied in various contexts previously [8,14,15]. In particular, we refer to the pioneering work [10], followed by the important contributions [11,12], which revealed the existence of the aforementioned low-dimensional dynamical scenarios for second-order quintic CGL models. The results of [10][11][12] were established by numerical and even analytical reductions to suitable finite dimensional dynamical systems, capturing the long-time dynamics of the original infinite dimensional one.…”

Transitional Behaviors of Cqgle Solitons Across Boundaries on a Phase Plane

“…Bifurcation phenomena have been observed and discussed [13][14][15]. In [13], an experiment was set up to verify the prediction with the CQGLE.…”

“…As the controlling parameters are changed continuously on a parametric plane, the solution may transit from pulsating to period-doubling, period-quadrupling, and eventually to chaotic type. In [15], different parametric planes were used to categorize solitons obtained by solving the CQGLE, and boundaries dividing different types of solitons were drawn [14].…”