Cubic-quintic Complex Ginzburg-Landau Equation Research Articles

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.

Stability of few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation

This letter explores stable few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation. Using a (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation, the letter explores solitonic behaviors of light bullets in materials with a negative refractive index under the few-cycle regime. The analysis involves a Lagrangian variational approach designed for dissipative nonlinear systems and utilizes direct numerical simulations to demonstrate the possibility of stable light bullets forming in metamaterials and corroborate the stability analysis. Based on the stability conditions for the system parameters using the Routh-Hurwitz criterion, optimal values of the cubic and quintic self-steepening effects promote stable propagation of light bullets, while an imbalance may lead to collapse. These findings suggest that combining few-cycle light bullets with metamaterial-based optics may enable new possibilities for ultrafast and nonlinear optics.

Propagation dynamics of multipole solitons generated in dissipative systems

The propagation dynamics of multipole solitons generated in dissipative systems are investigated numerically based on the fractional complex cubic-quintic Ginzburg–Landau equation using the Airy beam as the input beam. The effect of different parameter values on the generation of stable solitons is explored. In addition, we observe different resultant domains of the input beam evolving in the linear loss coefficient or cubic gain coefficient and Lévy index parameter planes. The results show that the evolution can lead to the formation of stable multipole solitons. It is also demonstrated that two solitons merge to form a single soliton. And, the relation between the merger distance and the initial amplitude is given.

Exact solutions to the fractional complex Ginzburg-Landau equation with cubic-quintic and Kerr law nonlinearities

The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved (G′/G2) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.

Exact periodic solution family of the complex cubic-quintic Ginzburg–Landau equation with intrapulse Raman scattering

Exact periodic solution family of the complex cubic-quintic Ginzburg–Landau equation with intrapulse Raman scattering

Analytic Solutions of 2D Cubic Quintic Complex Ginzburg-Landau Equation

The dynamical behaviour of traveling waves in a class of two-dimensional system whose amplitude obeys the two-dimensional complex cubic-quintic Ginzburg-Landau equation is deeply studied as a function of parameters near a subcritical bifurcation. Then, the bifurcation method is used to predict the nature of solutions of the considered wave equation. It is applied to reduce the two-dimensional complex cubic-quintic Ginzburg-Landau equation to the quintic nonlinear ordinary differential equation, easily solvable. Under some constraints of parameters, equilibrium points are obtained and phase portraits have been plotted. The particularity of these phase portraits obtained for new ordinary differential equation is the existence of homoclinic or heteroclinic orbits depending on the nature of equilibrium points. For some parameters, one has the orbits starting to one fixed point and passing through another fixed point before returning to the same fixed point, predicting then the existence of the combination of a pair of pulse-dark soliton. One has also for other parameters, the orbits linking three equilibrium points predicting the existence of a dark soliton pair. These results are very important and can predict the same solutions in many domains, particularly in wave phenomena, mechanical systems, or laterally heated fluid layers. Moreover, depending on the values of parameter systems, the analytical expression of the solutions predicted is found. The three-dimensional graphs of these solutions are plotted as well as their 2D plots in the propagation direction.

W-chirped solitons and modulated waves patterns in parabolic law medium with anti-cubic nonlinearity

In this work, investigation of the W-chirped solitons and modulation instability are addressed in nonlinear optic fibers where cubic-quintic and parabolic law parameters are used. For this reason, the modified cubic-quintic complex Ginzburg–Landau equation with parabolic law nonlinearity is employed. Using the sub-ODE method, the chirp component and combined chirped soliton solutions, including rational soliton solutions, are obtained. Using specific parameters of the nonlinear structure, one can identify dark and bright solitons. For a suitable constant parameter of the chirp component, the W-chirped bright as well as the combined bright-dark solitons are displayed via the 3D plots. The linear stability analysis of the plane wave is used to derive an expression of the modulation instability spectrum that is employed to show the effects of the nonlinear parameters in normal and anomalous dispersion regimes. It is pointed out that both cubic and quintic parameters can enlarge or reduce the bandwidth of the modulation instability. Through the numerical simulation of the continuous wave, it becomes apparent that for suitable optical fiber parameter values, modulated wave objects are formed to exhibit mild oscillations. For specific wavenumber excitation, another modulated wave object is depicted to show how the unstable mode can switch to the stable mode for the long-time evolution of the continuous wave. The obtained results could certainly open a new future for signal treatment in optical fiber, where parabolic law is considered.

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

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.

(Invited) Kink solutions of the complex cubic– quintic Ginzburg-Landau equation in the presence of intrapulse Raman scattering

We present a study of solutions of the complex cubic–quintic Ginzburg-Landau equation (CCQGLE) and CCQGLE perturbed with the term responsible for the intrapulse Raman scattering (IRS). In both cases, using the method of first-order autonomous sub-equations, novel exact kink and antikink solutions have been identified. These solutions are represented in explicit analytic form. The numerical investigation of the influence of the IRS on the shape, amplitude, and velocity of the right side of the observed "front" CCQGLE solution revealed a good correlation with the identical features predicted by the newly discovered exact kink solution.

The modulation instability of shallow wake flows based on the higher-order generalized cubic-quintic complex Ginzburg–Landau equation

In the context of the parallel flow hypothesis, we derive a higher-order generalized cubic-quintic complex Ginzburg–Landau (GCQ-CGL) equation to describe the amplitude evolution of shallow wake flow from the dimensionless shallow water equations by using multi-scale analysis, perturbation expansion, and weak nonlinear theory. The evolution model includes not only the slowly changing envelope approximation but also the influence of higher-order dissipation, dispersion, and cubic and quintic nonlinear effects. We give the analytical solution of the higher-order GCQ-CGL equation based on the ansatz and coordinate transformation methods, and we discuss the influence of the higher-order dissipation coefficient on the amplitude and frequency of the wake flow by means of three-dimensional diagrams, contour maps, and plane graphs. The subsequent linear stability analysis gives a theoretical basis for the modulation instability (MI) of plane waves, and the linear theory predicts the instability of any amplitude of the main waves. Finally, we focus on the MI of shallow wake flows. Results show that the MI gain function is internally related to the background wave number, disturbance wave number, background amplitude, disturbance expansion parameter, and dissipation coefficient. The area of the MI decreases as the higher-order dissipation coefficient decreases.

Propagation characteristics of bright and mixed solitons based on the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation

In the study of telecommunication system, the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation is used as the optical solitons transmission model, which not only explains the physical meaning of the existing model with quintic terms, but also has more nonlinear dynamics characteristics of the higher dimensional system than the lower dimensional system. In this paper, the analytical soliton solutions of the (3+1)-dimensional cubic-quintic CGL equations with variable coefficients are obtained by using the modified Hirota method. By selecting certain parameters of the nonlinear coefficients and spectral filtering terms, a special kind of mixed soliton solution is obtained, which has the characteristics of bright soliton, dark soliton and kinked soliton at the same time. Subsequently, the influence of changing the nonlinear, spectral filtering, linear loss parameters and other parameters on the transmission characteristics of solitons is discussed respectively, so as to realize the control of optical solitons, which can not only control the propagation of optical solitons in different forms, but also can realize the adjustment of the amplitude and pulse width of the pulse and control the propagation direction and energy of the pulse for the mixed solitons of a particular form. The research results of high dimensional CGL system in this paper can be applied to nonlinear optical system, ultra-fast optical digital logic system and other different experiments and application fields.

Linear Stability Analysis of Solitons Governed by the 2D Complex Cubic-Quintic Ginzburg-Landau Equation

Linear Stability Analysis of Solitons Governed by the 2D Complex Cubic-Quintic Ginzburg-Landau Equation

Impact of spectral filtering on the stability of a stationary dissipative soliton in the complex cubic–quintic Ginzburg–Landau equation in the presence of higher-order effects

In this work, we investigate the influence of linear and nonlinear effects on the stability of stationary solitons in the framework of the Complex Cubic–Quintic Ginzburg–Landau Equation (CCQGLE). In this regard, analytical and numerical results of the stationary soliton revealed the existence of two types of pulses in the stationary state for the given parameters. These pulses in the parameter space propagate with different amplitudes and frequencies. When the amplitude of a pulse increases (resp. decreases), its frequency decreases (resp. increases). In the first pulse (first fixed point), we obtained several dynamics (source, circle and sink) and we showed that there is a boundary between the source and sink dynamics. This boundary is the dynamics of an unstable limit cycle (subcritical bifurcation) representing the bifurcation in the sense of Poincaré–Andronov–Hopf (P. A. H). The second pulse (second fixed point), showed us a single dynamic called saddle. When we take a coordinate in the parameter space corresponding to a dynamics, we find that the structure of the soliton is preserved in the case of a sink, and is destroyed at a certain distance in the other cases. This proves that the soliton envelope is stable in the sink dynamics and unstable in the other dynamics (source, saddle and circle). We also plotted the bifurcation parameters as a function of the gain dispersion. This allowed us to highlight the areas corresponding to each dynamic. Direct numerical simulations of the perturbed CCQGLE confirmed our theoretical predictions. We observed the robustness of the stable soliton by superimposing the noise on the initial pulse and found that the soliton structure is preserved up to a certain limit. These results can allow us to transmit information over long distances.

Controllable transmission of Airy pulses in nonlinear dissipative system

The propagation characteristics of truncated Airy pulses have been numerically investigated in complex cubic-quintic Ginzburg-Landau equation and some interesting nonlinear phenomena are observed. The results show that the transmission trajectory and amplitude of the Airy pulses can be controlled by changing the gain dispersion. Due to the complex interaction between the side lobes, the double Airy pulses evolve into single soliton, double solitons and triple solitons, respectively. The interaction between linear loss and nonlinear gain has a significant effect on the propagation dynamics of Airy pulses, which can obtain stable bound state solitons or pulsating-like solitons. In addition, the period, structure and number of solitons can be controlled by properly choosing the parameters of the quintic nonlinear correction term.

A modification to the first integral method and its applications

In this article, we modify Feng’s first integral method (FIM) for the purpose of enlarging its applications. Compared with original FIM, our modified FIM is more straight and can also be employed to find first integral of higher-order ordinary differential equations (ODEs). We employ our modified FIM into five differential equations, namely, the density-dependent conformable fractional diffusion-reaction equation, the Duffing-van der Pol oscillator, the complex cubic-quintic Ginzburg–Landau equation, the well-known nonlinear evolution equation for description of surface waves in a convecting liquid, and the KdV–Burgers–Fisher equation. Consequently, we get the same first integral obtained by Feng’s FIM for the first equation; for the second equation, we reobtain certain important first integral reported previously; for the third equation, we construct a new first integral of complex cubic-quintic Ginzburg–Landau equation; and for the fourth and fifth equations, we show the effectiveness of our approach to third-order ODEs and reobtain the same first integral recently presented by Kudryashov for the fourth equation, and for the fifth equation, two new first integrals are presented. All the above fully reveal the effectiveness of our modification.

Dark soliton solutions of the cubic-quintic complex Ginzburg–Landau equation with high-order terms and potential barriers in normal-dispersion fiber lasers

In this paper, we numerically study dark solitons in normal-dispersion optical fibers described by the cubic-quintic complex Ginzburg–Landau equation. The effects of the third-order dispersion, self-steepening, stimulated Raman dispersion, and external potentials are also considered. The existence, chaotic content and interactions of these objects are analyzed, as well as the tunneling through a potential barrier and the formation of dark breathers aside from dark solitons in two dimensions and their mutual interactions as well as with periodic potentials. Furthermore, the homogeneous solutions of the model and the conditions for their stability are also analytically obtained.

Role of the quintic nonlinear refractive term in the stability of dissipative solitons of the complex Ginzburg–Landau equation

We revisit the role of the quintic terms of the complex cubic-quintic Ginzburg–Landau equation in the generation of stable dissipative solitons. Using direct numerical simulations and a qualitative analysis, we show that the presence of one of the two quintic terms is a sine qua non. However, this term is not necessarily the quintic gain saturation term as had been demonstrated by Moores [Opt. Commun. 96, 65 (1993)OPCOB80030-401810.1016/0030-4018(93)90524-9] but can be the higher-order (quintic) nonlinear refraction term. We prove that by numerically solving this equation, and we perform a qualitative analysis that shows that the negative soliton chirp, anomalous dispersion, and spectral filtering are the physical effects responsible for gain saturation in this case.

Dynamics of moving cavity solitons in two-level laser system from symmetric gaussian input: vectorial cubic-quintic complex Ginzburg–Landau equation

Dynamics of moving cavity solitons in two-level laser system from symmetric gaussian input: vectorial cubic-quintic complex Ginzburg–Landau equation

Soliton Rectangular Pulses and Bound States in a Dissipative System Modeled by the Variable-Coefficients Complex Cubic-Quintic Ginzburg–Landau EquationSupported by the National Natural Science Foundation of China (Grant Nos. 11875008 and 12075034), and the Beijing University of Posts and Telecommunications Excellent Ph.D. Students Foundation (Grant No. CX2021129).

The complex cubic-quintic Ginzburg–Landau equation (CQGLE) is a universal model for describing a dissipative system, especially fiber laser. The analytic one-soliton solution of the variable-coefficients CQGLE is calculated by a modified Hirota method. Then, phenomena of soliton pulses splitting and stable bound states of two solitons are investigated. Moreover, rectangular dissipative soliton pulses of the variable-coefficients CQGLE are realized and controlled effectively in the theoretical research for the first time, which breaks through energy limitation of soliton pulses and is expected to provide theoretical basis for preparation of high-energy soliton pulses in fiber lasers.

Q-switching bifurcation dynamics of passively mode-locked lasers.

We model Q-switched pulses in passively mode-locked lasers using the cubic-quintic complex Ginzburg-Landau equation (CGLE). We show that a wide set of parameters of this equation leads to Q-switched pulses of triangular shape that consist of a periodic sequence of evolving dissipative solitons. Bifurcation diagrams demonstrating various transformations of these pulses are calculated in terms of five major parameters of the CGLE. The diagrams show period doubling transformations as well as the transition to a chaotic evolution of the Q-switched pulses.