Cubic-quintic Ginzburg-Landau Equation Research Articles

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.

Discriminating the existence regions and dynamics between quartic self-similar pulse and dissipative pure quartic soliton in ultrashort fiber laser

Recently, both quartic self-similar pulses (QSSPs) and dissipative pure-quartic-solitons (DPQSs) pulses have been theoretically demonstrated in mode-locked fiber laser with pure positive fourth-order dispersion (FOD). Here, we investigate the existence regions and dynamics of those two soliton pulses by numerically solving the cubic-quintic Ginzburg-Landau equation. We find the positive FOD fiber laser tends to emit QSSPs in the presence of high nonlinear gain, while it is apt to generate DPQS pulses in the case of low nonlinear gain. An unstable pulse region separates the QSSP region from the DPQS region. The characteristics of QSSP and DPQS pulses are dependent on the nonlinear gain and FOD. The QSSP can carry more energy than that of DPQS pulse, because the DPQS pulse with large energy could be unstable due to the excessive pulse pedestal. Our results are not only useful for understanding the dynamics of the DPQS pulse and QSSP in the cubic-quintic Ginzburg-Landau equation with pure FOD, but also provide a guideline for generating high-energy pulses from the positive FOD fiber laser.

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.

Generation of double Sasa-Satsuma, double Kuznetsov-Ma and other exotic solutions of cubic-quintic Ginzburg-Landau equation in a left-handed material

We present new exotic numerical solutions and a strange periodical mixed Sasa-Satsuma and Kuznetsov-Ma waves’ train. Much more, we study a nonlinear Schrödinger (NLS) equation for absorption regime in a left-handed material. The complex coefficients coming from absorption regime transform this NLS equation into a Cubic-Quintic Ginzburg–Landau (CQGL) equation. Moreover, we employ variational theory with Type I Ansatz function containing seven coordinates compared to six coordinates often used. Some interesting results are found such as one, two, three, four, five, six and seven peak solutions. Among those solutions specific ones are outlined as follows. (i) Double Kuznetsov-Ma breathers and tree structure. (ii) Conventional, random, double, triple and new forms of Sasa-Satsuma solutions. (iii) Stair-, Pyramid-, flower-shaped three and five peak solutions. (iv) First-and second-forms of seven peak solutions. Further, we also present the physical conditions and the internal excitation leading to the generation of those new exotic solutions.

Extension of the stability criterion for dissipative vector solitons of a laser coupled two-dimensional Ginzburg–Landau Equation generated from vector asymmetric inputs

Pulses propagating in inhomogeneous nonlinear media with linear/nonlinear gain and loss described by the vector (2+1)−dimensional cubic-quintic Ginzburg−−Landau equations (CQCGLEs) are considered. The evolution and the stability of the vector dissipative optical solitons generated from an elliptic input are studied. Based on the variational approach, we analyze the influence of various physical parameters on the dynamics of the propagating signal and its relevant parameters. Following a suitable choice of the test function, the stability analysis has been fulfilled by the analysis of the total energy. Numerical results have confirmed the analytical prediction of the collision on the stability of the vector dissipative solitons.

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

Chaotic diffusion of dissipative solitons: From anti-persistent random walks to Hidden Markov Models

In previous publications, we showed that the incremental process of the chaotic diffusion of dissipative solitons in a prototypical complex Ginzburg-Landau equation, known, e.g., from nonlinear optics, is governed by a simple Markov process leading to an Anti-Persistent Random Walk of motion or by a more complex Hidden Markov Model with continuous output densities. In this article, we reveal the transition between these two models by examining the dependence of the soliton dynamics on the main bifurcation parameter of the cubic-quintic Ginzburg-Landau equation, and by identifying the underlying hidden Markov processes. These models capture the non-trivial decay of correlations in jump widths and sequences of symbols representing the symbolic dynamics of short and long jumps, the statistics of anti-persistent walk episodes, and the multimodal density of the jump widths. We demonstrate that there exists a physically meaningful reduction of the dynamics of an infinite-dimensional deterministic system to one of a probabilistic finite state machine and provide a deeper understanding of the soliton dynamics under parameter variation of the underlying nonlinear dynamics.

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.

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.

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.

Analysis of the 2D complex Ginzburg-Landau Equation using Singular Value Decomposition

The cubic-quintic Ginzburg-Landau equation (CQGLE) governs the dynamics of solitons in lasers and many optical systems. Using data obtained from the simulations of the CQGLE, we performed a singular value decomposition (SVD) to create a low dimensional model that qualitatively predicts the stability of the solitons as a function of the energy gain constant. It was found both in the full simulations and in the low dimensional model that the soliton becomes unstable when the gain exceeds a certain threshold value. Both the low dimensional model and the full simulation demonstrated the same qualitative behavior when the soliton loses stability.

Concurrent Passive Mode-Locked and Self-Q-Switched Operation in Laser Systems.

Concurrent passive mode-locked and self-Q-switched operation of laser devices is modeled using the complex cubic-quintic Ginzburg-Landau equation. Experimental observations use a passively mode-locked fiber ring laser with a waveguide array as a fast saturable absorber. The shape of each individual self-Q-switched pulse and the periodic trains of pairs of such pulses are in a good qualitative agreement with the numerical results.

Interaction of dissipative solitons stabilized by nonlinear gradient terms.

We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possible outcomes of theses collisions: Stationary bound states, oscillatory bound states, meandering oscillatory bound states, bound states with large-amplitude oscillations, partial annihilation, complete annihilation, and interpenetration. Detailed results and their analysis are presented for one value of the corresponding nonlinear gradient term, while the results for two other values are just mentioned briefly. We compare our results with those obtained for coupled cubic-quintic complex Ginzburg-Landau equations and with the cubic-quintic complex Swift-Hohenberg equation. It turns out that both meandering oscillatory bound states as well as bound states with large-amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term. Remarkably, we find for the large-amplitude oscillations a linear relationship between oscillation amplitude and period.

Transition between optical turbulence and dissipative solitons in a complex Ginzburg-Landau equation with quasi-cw noise

Noise-induced phase transition between chaos and ordered states plays an important role in the self-organization of dissipative structures. Here, we numerically simulate the phase transitions between statistically stationary regimes in a laser cavity by using a stochastic complex cubic-quintic Ginzburg-Landau equation with a phase-diffusion model. Depending on the linear and nonlinear gains, the dissipative system converges from the initial noise floor to different regimes, including continuous wave, phase turbulence, amplitude turbulence (AT), and stable soliton structure. Around the regime boundaries, we found that only certain phase noise can cause the instability of phase turbulence. The defects formed due to the noise will continue to spread out and self-organize to form AT or stabilize the soliton structure in the defect area. Our results are helpful in understanding the dynamics of the buildup of mode locking and noiselike pulses in passively mode-locked lasers.

Engineering chirped Lambert W-kink signals in a nonlinear electrical transmission network with dissipative elements

The dynamics of modulated waves in a nonlinear discrete transmission network with dissipative elements are investigated analytically. The generalized cubic-quintic Ginzburg-Landau (GCQ-GL) equation with intrapulse Raman scattering (IRS) terms governing spatially damped slowly modulated wave propagation is derived. Considering modulated Stokes wave propagating in the network, we investigate their baseband modulational instability (MI) and show that there are significant changes for the bandwidth frequency where the network may exhibit baseband MI. We show that the MI growth rate increases with the propagating frequency. The analytical chirped Lambert W-kink soliton-like solution of the derived GCQ-GL equation is presented and used to analyze the dissipative effects on chirped Lambert W-kink signals propagating through our network system. We show that the amplitude decreases both in space and time, while the width and the velocity remains constant when the Lambert W-kink signal propagates along the dissipative network. The effects of the cubic-Raman contributions on (i) the propagating frequency, (ii) the baseband MI, as well as (iii) the dynamics of Lambert W-kink signals are also investigated. In particular, our results reveal that a decrease in the cubic-Raman contributions enhances the baseband MI. The results of this paper may be useful for theoretical study of the propagation of ultrashort pulses in a single-mode optical fiber, as well as for experimental realization of undistorted transmission of optical pulses in optical fibers and further understanding their optical transmission properties.