Cubic-quintic Complex Ginzburg-Landau Equation Research Articles

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.

Modulation instability in nonlinear metamaterials modeled by a cubic-quintic complex Ginzburg-Landau equation beyond the slowly varying envelope approximation.

Considering the theory of electromagnetic waves from the Maxwell's equations, we introduce a (3+1)-dimensionsal cubic-quintic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in nonlinear metamaterials. The model equation, which is derived beyond the slowly varying envelope approximation, includes the effects of diffraction, dispersion, loss, gain, cubic, and quintic nonlinearities, as well as cubic and quintic self-steepening effects. The modulational instability of the plane waves is studied both theoretically, using the linear stability analysis, and numerically, using direct simulations of the Fourier space of the proposed nonlinear wave equation, based on the Drude model. The linear theory predicts instability for any amplitude of the primary wave. Also, in the linear stability analysis, self-steepening effects of different orders are confronted and one discusses their effects on the behavior of the gain spectrum under both normal and anomalous group-velocity dispersion regimes. Analytical results are equally confronted to direct numerical simulations and fully agree with the predictions from the gain spectra. Modulational instability is manifested by clusters of solitons and multihump and dromion-like structures, whose emergence and features depend not only on system parameters, such as the cubic and quintic self-steepening coefficients, but also on the propagation distance under a suitable balance between nonlinear and dispersive effects.

Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering

In this paper, an exact explicit solution for the complex cubic-quintic Ginzburg-Landau equation is obtained, by using Lambert W function or omega function. More pertinently, we term them as Lambert W-kink-type solitons, begotten under the influence of intrapulse Raman scattering. Parameter domains are delineated in which these optical solitons exit in the ensuing model. We report the effect of model coefficients on the amplitude of Lambert W-kink solitons, which enables us to control efficiently the pulse intensity and hence their subsequent evolution. Also, moving fronts or optical shock-type solitons are obtained as a byproduct of this model. We explicate the mechanism to control the intensity of these fronts, by fine tuning the spectral filtering or gain parameter. It is exhibited that the frequency chirp associated with these optical solitons depends on the intensity of the wave and saturates to a constant value as the retarded time approaches its asymptotic value.

Control of dissipative rogue waves in nonlinear cavity optics: Optical injection and time-delayed feedback.

We investigate and review the formation of two-dimensional dissipative rogue waves in cavity nonlinear optics with transverse effects. Two spatially extended systems are considered for this purpose: the driven Kerr optical cavities subjected to optical injection and the broad-area surface-emitting lasers with a saturable absorber. We also consider a quasi-two-dimensional system (the two dimensions being space and time) of a fiber laser describing the complex cubic-quintic Ginzburg-Landau equation. We show that rogue waves are controllable by means of time-delayed feedback and optical injection. We show that without delayed feedback, transverse structures are stationary or oscillating. However, when the strength of the delayed feedback is increased, all the systems generate giant two-dimensional pulses that appear with low probability and suddenly appear and disappear. We characterize their formation by computing the probability distribution, which shows a long tail. Besides, we have computed the significant wave height, which measures the mean wave height of the highest third of the waves. We show that for all systems, the distribution tails expand beyond two times the significant wave height. Furthermore, we also show that optical injection may suppress the rogue wave formation in a semiconductor laser with a saturable absorber.

Breaking of symmetry of interacting dissipative solitons can lead to partial annihilation.

We show that for a large range of approach velocities and over a large interval of stabilizing cubic cross-coupling between counterpropagating waves, a collision of stationary pulses leads to a partial annihilation of pulses via a spontaneous breaking of symmetry. This result arises for coupled cubic-quintic complex Ginzburg-Landau equations for traveling waves for sufficiently large values of the stabilizing cubic cross-coupling and for large enough approach velocities of the pulses. Briefly, we show in addition that the collision of counterpropagating pulses in a system of two coupled cubic Ginzburg-Landau equations with nonlinear gradients (Raman effect) might also lead to partial annihilation, indicating that this breaking of symmetry is generic. Systems of experimental interest include surface reactions, convective onset, biosolitons, and nonlinear optics.

Cascade replication of soliton solutions in the one-dimensional complex cubic-quintic Ginzburg-Landau equation

In the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation, we demonstrate two methods to achieve cascade replication of dissipative solitons from a single one by using external forces whose positions and magnitudes are being precisely controlled. The first method combines high amplitude positive currents and negative currents to increase the efficiency of the process significantly and to make the process repeatable. By using two repulsive forces whose locations are time dependent, the second method enables one to obtain multiple dissipative solitons in only one cascade replication process.

Spatiotemporal dissipative solitons and vortices in a multi-transverse-mode fiber laser.

We introduce a model for spatiotemporal modelocking in multimode fiber lasers, which is based on the (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation (cGLE) with conservative and dissipative nonlinearities and a 2-dimensional transverse trapping potential. Systematic numerical analysis reveals a variety of stable nonlinear modes, including stable fundamental solitons and breathers, as well as solitary vortices with winding number n = 1, while vortices with n = 2 are unstable, splitting into persistently rotating bound states of two unitary vortices. A characteristic feature of the system is bistability between the fundamental and vortex spatiotemporal solitons.

Mechanism of dissipative soliton stabilization by nonlinear gradient terms.

We present a feedback mechanism for dissipative solitons in the cubic complex Ginzburg-Landau (CGL) equation with a nonlinear gradient term. We are making use of a mechanical analog containing contributions from a potential and from a nonlinear viscous term. The feedback mechanism relies on the continuous supply of energy as well as on dissipation of the stable pulse. Our picture is corroborated by the numerical solution of the full equation. A quintic contribution is not necessary for stabilization in the presence of a suitable nonlinear gradient term as we show by using a linear stability analysis for the stationary pulses. We find that the limit of vanishing magnitude of the nonlinear gradient term is singular: For exactly vanishing nonlinear gradient terms stable pulses do not exist. This situation is qualitatively different from that found for the cubic-quintic complex Ginzburg-Landau equation with a nonlinear gradient term: In this case the limit of a vanishing nonlinear gradient contribution is completely smooth. We demonstrate that, for small magnitude of the nonlinear gradient term simple types of scaling behavior are found for the amplitude, the full width at half maximum (FWHM), the velocity, and the effective frequency of the stable pulse as a function of the magnitude of the nonlinear gradient term.

Dromion-like structures and periodic wave solutions for variable-coefficients complex cubic–quintic Ginzburg–Landau equation influenced by higher-order effects and nonlinear gain

In this work, the variable-coefficients complex cubic–quintic Ginzburg–Landau equation (CCQGLE) influenced by higher-order effects and nonlinear gain is considered. Based on the asymmetric method, analytic one-soliton solution for the variable-coefficients CCQGLE is constructed for the first time. In addition, with some certain conditions, the periodic wave and dromion-like structures are derived. The results obtained may be helpful in understanding the solitons amplification and solitons management in optical fiber.

High-amplitude dissipative solitons in the normal and anomalous dispersion regimes

In this work, using the method of moments and starting from a singularity found by Akhmediev and co-workers, very high-amplitude solitons of the cubic-quintic complex Ginzburg–Landau equation are predicted in both the normal and anomalous dispersion regimes. The propagation and the main characteristics of such very high-amplitude solitons are investigated in both dispersion regimes. Moreover, the region of existence of these pulses is numerically found in the plane defined by the dispersion and the nonlinear gain saturation parameters. In general, numerical computations are in good agreement with the predictions based on the method of moments when a quartic trial function is assumed. We show that choosing an appropriate signal for the reactive quintic nonlinearity parameter can provide an extension of the region of existence of the above pulses. High-energy ultrashort pulses are found mainly in the normal dispersion region, which is in agreement with the experimental observations reported by other authors.

A hidden Markov model for the dynamics of diffusing dissipative solitons

We investigate the dynamics of dissipative solitons in the cubic-quintic complex Ginzburg–Landau equation in one spatial dimension for different values of the bifurcation parameter . We consider a certain range of the parameter where dissipative solitons show explosions, i.e. transient enlargements of the soliton that lead to spatial shifts if the explosions are asymmetric. We find that depending on the parameter , the arising sequence of spatial shifts can be modeled by a simple anti-persistent random walk or by a more complicated hidden Markov model. We show with the help of exact analytical calculations that these models are able to reproduce several statistics of the soliton motion such as the distribution of spatial shifts, the correlation of spatial shifts, and the distribution of zig-zag streaks.

Simultaneous influence of additive and multiplicative noise on stationary dissipative solitons.

We investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially δ-correlated additive noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. Depending on the ratio between the strength of additive and multiplicative noise we find a number of distinctly different types of behavior including explosions, collapse, filling in, and spatio-temporal disorder as well as intermittent behavior of all types listed. Techniques used to analyze the results include snapshots, x-t plots and plots of the spatially and temporally averaged amplitude as a function of the strength of multiplicative noise while keeping the strength of additive noise fixed. Typically 50 realizations are used for averaging to obtain the corresponding data points in these diagrams. For the widths of these distribution as a function of additive noise we obtain a linear decrease in the limit of fairly large, but fixed values of the multiplicative noise. To summarize our findings concisely we show three-dimensional plots of the mean pattern amplitude and the generalized susceptibility as a function of the strengths of additive and multiplicative noise. We critically compare the results of our investigations with those obtained in the two limiting cases of purely additive and of purely multiplicative noise.

Impact of high-order effects on soliton explosions in the complex cubic-quintic Ginzburg-Landau equation

We investigate the impact of higher-order nonlinear and dispersive effects on the onset of soliton explosions in the complex cubic-quintic Ginzburg-Landau equation. We show how the interplay of the high order effects (HOEs) results in the splitting of symmetric explosion modes and to the formation of right- or left-side periodic explosions. In addition, we demonstrate that HOEs induce a series of pulsating instabilities, leading to a significant reduction of the stability region of the single soliton solution.

Stable transmission of solitons in the complex cubic–quintic Ginzburg–Landau equation with nonlinear gain and higher-order effects

In this paper, a complex cubic–quintic Ginzburg–Landau equation (CCQGLE) is investigated. Using the asymmetric method, the analytic one-soliton solution of the CCQGLE is obtained for the first time. Through analyzing the solutions obtained, the transmission of the soliton is controlled by changing the values of related parameters. Results of this paper contribute to obtain the analytic soliton solution of the higher-order CCQGLE.

Generation of stable multi-vortex clusters in a dissipative medium with anti-cubic nonlinearity

We demonstrate the generation of vortex solitons in a model of dissipative optical media with the singular anti-cubic (AC) nonlinearity, by launching a vorticity-carrying Gaussian input into the medium modeled by the cubic-quintic complex Ginzburg-Landau equation. The effect of the AC term on the beam propagation is investigated in detail. An analytical result is produced for the asymptotic form of fundamental and vortical solitons at the point of r→0, which is imposed by the AC term. Numerical simulations identify parameter domains that maintain stable dissipative solitons in the form of vortex clusters. The number of vortices in the clusters is equal to the vorticity embedded in the Gaussian input.

The solutions with recurrence property for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations

Stochastic periodic type solution is a powerful tool for studying qualitative analysis of stochastic dynamical systems. In this paper, we will establish the bounded solutions, stationary solutions, periodic solutions, almost periodic solutions, almost automorphic solutions for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations under suitable conditions. The main novelty of this paper is dealing with cubic nonlinear terms and the quintic nonlinear terms which are not Lipschitz. We overcome this difficulty by the semigroup approach, stochastic analysis techniques, energy estimate method and refined inequality technique.

Replication of dissipative vortices modeled by the complex Ginzburg-Landau equation

Dissipative vortices are stable two-dimensional localized structures existing due to balance between gain and loss in nonlinear systems far from equilibrium. Being resistant to the dispersion and nonlinear distortions they are considered as promising information carriers for new optical systems. The key challenge in the development of such systems is getting control over vortex waveforms. In this paper we report on replication of two-dimensional fundamental dissipative solitons and vortices due to their scattering on a locally applied potential in the cubic-quintic complex Ginzburg-Landau equation. It has been found that an appropriate potential non-trivially splits both fundamental solitons and vortices into a few exact copies without losing in their amplitude levels. A remarkably simple potential having a finite supporter along the longitudinal coordinate and a double peaked dependence on a single transverse coordinate is found to be suitable for the replication of the two-dimensional localized structures.

Dynamic evolution of pulsating solitons in a dissipative system with the gain saturation effect.

We numerically investigate the dynamic evolution of pulsating solitons based on complex cubic-quintic Ginzburg-Landau equation with gain dynamics effects. We show that an additional soliton can be generated by the disturbance caused by a dispersion wave emitted by a single-period pulsating soliton and these solitons form soliton molecule. More complicated oscillating processes, such as snaking pulsation and double-periodic pulsation are actuated by periodic collision of the entangled solitons. Moreover, the dispersive wave, caused by high gain parameters and the soliton collision, appears periodically which is in sync with the pulsating process. These results are consistent with the recent experiments of soliton pulsations measured by dispersive Fourier transform techniques, and will stimulate further experimental research of the complex multi-soliton bunches in dissipative systems.