Generalized Complex Ginzburg-Landau Equation Research Articles

Suppression of chaos in the periodically perturbed generalized complex Ginzburg–Landau equation by means of parametric excitation

Suppression of chaos in the periodically perturbed generalized complex Ginzburg–Landau equation by means of parametric excitation

Modulational instability and nonlinear dynamics of femtosecond lasers in transparent materials with non-Kerr nonlinearities

Femtosecond laser inscription in transparent materials is a physical process that finds widespread applications in material engineering, particularly in laser micromachining technology. In this process, the nonlinear optical response of the transparent material can be either intrinsic or induced by multiphoton ionization processes. In this work, a generic model is considered to describe the dynamics of femtosecond laser filamentation in transparent materials characterized by non-Kerr nonlinearities, focusing on the influence of multiphoton ionization processes in the generation of an electron plasma of inhomogeneous density. The mathematical model consists of a complex Ginzburg–Landau equation with a generalized saturable nonlinearity, besides the residual nonlinearity related to multiphoton ionization processes. This generalized complex Ginzburg–Landau equation is coupled to a rate equation for time evolution of the electron plasma density, where multiphoton ionizations are assumed to be the sole processes controlling the generation of the electron plasma. Dynamical properties of the model are discussed starting from the continuous-wave regime, where a modulational-instability analysis enables us to determine the stability conditions of continuous-wave modes in the system. The analysis reveals a dominant tendency of continuous-wave stability for relatively large values of the multiphoton ionization order K, provided the femtosecond laser operates in the anomalous dispersion regime. Numerical simulations of the mathematical model feature a family of wavetrains composed of self-focused, well-separated, pulse-shaped optical filaments whose repetition rates are shortened but amplitudes are increased, with an increase in K. Simulations suggest that such nonlinear wavetrain structures do not need the transparent material to be intrinsically nonlinear and that they may also be favored solely by the nonlinearity induced by multiphoton ionization processes in a linear transparent material.

Transition to turbulence or to periodic patterns in parallel flows

We propose a scenario for the formation of localized “turbulent” spots in transition flows, which is known as resulting from the subcritical character of the transition. We show that it is not necessary to add “by hand” a term of random noise in the equations, in order to describe the existence of long wavelength fluctuations as soon as the bifurcated state is beyond the Benjamin–Feir instability threshold. We derive the instability threshold for generalized complex Ginzburg–Landau equation which displays subcriticality. Beyond but close to the Benjamin–Feir threshold we show that the dynamics is mainly driven by the phase of the complex amplitude which obeys Kuramoto–Sivashinsky equation. As already proved for the supercritical case, the fluctuations of the intensity are smaller than those of the phase and slaved to the phase. On the opposite, below the Benjamin–Feir instability threshold, the bifurcated state does lose the randomness associated to turbulence so that the transition becomes of the mean-field type as in noiseless reaction–diffusion systems and leads to pulse-like patterns.

New computational optical solitons for generalized complex Ginzburg–Landau equation by collective variables

The purpose of the paper is to implement the collective variable method to investigate the generalized complex Ginzburg–Landau equation, which characterizes the kinetics of solitons in respect of pulse parameters for fiber optics. The statistical simulations of the interacting system of ordinary differential equations that reflect all the collective variables included in the pulse ansatz have been successfully carried out using a well-known numerical methodology, the fourth-order Runge–Kutta technique. The collective variable method is employed to plot the pulse variation characteristics as a function of propagation distance. The amplitude, temporal position, width, chirp, frequency, and phase of the pulse are all depicted against the propagated coordinate, where the width, phase of soliton, amplitude, and chirp all show a strong periodicity. The numerical dynamics of solitons have also been exhibited against varying values of pulse parameters to highlight differences in collective variables. Other key bits of the current investigation are also determined.

A generalized complex Ginzburg-Landau equation: Global existence and stability results

<p style='text-indent:20px;'>We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.

Nonlinearity managed dissipative solitons

We study the dynamics of localized pulses in the complex cubic–quintic Ginzburg–Landau (GL) equation with strong nonlinearity management. The generalized complex GL equation, averaged over rapid modulations of the nonlinearity, is derived. We employ the variational approach to this averaged equation to derive a dynamical system for the dissipative soliton parameters. By using numerical simulations of this dynamical system and the full GP equation, we show that nonlinearity-managed dissipative solitons exist in the model. The parameter regions for stabilization of exploding soliton are obtained.

Effect of time-varying flow-shear on the nonlinear stability of the boundary of magnetized toroidal plasmas

We propose a phenomenological yet general model in a form of extended complex Ginzburg-Landau equation to understand edge-localized modes (ELMs), a class of quasi-periodic fluid instabilities in the boundary of toroidal magnetized high-temperature plasmas. The model reproduces key dynamical features of the ELMs (except the final explosive relaxation stage) observed in the high-confinement state plasmas on the Korea Superconducting Tokamak Advanced Research: quasi-steady states characterized by field-aligned filamentary eigenmodes, transitions between different quasi-steady eigenmodes, and rapid transition to non-modal filamentary structure prior to the relaxation. It is found that the inclusion of time-varying perpendicular sheared flow is crucial for reproducing all of the observed dynamical features.

Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers

Generalized complex Ginzburg–Landau equation (GCGLE) can be used to describe the nonlinear dynamic characteristics of fiber lasers and has riveted much attention of researchers in ultrafast optics. In this paper, analytic solutions of the GCGLE are obtained via the modified Hirota bilinear method. Kink waves and period waves are presented by selecting the relevant parameters. The influence of the related parameters on them is analyzed and studied. The results indicate that the desired pulses can be demonstrated by effectively controlling the dispersion and nonlinearity of fiber lasers.

Modulation instability and dissipative ion-acoustic structures in collisional nonthermal electron-positron-ion plasma: solitary and shock waves55The project is supported by NSF of China (11501441, 11371289, 11371288), National Natural Science Foundation of China (U1261112), China Postdoctoral Science Foundation (2014M560756), and Fundamental Research Funds for the Central Universities (xjj2015067).

The nonlinear behavior of an ion-acoustic wave packet is investigated in a three-component plasma consisting of warm ions, nonthermal electrons and positrons. The nonthermal components are assumed to be inertialess and hot where they are modeled by the kappa distribution. The relevant processes, including the kinematic viscosity amongst the plasma constituents and the collision between ions and neutrals, are taken into consideration. It is shown that the dynamics of the modulated ion-acoustic wave is governed by the generalized complex Ginzburg–Landau equation with a linear dissipative term. The dispersion relation and modulation instability criterion for the generalized complex Ginzburg–Landau equation are investigated numerically. In the general dissipation regime, the effect of the plasma parameters on the dissipative solitary (dissipative soliton) and shock waves is also discussed in detail.

Active Curved Polymers Form Vortex Patterns on Membranes.

Recent invitro experiments with FtsZ polymers show self-organization into different dynamic patterns, including structures reminiscent of the bacterial Z ring. We model FtsZ polymers as active particles moving along chiral, circular paths by Brownian dynamics simulations and a Boltzmann approach. Our two conceptually different methods point to a generic phase behavior. At intermediate particle densities, we find self-organization into vortex structures including closed rings. Moreover, we show that the dynamics at the onset of pattern formation is described by a generalized complex Ginzburg-Landau equation.

Bifurcation to traveling waves in the cubic–quintic complex Ginzburg–Landau equation

We consider the one-dimensional complex Ginzburg–Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg–Landau equation, Pure Appl. Anal. 7(5) (2008) 1237–1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling waves is examined and it is proved by means of the center manifold reduction method and some perturbation arguments, that the variable amplitude traveling waves are quasi-periodic and they connect two constant amplitude traveling waves.

Impact of higher-order effects on pulsating and chaotic solitons in dissipative systems

We investigate numerically, both in time and frequency domains, the influence of some higher-order effects, namely the third-order dispersion, intrapulse Raman scattering, and self-steepening, on the dynamics of different pulsating and chaotic solitons in dissipative systems, which are described by a generalized complex Ginzburg-Landau equation. We show that the higher-order effects can have a dramatic impact on the dynamics of such pulses and that, for some ranges of the parameter values, they can be transformed into fixed-shape solitons. This paper is dedicated to Prof. Helmut Brand on the occasion of his 60th birthday.

Influence of the Frequency-chirp on Pulse in Passively Mode-locking Optical Fiber Ring Laser

We consider a model of a passively mode-locking fiber ring laser bult using a saturable absorber and a chirped fiber Bragg grating to balance dispersion and nonlinearity. The evolution of the slowly envelope of the optical field in a loop fiber subject to dispersion , Kerr nonlinearity, frequency- chirp and nonlinear absorption is given by the generalized complex Ginzburg-Landau equation. The influence of the frequency-chirp on the pulse is simulated and discussed, and the stationary conditions concerning the chirp parameter are found out for our laser.

Stability of nonequilibrium polariton superflow in the presence of a cylindrical defect

We make a theoretical study of the stability of nonequilibrium polariton superflows that interact with a cylindrical defect. The nonresonantly pumped polariton condensate is modeled with a generalized complex Ginzburg-Landau equation. At low pump intensities the dissipation is found to stabilize the superflow. At large pump intensities, we find an instability that sets a lower critical speed for superfluidity. For even larger pump power, the lower and upper critical speeds meet and stable superflows are no longer possible.

Impact of higher-order effects on pulsating, erupting and creeping solitons

We investigate numerically the dynamics of pulsating, erupting and creeping soliton solutions of a generalized complex Ginzburg–Landau equation (CGLE), including the third-order dispersion, intrapulse Raman scattering and self-steepening effects. We show that these higher-order effects (HOEs) can have a dramatic impact on the dynamics of the above mentioned CGLE solitons. For some ranges of the parameter values, the periodic behavior of some of these pulses is eliminated and they are transformed into fixed-shape solitons. Some particular interesting cases are discussed concerning the combined action of the three HOEs.

Generation of pulse trains in nonlinear optical fibers through the generalized complex Ginzburg-Landau equation

We consider a higher-order complex Ginzburg-Landau equation, with the fourth-order dispersion and cubic-quintic nonlinear terms, which can describe the propagation of an ultrashort subpicosecond or femtosecond optical pulse in an optical fiber system. We investigate the modulational instability (MI) of continuous wave solution of this equation. Several types of modulational instability gains are shown to exist in both the anomalous and normal dispersion regimes. We find that depending on the sign of the fourth-order dispersion coefficient, the MI appears for normal and anomalous dispersion regime. Simulations of the full system demonstrate that the development of the MI leads to establishment of a regular or chaotic array of pulses, a chain of well-separated peaks with continuously growing or decaying amplitudes depending on the sign of the loss/gain coefficient and higher-order dispersions terms. Comparison of the calculations with reported numerical results shows a satisfactory agreement.

A Dirichlet inhomogeneous boundary value problem for a generalized Ginzburg–Landau equation

The Ginzburg–Landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study a generalized complex Ginzburg–Landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives. We prove a global existence and uniqueness theorem.

Long time behaviour for generalized complex Ginzburg–Landau equation

In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL) u t = ρ u − Δ φ ( u ) − ( 1 + i γ ) Δ u − ν Δ 2 u − ( 1 + i μ ) | u | 2 σ u + α λ 1 ⋅ ∇ ( | u | 2 u ) + β ( λ 2 ⋅ ∇ ) | u | 2 is studied. The existence of global attractor for this equation with periodic boundary condition is established and upper bounds of Hausdorff and fractal dimensions of attractor are obtained.

Motion of clusters of weakly coupled two-dimensional cavity solitons

An analysis of clusters of weakly coupled two-dimensional spatial optical solitons in a large-aperture class A laser with a saturable absorber is developed. The symmetries that control the transverse motion of the clusters are described. Numerical solutions of the governing generalized complex Ginzburg-Landau equation demonstrate the existence of four types of clusters of weakly coupled cavity solitons that correspond to symmetries of transverse intensity distributions and energy flows: (1) stationary (with two mirror symmetry axes), (2) rotating about a stationary center of mass (invariant under rotation), (3) translating without rotation (with a single mirror symmetry axis), and (4) asymmetric ones rotating about a center of mass that moves around a circle (with equal periods of rotation and circular motion).

Streamer formation and collapse in electron temperature gradient driven turbulence

A simple model is useful to understand the formation and persistence of radially elongated structures (streamers) in electron temperature gradient (ETG) driven modes. The ETG model is very similar to the thermal Rossby wave model, a system of broad interest. The detailed correspondence of these two models is discussed. Streamer formation in this simple model is analyzed using the modulational stability method. In the inviscid limit of the model, an amplitude equation similar to the nonlinear Schrödinger equation (NLS) is derived. This equation has a second derivative cubic nonlinearity and is identified as a special case of a more general higher order NLS. Analytical solutions are found in the form of travelling waves and a localized thorn. Using the Lagrangian structure of the amplitude equation, it is shown that one-dimensional collapse in the poloidal direction is possible in this system for certain parameter values, and for sufficiently localized inital flow. This identifies a parameter regime basin in which there is an attractor with the structure of a thin extended streamer. In the viscous limit, another amplitude equation, which is a certain special case of the generalized complex Ginzburg–Landau equation, is obtained. Fixed points of the corresponding dynamical system are identified and their stability is investigated.