Complex Ginzburg Landau System Research Articles

Evolution of periodic wave and dromion-like structure solutions in the variable coefficients coupled high-order complex Ginzburg–Landau system

Evolution of periodic wave and dromion-like structure solutions in the variable coefficients coupled high-order complex Ginzburg–Landau system

Bifurcation analysis and optical soliton solutions for the fractional complex Ginzburg–Landau equation in communication systems

The nonlinear fractional complex Ginzburg–Landau system is a classical equation which has been developed vigorously in the fields of the combustion theory, nonlinear optics, signal processing, statistical mechanics and so on. This paper mainly concentrates on the bifurcations and single dispersive optical solitons for the fractional complex Ginzburg–Landau equation in communication systems. Starting with the traveling wave transformations and some other suitable transformations, the fractional complex Ginzburg–Landau system is converted into an equivalent ordinary differential system. Secondly, the bifurcation phase diagram and Hamiltonian of fractional complex Ginzburg–Landau system are also given. In addition to, the dispersive optical solitons and traveling wave solutions are derived via the dynamical theory method, the polynomial complete discriminant system technique and symbolic computation. It is notable that the obtained optical soliton solutions may improve or complement the corresponding results in the literature (Akram et al., 2022, Ouahid et al., 2021). Finally, in order to further understand the propagation of the fractional complex Ginzburg–Landau equation in nonlinear optics, three-dimensional and two-dimensional diagrams are given.

Iterative learning control of complex Ginzburg–Landau system

Different from the existing research on iterative learning control (ILC) real-valued systems, complex-valued partial differential system is studied in this paper. All variables and parameters of the system are complex, and distributed control is imposed on the system. First, the original complex system is transformed into two coupled real systems, which represent the real and imaginary parts of the original complex system, respectively. Second, a complex [Formula: see text]-type ILC algorithm is designed that the real and imaginary parts of the learning law are coupled to each other. Then, the contraction mapping principle and analytical techniques are used to ensure that the tracking error converges to zero in the sense of [Formula: see text]-norm. Finally, a numerical simulation is presented to show the effectiveness of the proposed algorithm.

Lie symmetry properties of the two-component complex Ginzburg–Landau system with variable coefficients

We classify the Lie point symmetries of a system of nonlinearly coupled complex Ginzburg–Landau equations with time and space dependent potentials and nonlinearities. New families of systems with exact solutions and conservation laws are identified. When there is only space dependence, the system can be reduced to a set of ordinary differential equations for the amplitude and phase components, for which integrability and first integrals are discussed.

Non-Lagrangian approach for coupled complex Ginzburg-Landau systems with higher order-dispersion

It is known that after a particular distance of evolution in fiber lasers, two (input) asymmetric soliton like pulses emerge as two (output) symmetric pulses having same and constant energy. We report such a compensation technique in dispersion managed fiber lasers by means of a semi-analytical method known as collective variable approach (CVA) with including third-order dispersion (TOD). The minimum length of fiber laser, at which the output symmetric pulses are obtained from the input asymmetric ones, is calculated for each and every pulse parameters numerically by employing Runge-Kutta fourth order method. The impacts of intercore linear coupling, asymmetric nature of initial parameters and TOD on the evolution of pulse parameters and on the minimum length are also investigated. It is found that strong intercore linear coupling and asymmetric nature of input pulse parameters result in the reduction of fiber laser length. Also, the role of TOD tends to increase the width of the pulses as well as their energies. Besides, chaotic patterns and bifurcation points on the minimum length of the fiber owing to the impact of TOD are also reported in a nutshell.

Amplitude death criteria for coupled complex Ginzburg–Landau systems

Amplitude death, which occurs in a system when one or more macroscopic wavefunctions collapse to zero, has been observed in mutually coupled solid-state lasers, analog circuits, and thermoacoustic oscillators, to name a few applications. While studies have considered amplitude death on oscillator systems and in externally forced complex Ginzburg–Landau systems, a route to amplitude death has not been studied in autonomous continuum systems. We derive simple analytic conditions for the onset of amplitude death of one macroscopic wavefunction in a system of two coupled complex Ginzburg–Landau equations with general nonlinear self- and cross-interaction terms. Our results give a more general theoretical underpinning for recent amplitude death results reported in the literature, and suggest an approach for tuning parameters in such systems so that they either permit or prohibit amplitude death of a wavefunction (depending on the application). Numerical simulation of the coupled complex Ginzburg–Landau equations, for example, including cubic, cubic–quintic, and saturable nonlinearities, is used to illustrate the analytical results.

Double Hopf bifurcation induces coexistence of periodic oscillations in a diffusive Ginzburg–Landau model

We perform bifurcation analysis in a complex Ginzburg–Landau system with delayed feedback under the homogeneous Neumann boundary condition. We calculate the amplitude death region, and it turns out that the boundary of the amplitude death region consists of two Hopf bifurcation curves with wave number zero. The existence conditions for double Hopf bifurcations are established. Taking the feedback strength and time delay as bifurcation parameters, normal forms truncated to the third order at double Hopf singularity are derived, and the unfolding near the critical points is given. The bifurcation diagram near the double Hopf bifurcation is drawn in the two-parameter plane. The phenomena of amplitude death, the existence of stable bifurcating periodic solutions, and the coexistence of two stable periodic solutions with fast oscillation and slow oscillation respectively are simulated.

Sensor and actuator placement trade-offs for a linear model of spatially developing flows

We consider feedback flow control of the linearised complex Ginzburg–Landau system. The particular focus is on any trade-offs present when placing a single sensor and a single actuator. The work is presented in three parts. First, we consider the estimation problem in which a single sensor is used to estimate the entire flow field (without any control). Second, we consider the full information control problem in which the entire flow field is known, but only a single actuator is available for control. By considering the optimal sensor placement and optimal actuator placement while varying the stability of the system, a fundamental trade-off for both problems is made clear. Third, we consider the overall feedback control problem in which only a single sensor is available for measurement; and only a single actuator is available for control. By varying the stability of the system, similar fundamental trade-offs are made clear. We discuss implications for effective feedback control with a single sensor and a single actuator and compare it to previous placement methods.

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

We formulate and study dynamics from a complex Ginzburg–Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg–Landau systems under the added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg–Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schrödinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). We also find a collection of transient features, including transient defects and what appear to be rogue waves, while in two spatial dimensions we observe highly localized pattern formation. While saturation will often regularize the dynamics, such transient dynamics can still be observed – and in some cases even prolonged – as the saturability of the media is increased, as the saturation may act to slow the timescale.

Amplitude death in a pair of one-dimensional complex Ginzburg-Landau systems coupled by diffusive connections.

This paper shows that, in a pair of one-dimensional complex Ginzburg-Landau (CGL) systems, diffusive connections can induce amplitude death. Stability analysis of a spatially uniform steady state in coupled CGL systems reveals that amplitude death never occurs in a pair of identical CGL systems coupled by no-delay connection, but can occur in the case of delay connection. Moreover, amplitude death never occurs in coupled identical CGL systems with zero nominal frequency. Based on these analytical results, we propose a procedure for designing the connection delay time and the coupling strength to induce spatial-robust stabilization, that is, a stabilization of the steady state for any system size and any boundary condition. Numerical simulations are performed to confirm the analytical results.

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto-Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.

Anticipated synchronization in coupled complex Ginzburg-Landau systems.

We study the occurrence of anticipated synchronization in two complex Ginzburg-Landau systems coupled in a master-slave configuration. Master and slave systems are ruled by the same autonomous function, but the slave system receives the injection from the master and is subject to a negative delayed self-feedback loop. We give evidence that the magnitude of the largest anticipation time, obtained for complex-valued coupling constants, depends on the dynamical regime where the system operates (defect turbulence, phase turbulence, or bichaos) and scales with the linear autocorrelation time of the system. We also provide analytical conditions for the stability of the anticipated synchronization manifold that are in qualitative agreement with those obtained numerically. Finally, we report on the existence of anticipated synchronization in coupled two-dimensional complex Ginzburg-Landau systems.

Tanh-series representation of stationary dissipative solitons in complex Ginzburg-Landau systems

In terms of a tanh-series representation, the stationary dissipative solitons such as arbitrary-amplitude solitons, plain solitons, flattop solitons, and composite solitons in complex Ginzburg-Landau (CGL) systems can be exactly described, as confirmed by extensive simulations. It thus suggests another route to understanding solitons in CGL-like systems that is highly accurate.

Length scale of interaction in spatiotemporal chaos

Extensive systems have no long scale correlations and behave as a sum of their parts. Various techniques are introduced to determine a characteristic length scale of interaction beyond which spatiotemporal chaos is extensive in reaction-diffusion networks. Information about network size, boundary condition, or abnormalities in network topology gets scrambled in spatiotemporal chaos, and the attenuation of information provides such characteristic length scales. Space-time information flow associated with the recovery of spatiotemporal chaos from finite perturbations, a concept somewhat opposite to the paradigm of Lyapunov exponents, defines another characteristic length scale. High-precision computational studies of asymptotic spatiotemporal chaos in the complex Ginzburg-Landau system and transient spatiotemporal chaos in the Gray-Scott network show that these different length scales are comparable and thus suitable to define a length scale of interaction. Preliminary studies demonstrate the relevance of these length scales for stable chaos.

Analysis of stationary droplets in a generic Turing reaction-diffusion system

Solitonlike structures called "droplets" are found to exist within a paradigm reaction-diffusion model that can be used to describe patterning in a number of biological systems, for example, on the skin of various fish species. They have also been found in many other systems that can be modeled with a complex Ginzburg-Landau system. These droplets can be analyzed in the biological paradigm model because the system has two nonzero stable steady states that are symmetric; however, the asymmetric case is more challenging. We first review the properties of the paradigm system and then extend a recently developed perturbation technique [D. Gomila, J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004)] to investigate the weakly asymmetric case. We compare the results of our mathematical analysis with numerical simulations and show good agreement in the region where the assumptions hold.

Nonlinear waves with negative phase velocity

Recently, waves propagating with negative phase velocity [simply called antiwaves (AWs)] have attracted great attention in the area of nonlinear oscillatory systems. In the present work we investigate the parameter conditions for AWs. So far AWs have been revealed from systems slightly beyond Hopf bifurcation or some other instabilities, and from some wave sources with certain restricted frequencies. Here we study general oscillatory media (including generalized complex Ginzburg-Landau systems and Brusselator model) and specify the parameter conditions of AWs by certain characteristic behaviors of the dispersion relation of the systems. Moreover, we predict that AWs and NWs (normal waves with positive phase velocity) can be realized at a same intrinsic parameter values but different pacing frequencies in parameter regions where the dispersion relation exhibits a maximum or minimum. All numerical simulations are perfectly consistent with these theoretical predictions where the oscillatory systems are driven by external periodic pacings with 1:1 frequency locking responses.

A remark on the dimension of the attractor for the Dirichlet problem of the complex Ginzburg–Landau equation

Using the improved lower bound on the sum of the eigenvalues of the Dirichlet Laplacian proven by Melas [Proc. Am. Math. Soc. 131, 631 (2003)], we remark on a modified estimate of the dimension of the global attractor associated with the complex Ginzburg–Landau (CGL) equation ∂tu−(λ+iα)Δu+(κ+iβ)|u|2u−γu=0, supplemented with Dirichlet boundary conditions. This modified estimate could be of significance depending on β∊R being a small parameter, an important case for the CGL system.

MODULATIONAL INSTABILITY IN NONLINEAR BI-INDUCTANCE TRANSMISSION LINE

A nonlinear dissipative transmission line is considered and by performing the complex demodulation technique of the signal which allows, in particularly, to separate the right traveling and left traveling waves, we show that the amplitudes of these waves can be described by a complex coupled Ginzburg–Landau equations (CG-LE). The so-called phase winding solutions of the constructed CG-LE is analyzed. We also study the coherent structures in the obtained complex Ginzburg–Landau system. We show that the constructed CG-LE possesses nonlinear plane wave solutions and the modulational instability of these solutions is analyzed. The condition of the modulational instability is given in term of the coefficients of the constructed CG-LE and then in term of the wavenumber of the two right traveling and left traveling waves in the considered transmission line. The results obtained here show that the nonlinear plane wave solutions of the CG-LE under perturbation with zero wavenumber cannot be stable under modulation.

Front explosion in a resonantly forced complex Ginzburg–Landau system

Periodically forced oscillatory reaction–diffusion systems near the Hopf bifurcation can be modeled by the resonantly forced complex Ginzburg–Landau equation. In the 3:1 resonant locking regime this equation has three stable fixed points corresponding to the phase-locked states in the underlying reaction–diffusion system. Phase fronts separate spatial domains containing the phase-locked states. When the Ginzburg–Landau equation parameters lie in the Benjamin–Feir-unstable regime, the phase fronts have a turbulent internal spatio-temporal structure. As the forcing intensity is changed, the intrinsic width of a front grows until, at a critical value, the front “explodes” and the turbulent interfacial zone expands to fill the entire domain. The scaling properties of this transition are explored and it is shown that front width and spatial and temporal correlations diverge as the critical forcing intensity is approached. These results are compared with similar behavior seen in a coupled map model with period-3 local dynamics. The prospects for observation of these phenomena in experiments on periodically forced reaction–diffusion systems are discussed.