Complex Ginzburg-Landau Equation Research Articles

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.

Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation

In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that the difference scheme is stable and convergent under mild conditions. The optimal convergence order $\mathcal {O}(\tau ^{2}+{h_{x}^{2}}+{h_{y}^{2}})$ is obtained in the pointwise sense by developing a new two-dimensional fractional Sobolev imbedding inequality based on the work in Kirkpatrick et al. (Commun. Math. Phys. 317, 563–591 2013), an energy argument and careful attention to the nonlinear term. Numerical examples are presented to verify the validity of the theoretical results for different choices of the fractional orders α and β.

Arnold tongues in oscillator systems with nonuniform spatial driving.

Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially coupled oscillators: the complex Ginzburg-Landau equation (CGLE) and a generalization of the CGLE in which oscillators are coupled through an external medium (emCGLE). We focus on external control drives that vary in both space and time. We find that the spatial distribution of the drive signal controls the frequency ranges over which oscillators synchronize to the drive and that boundary conditions strongly influence synchronization to external drives for the CGLE. Our calculations also show that the emCGLE has a low density regime in which a broad range of frequencies can be synchronized for low drive amplitudes. We study the bifurcation structure of these models and find that they are very similar to results for the driven Kuramoto model, a system with no spatial structure. We conclude by discussing qualitative implications of our results for controlling coupled oscillator systems such as the social amoebae Dictyostelium and populations of Belousov Zhabotinsky (BZ) catalytic particles using spatially structured external drives.

Modulational instability and soliton propagation in an alternate right-handed and left-handed multi-coupled nonlinear dissipative transmission network

Abstract We investigate analytically the dynamics of modulated waves in an alternate right-handed and left-handed multi-coupled nonlinear dissipative discrete electrical lattice, made of several of the well-known modified Noguchi electrical transmission network that are transversely coupled to one another by a linear capacitor C 2 . The damped two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) governing slowly modulated wave propagation in this network system is derived. Considering modulated Stokes waves propagating in the network, we establish their criteria of instability, derive the analytical expression of the modulational instability (MI) growth rate, and show that the presence of the dispersive element C S softens the instability of the network system. Through the MI growth rate, we establish that the instability features of our network system show the famous donut Benjamin–Feir instability region. With the help of the linear dispersion law, we show that the behavior of our network model depends on that of the propagating frequency f p . It can adopt purely right-handed (corresponding to monotone increasing f p ), purely left-handed (associated to monotone decreasing f p ) or composite right-/left-handed (for composite monotone increasing and monotone decreasing f p ) behaviors without changing its structure. Also, the network system under consideration supports purely backward (when f p is positive), purely forward (when f p is negative) or composite backward/forward (when f p changes its sign) traveling waves. It appears that the network is right-handed (RH) and/or composite right-/left-handed (RH/LH) and supports backward traveling waves only for very low frequencies and becomes left-handed and supports only forward traveling waves for only high frequencies. Though the exact and approximative soliton-like solutions of the derived CGLE of the network, we investigate analytically RH, LH, as well as RH/LH behaviors of the system and show how to manipulate the transversal wave number to modify the behavior of the network, as well as the width and the motion of the bright solitary voltage signals in the network. We show that our network system supports RH and LH bright and kink soliton signals that simultaneously propagate at the same frequency; we also show that it supports backward and forward LH bright and kink soliton signals which simultaneously propagate at opposite frequency.

A linearized element-free Galerkin method for the complex Ginzburg–Landau equation

In this paper, an effective linearized element-free Galerkin (EFG) method is developed for the numerical solution of the complex Ginzburg–Landau (GL) equation. To deal with the time derivative and the nonlinear term of the GL equation, an explicit linearized procedure is presented. The unconditional stability and the error estimate of the procedure are analyzed. Then, a stabilized EFG method is proposed to establish linear algebraic systems. In the method, the penalty technique is used to facilitate the satisfaction of boundary conditions, and the stabilized moving least squares approximation is used to enhance the stability and performance. The linearized EFG method is a meshless method and possesses high precision and convergence rate in both space and time. Theoretical error and convergence of the linearized EFG method are analyzed. Finally, some numerical results are provided to demonstrate the efficiency of the method and confirm the theoretical results.

The existence of exponentially decreasing solutions to time dependent hyperbolic systems

We consider the time-dependent hyperbolic system which can be reduced from the time dependent PDE equations, such as the time-dependent complex Ginzburg-Landau equations, Boussinesq equations and sublinear Duffing equations, which are infinite-dimensional systems and finite-dimensional systems respectively. Note that the solution to the hyperbolic system is very sensitive to initial datum, by carefully choosing the initial datum, we construct some exponentially decreasing solution with respect to time-variable t∈R. This is different from the previous results, which proved the existence of exponentially decreasing solution only for t∈R+. As a consequence, we also construct exponentially decreasing solutions for time dependent PDE systems.

Thermocapillary instability on a film falling down a non-uniformly heated slippery incline

A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number Mn as well as slip length δ. This depicts that δ and Mn both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of Mn and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of Mn, when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude (ζa) decreases (increases) with the increment of Mn and δ. Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum hmax and minimum hmin thickness amplifies, for small change of Mn and δ. Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ. The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as Mn increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ>max16Mn−13,12Mn−23−Mn.

Laser dynamics in nonlinear transparent media with electron plasma generation: effects of electron–hole radiative recombinations

The performance of optical devices manufactured via laser micromachining on nonlinear transparent materials usually relies on three main factors, which are the characteristic laser parameters (i.e. the laser power, pulse duration and pulse repetition rate), the characteristic properties of host materials (e.g. their chromatic dispersions, optical nonlinearities or self-focusing features) and the relative importance of physical processes such as the avalanche impact ionization, multiphoton ionization and electron–hole radiative recombination processes. These factors act in conjunction to impose the regime of laser operation; in particular, their competition determines the appropriate laser operation regime. In this work a theoretical study is proposed to explore the effects of the competition between multiphoton absorption, plasma ionization and electron–hole radiative recombination processes on the laser dynamics in transparent materials with Kerr nonlinearity. The study rests on a model consisting of a K-order nonlinear complex Ginzburg–Landau equation, coupled to a first-order equation describing time variation of the electron plasma density. An analysis of the stability of continuous waves, following the modulational instability approach, reveals that the combination of multiphoton absorption and electron–hole radiative recombination processes can be detrimental or favorable to continuous-wave operation, depending on the group-velocity dispersion of the host medium. Numerical simulations of the model equations in the fully nonlinear regime reveal the existence of pulse trains, the amplitudes of which are enhanced by the radiative recombination processes. Numerical results for the density of the induced electron plasma feature two distinct regimes of time evolution, depending on the strength of the electron–hole radiative recombination processes.

New Exact Solutions of the Fractional Complex Ginzburg–Landau Equation

In this paper, we apply the complete discrimination system method to establish the exact solutions of the fractional complex Ginzburg–Landau equation in the sense of the conformable fractional derivative. Firstly, by the fractional traveling wave transformation, time-space fractional complex Ginzburg–Landau equation is reduced to an ordinary differential equation. Secondly, some new exact solutions are obtained by the complete discrimination system method of the three-order polynomial; these solutions include solitary wave solutions, rational function solutions, triangle function solutions, and Jacobian elliptic function solutions. Finally, two numerical simulations are imitated to explain the propagation of optical pulses in optic fibers. At the same time, the comparison between the previous results and our results are also given.

Parameter estimation in systems exhibiting spatially complex solutions via persistent homology and machine learning

We use persistent homology to extract topological information from complex spatio-temporal data generated by differential equations and use this information to estimate the corresponding parameters of the differential equation using regression methods in machine learning. We apply this technique to a predator–prey system and to the complex Ginzburg–Landau equation.

Asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains

This paper mainly investigates the asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains. We first prove the existence and uniqueness of random attractors for the considered equation and its limit equation. Due to the non-compactness of Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of such a stochastic equation is proved by the tail-estimate method. Then, we show the upper semi-continuity of random attractors when thin domains collapse onto the real space R.

Cubic-quartic optical soliton perturbation in polarization-preserving fibers with complex Ginzburg-Landau equation having five nonlinear refractive index structures

In this research, we present a comprehensive coverage of soliton solutions to perturbed cubic-quartic (CQ) nonlinear complex Ginzburg-Landau (NLCGL) equation with five nonlinear laws to the refractive index. We specialize in studying the nonlinearity in the five different forms: Kerr law, parabolic law, polynomial law, quadratic-cubic law and parabolic non-local law. The integration scheme, namely the addendum to Kudryashov's approach has made this retrieval possible. For five laws nonlinearity, the analytical solutions are clearly constructed the combo-bright-singular, bright, dark and singular optical soliton solutions.

New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extendedG′/G-expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

In–Out Intermittency with Nested Subspaces in a System of Globally Coupled, Complex Ginzburg–Landau Equations

Chaos and intermittency are studied for the system of globally coupled, complex Ginzburg–Landau equations governing the dynamics of extended, two-dimensional anisotropic systems near an oscillatory (Hopf) instability of a basic state with two pairs of counterpropagating, oblique traveling waves. Parameters are chosen such that the underlying normal form, which governs the dynamics of the spatially constant modes, has two symmetry-conjugated chaotic attractors. Two main states residing in nested invariant subspaces are identified, a state referred to as Spatial Intermittency ([Formula: see text]) and a state referred to as Spatial Persistence ([Formula: see text]). The [Formula: see text]-state consists of laminar phases where the dynamics is close to a normal form attractor, without spatial variation, and switching phases with spatiotemporal bursts during which the system switches from one normal form attractor to the conjugated normal form attractor. The [Formula: see text]-state also consists of two symmetry-conjugated states, with complex spatiotemporal dynamics, that reside in higher dimensional invariant subspaces whose intersection forms the 8D space of the spatially constant modes. We characterize the repeated appearance of these states as (generalized) in–out intermittency. The statistics of the lengths of the laminar phases is studied using an appropriate Poincaré map. Since the Ginzburg–Landau system studied in this paper can be derived from the governing equations for electroconvection in nematic liquid crystals, the occurrence of in–out intermittency may be of interest in understanding spatiotemporally complex dynamics in nematic electroconvection.

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.

Solutions to Space-Time Fractional Complex Ginzburg-Landau Equations With the Complete Discrimination System for Polynomial Method

The space-time fractional complex Ginzburg-Landau equation was studied. Firstly, the space-time fractional complex Ginzburg-Landau equation was transformed into the ordinary differential equation through the fractional complex transform. Secondly, the ordinary differential equation was reduced to an elementary integral form. Finally, a series of exact solutions including solitary wave solutions, rational function type solutions, triangle function type periodic solutions, and Jacobian elliptic function doubly-periodic solutions, were constructed with the complete discrimination system for polynomial method.

Ginzburg--Landau Patterns in Circular and Spherical Geometries: Vortices, Spirals, and Attractors

This paper consists of three results on pattern formation of Ginzburg-Landau $m$-armed vortex solutions and spiral waves in circular and spherical geometries. First, we completely describe the global bifurcation diagram of vortex equilibria. Second, we prove persistence of all bifurcation curves under perturbations of parameters, which yields the existence of spiral waves for the complex Ginzburg-Landau equation. Third, we explicitly construct the global attractor of $m$-armed vortex solutions. Our main tool is a new shooting method that allows us to prove hyperbolicity of vortex equilibria in the invariant subspace of vortex solutions.

Ginzburg--Landau Spiral Waves in Circular and Spherical Geometries

We prove the existence of $m$-armed spiral wave solutions for the complex Ginzburg-Landau equation in the circular and spherical geometries. We establish a new global bifurcation approach and generalize the results of existence for rigidly-rotating spiral waves. Moreover, we prove the existence of two new patterns: frozen spirals in the circular and spherical geometries, and 2-tip spirals in the spherical geometry.

Traveling Wave Solutions of the Quintic Complex One-Dimensional Ginzburg-Landau Equation

A subset of traveling wave solutions of the quintic complex Ginzburg-Landau equation (QCGLE) is presented in compact form. The approach consists of the following parts: 1) Reduction of the QCGLE to a system of two ordinary differential equations (ODEs) by a traveling wave ansatz; 2) Solution of the system for two (ad hoc) cases relating phase and amplitude; 3) Presentation of the solution for both cases in compact form; 4) Presentation of constraints for bounded and for singular positive solutions by analysing the analytical properties of the solution by means of a phase diagram approach. The results are exemplified numerically.

Dynamics of high-frequency modulated waves in a nonlinear dissipative continuous bi-inductance network

This paper presents intensive investigation of dynamics of high frequency nonlinear modulated excitations in a damped bimodal lattice. The effects of the dissipation are considered through a linear dissipation coefficient whose evolution in terms of the carrier wave frequency is checked. There appears that the dissipation coefficient increases with the carrier wave frequency. In the linear limit and for high frequency waves, study of the asymptotic behavior of plane waves reveals the existence of two additional regions in the dispersion curve where the modulational phenomenon is observed compared to the lossless line. Based on the multiple scales method exploited in the continuum approximation using an appropriate decoupling ansatz for the voltage of the two different cells, it appears that the motion of modulated waves is described by a dissipative complex Ginzburg–Landau equation instead of a Korteweg–de Vries equation. We also show that this amplitude wave equation admits envelope and hole solitons in the high frequency mode. From basic sources, we design a programmable electronic generator of complex signals with desired characteristics, which delivers signals exploited as input waves for all our numerical simulations. These simulations are performed in the LTspice software that uses realistic components and give the results that corroborate perfectly our analytical predictions.