Complex Ginzburg-Landau Equation Research Articles

Frequency modes of unstable spiral waves in two-dimensional Rosenzweig–MacArthur ecological networks

The existence of two frequency regimes in a two-dimensional (2D) Rosenzweig–MacArthur ecological network is debated. The semi-discrete approximation differentiates the two regimes, each described by a 2D complex Ginzburg–Landau equation. Using the standard theory of the linear stability analysis, a generalized expression for the modulational instability growth rate is derived for each frequency mode. The parametric study of the growth rate of modulational instability reveals its sensitivity to the changes in the recruitment rate of the resources. Moreover, direct numerical simulations are carried out to confirm our analytical results. Over the prolonged evolution of the perturbed plane wave solution, the high-frequency mode entertains spiral wave patterns. In contrast, the appearance of target waves manifests the low-frequency regime. In that context, we further explore the impact of the recruitment rate of resources and give the qualitative meaning of the obtained dynamical behaviors and their ecological implications. This work may additionally provide more insight into the mechanism leading to spiral and target waves in environmental systems.

A Note on the Well-Posedness of Control Complex Ginzburg-Landau Equations in Zhidkov Spaces

In this note, we consider the Complex Ginzburg-Landau equations with a bilinear control term in the real line. We prove well-posedness results concerned with the initial value problem for these equations in Zhidkov spaces using Splitting methods.

Optical solitons for the complex Ginzburg–Landau equation with Kerr law and non-Kerr law nonlinearity

Optical solitons for the complex Ginzburg–Landau equation with Kerr law and non-Kerr law nonlinearity

Dynamics of stable solitons in Complex Ginzburg–Landau equation with [formula omitted]-symmetric Gaussian potential

In this paper, we investigate the stable propagation of soliton in Complex Ginzburg–Landau (CGL) equation with self focusing nonlinear mode in the presence of PT-symmetric Gaussian potential. In our model, we find the required condition to obtain the stable solution is that the value of spectral filtering is negative and the positive values of diffraction, Kerr nonlinearity and nonlinear gain/loss. By satisfying this condition, one can manipulate the stable propagation, intensity and power conservation of the soliton by simply varying the strength of the imaginary part of the complex PT-symmetric potential. In other words, the dynamics of the soliton depend on the strength of the imaginary part of potential irrespective of the value of propagation constant.

Multidimensional dissipative solitons and solitary vortices

This article offers a review of (chiefly, theoretical) results for self-trapped states (solitons) in two- and three-dimensional (2D and 3D) models of nonlinear dissipative media. The existence of such solitons requires to maintain two stable balances: between nonlinear self-focusing and linear spreading (diffraction and/or dispersion) of the physical fields, and between losses and gain in the medium. Due to the interplay of these conditions, dissipative solitons exist, unlike solitons in conservative models, not in continuous families, but as isolated solutions (attractors). The main issue in the theory is stability of multidimensional dissipative solitons, especially ones with embedded vorticity. First, stable 2D dissipative solitons are presented in the framework of the complex Ginzburg–Landau equation with the cubic–quintic nonlinearity, which combines linear loss, cubic gain, and quintic loss (the linear loss is necessary to stabilize zero background around dissipative solitons, while the quintic loss provides for the global stability of the setting). In addition to fundamental (zero-vorticity) solitons, stable spiral solitons produced by the CGL equation are produced too, with intrinsic vorticities S=1 and 2. Stable 2D solitons were also found in a system built of two linearly-coupled optical fields, with linear gain acting in one and linear loss, which plays the stabilizing role, in the other. In this case, the inclusion of the cubic loss (without quintic terms) is sufficient for the creation of stable fundamental and vortical dissipative solitons in the linearly-coupled system. In addition to truly localized states, weakly localized ones are presented too, in the single-component model with nonlinear losses, which does not include explicit gain. In that case, the losses are compensated by the influx of power from the reservoir provided, at the spatial infinity, by the weakly localized structure of the solution. Other classes of 2D models which are considered in this review make use of spatially modulated loss or gain to predict many species of robust dissipative solitons, including localized dynamical states featuring complex periodically recurring metamorphoses. Stable fundamental and vortical solitons are also produced by models including a trapping or spatially periodic potential. In the latter case, the consideration addresses 2D gap dissipative solitons as well. 2D two-component dissipative models including spin–orbit coupling are considered too. They give rise to stable states in the form of semi-vortex solitons, with vorticity carried by one component. In addition to the 2D solitons, the review includes 3D fundamental and vortical dissipative solitons, stabilized by the cubic–quintic nonlinearity and/or external potentials. Collisions between 3D dissipative solitons are considered too.

Soliton-mediated ionic pulses and coupled ionic excitations in a dissipative electrical network model of microtubules

Many cellular activities are mediated by microtubules (MTs), with most electrophysiological processes depending on ionic flow through MT cylinders. This paper addresses such conductive features by representing the MT as a nonlinear electrical transmission line composed of capacitive and dissipative properties. Thanks to the semi-discrete approximation near the continuum limit, coupled ionic pulses flowing along cellular microtubules are described by a set of coupled cubic complex Ginzburg-Landau (CGL) equations whose one of the solutions is a plane wave. The stability of the latter is checked, under weak modulation, using an explicit analytical expression for the modulational instability (MI) growth rate. The parametric analysis of the instability growth rate allows detecting parameter regions where ionic conductivity, through modulated waves, in the MT network is likely to occur. Dissipative bright-bright pulse soliton solutions for the nonlinearly coupled cubic CGL equations are constructed using a modified Hirota's bilinear method. A generalized coupled mode for the discrete ionic signal is proposed and used as the initial condition to be propagated in the nonlinear electrical transmission lattice of MT under direct numerical simulations. The ionic pulse transfer, mediated by the nonlinear interaction between oscillatory modes, is manifested by the formation of trains of modulated waves whose behaviors depend on the right choice of system parameters. Those results theoretically suggest that coupled ionic signals may facilitate information processing involving MTs.

Comment on “A comparative study of the optical solitons for the fractional complex Ginzburg–Landau equation using different fractional differential operators, Optik 256 (2022) 168626”

In a recently published article, G. Akram et al proposed a new extended ϕ6-model expansion method to construct optical solitons for the fractional complex Ginzburg–Landau equation with Kerr law nonlinearity using different fractional differential operators. In this article, we point out that there is additional constraint condition in the proposed extended ϕ6-model expansion method. Neglecting the additional constraint condition will cause mistakes.

Study of primary and secondary instabilities arising due to a chemical reaction in a two-component Rayleigh–Bénard system

Manifestation of stationary and oscillatory convection and secondary instabilities due to a chemical reaction in a two-component convective fluid system is reported in the paper by considering idealistic as well as physically realistic boundaries. Using a normal mode solution, analytical expression of the critical Rayleigh number for a stationary and oscillatory disturbances, and the natural frequency are reported. The range of parameters is identified where oscillatory motion happens. Further, the parameters’ range for existence of oscillatory regime is found to be larger for rigid boundaries compared to that of free boundaries. Furthermore, for both the boundaries, parameters’ range for this regime increases when the chemical reaction rate increases, leading to the conclusion that the oscillatory motion emerges as the most preferred mode in the two-component system due to the presence of a chemical reaction and the size of this domain is directly proportional to the chemical reaction rate. The marginal stability plots depict that the oscillatory and stationary regimes respectively correspond to Hopf and direct pitchfork bifurcations. The critical Rayleigh number and the wave number where codimension two bifurcation exists are documented in the paper for fixed values of parameters. It is shown that the codimension two bifurcation that arose in the problem is not a Takens–Bogdanov bifurcation. In a stationary regime, the domain for secondary instabilities of Eckhaus and zigzag is obtained using the spatio-temporal Newell–Whitehead–Segel equation. These instabilities grow with increasing chemical reaction rate. In the oscillatory regime, the complex Ginzburg–Landau equation is used to predict the appearance of the Benjamin–Feir instability.

Highly dispersive optical solitons in birefringent fibers for complex-Ginzburg–Landau equation with parabolic law of nonlinearity using two integration techniques

Object:This paper aims to deduce optical soliton solutions of the highly dispersive optical solitons with perturbed complex-Ginzburg–Landau equation having parabolic law nonlinear refractive index in birefringent fibers. Methods:The addendum Kudryashov’s method and algebraic sub-equation expansion method are used for solving the given system. Results:Combo soliton solutions, bright soliton solutions and singular soliton solutions as well as rational and periodic wave solutions have been deduced.

Exact solutions of the complex Ginzburg–Landau equation with law of four powers of nonlinearity

The complex Ginzburg–Landau equation having polynomial law of nonlinearity with four powers is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform. However the partial differential equation admits the translation groups in two independent variables and we look for the general solution using the traveling wave reduction. The first integral of nonlinear ordinary differential equation corresponding to the complex Ginzburg–Landau equation with the four powers of nonlinearity is found. This first integral is reduced to the nonlinear ordinary differential equation of the first order with the general solution expressed via the elliptic function. We demonstrate that the direct method allows us to obtain exact solutions without constraints on the parameters of the mathematical model. Partial cases of bright and dark optical solitons of the equation are given.

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.

Oscillatory dissipative solitons stabilized by nonlinear gradient terms: The transition to localized spatiotemporal disorder.

We investigate properties of oscillatory dissipative solitons (DSs) in a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. As a main result we find a transition to dissipative solitons with spatiotemporal disorder as a function of the diffusion coefficient. This transition proceeds via quasiperiodicity and shows incommensurate satellites next to the fundamental frequency and its harmonics indicating a possible route to localized spatiotemporal chaos. The transition back to oscillatory DSs follows a similar scenario.

Two-Dimensional Gross–Pitaevskii Equation With Space-Time White Noise

Abstract In this paper we consider the two-dimensional stochastic Gross–Pitaevskii equation, which is a model to describe Bose–Einstein condensation at positive temperature. The equation is a complex Ginzburg–Landau equation with a harmonic potential and an additive space-time white noise. We study the global well posedness of the model using an inhomogeneous Wick renormalization due to the potential and prove the existence of an invariant measure.

Linear and weakly non-linear stability analysis of oscillatory convection in rotating ferrofluid layer

In the present paper, the problem of ferroconvection in the presence of uniform vertical rotation is considered to investigate the linear and weakly non-linear oscillatory stability. Two-dimensional convective roll instability is discussed with finite-amplitude disturbances. For linear stability, as a first-order problem, the expressions for Rayleigh numbers for stationary and oscillatory convection are derived and the effects of Coriolis force and magnetic parameters on the onset of ferromagnetic convection are studied, numerically. In weakly nonlinear oscillatory analysis, the second-order and third-order stability problems are discussed and the complex Ginzburg-Landau equation describing the amplitude of convection cell in rotating ferrofluid is derived and consequently, the expression for Nusselt number representing the heat transfer rate is obtained. From the present analysis, we observed that the rotation has the usual stabilizing effect on the linear stability in ferroconvection, however, the magnetic number (M1) and the measure of nonlinearity of magnetization (M3) both have a destabilizing effect on the onset of linear instability. Also, we found that for non-linear convection, the heat transfer rate (the Nusselt number) increases with increasing values of Taylor number, magnetic number, and the measure of nonlinearity of magnetization.

Analytical study of complex Ginzburg–Landau equation arising in nonlinear optics

In this paper, some optical solitons and other solutions of the Complex Ginzburg–Landau (CGL) equation with the beta time derivative are analyzed in nonlinear optics. The kink, bright, and dark solitons of the model are acquired using the [Formula: see text]-expansion method. It is worth mentioning that the obtained and precise solutions are graphically displayed. To summarize, some relatively novel solutions and conclusions are obtained from the analysis of the model, which serves as a reference for the development of the Ginzburg–Landau equation in nonlinear optics.

Highly dispersive optical solitons in birefringent fibers for perturbed complex Ginzburg-Landau equation having polynomial law of nonlinearity

Objective:In our article, we deduce the soliton solutions for highly dispersive perturbed complex-Ginzburg–Landau equation in birefringent fibers having polynomial law of nonlinearity. Methods:In this study, we use two methods, namely the G′/G–expansion method and the addendum Kudryashov’s method. Results:We have found the straddled, combo bright–singular, dark, bright and singular solitons solutions as long as the rational solutions of the proposed equation.

Stationary optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion and Kudryashov's refractive index structures

This work obtains stationary optical solitons for complex Ginzburg–Landau equation that is structured with Kudryashov's self–phase modulation structures. The chromatic dispersion is also taken to be nonlinear. Six forms of nonlinear refractive index are considered. The adopted integration scheme is the generalized G′/G-expansion approach that yields solutions in terms of Jacobi's elliptic functions. The limiting approach, when applied with the modulus of ellipticity leads to the stationary optical solitons that finally emerge from the model.

Unconditional optimal error estimates of a modified finite element fully discrete scheme for the complex Ginzburg-Landau equation

In this paper, based on the 2-step backward differentiation formula (BDF2 for short) in time and the nonconforming Wilson element in space, a modified Galerkin finite element method named BDF2-MG FEM is proposed to solve the nonlinear complex Ginzburg-Landau equation (GLE for short). On one hand, a modified Ritz projection operator Rh is introduced and analyzed, which plays an important role in getting the unconditional optimal error estimates. On the other hand, a time-discrete system is constructed with the linearized BDF2 and the regularity is derived with the temporal error results. Combining these two aspects, the errors between RhUn and Uhn with order O(h3+h2△t) in L2-norm and O(h2+h2△t) in the modified energy norm are deduced, where h is the subdivision parameter, △t is the time step, Un and Uhn denote the solutions of the time-discrete system and the BDF2-MG FEM respectively. Therefore the boundedness of ‖Uhn‖0,∞ is proven without any restriction on the time-space grid ratio. Furthermore, by using the properties of Rh, unconditional optimal error estimates of order O(h3+(△t)2) in L2-norm and O(h2+(△t)2) in the modified energy norm are obtained directly. It should point out the spatial discrete errors of the BDF2-MG FEM are all one order higher than that of the BDF2 traditional Galerkin finite element method with Wilson element for the GLE. At last, a numerical experiment is presented to verify the validity of the theoretical analysis.

Predicting extreme events from data using deep machine learning: When and where

We develop a framework based on the deep convolutional neural network (DCNN) for model-free prediction of the occurrence of extreme events both in time (``when'') and in space (``where'') in nonlinear physical systems of spatial dimension two. The measurements or data are a set of two-dimensional snapshots or images. For a desired time horizon of prediction, a proper labeling scheme can be designated to enable successful training of the DCNN and subsequent prediction of extreme events in time. Given that an extreme event has been predicted to occur within the time horizon, a space-based labeling scheme can be applied to predict, within certain resolution, the location at which the event will occur. We use synthetic data from the two-dimensional complex Ginzburg-Landau equation and empirical wind speed data from the North Atlantic Ocean to demonstrate and validate our machine-learning-based prediction framework. The trade-offs among the prediction horizon, spatial resolution, and accuracy are illustrated, and the detrimental effect of spatially biased occurrence of extreme events on prediction accuracy is discussed. The deep learning framework is viable for predicting extreme events in the real world.

Long-wave instabilities of evaporating/condensing viscous film flowing down a wavy inclined wall: Interfacial phase change effect of uniform layers

The interfacial phase change effect on a thin film flowing down an undulated wall has been investigated in the present study. The study is performed for a general periodic undulated bottom of moderate steepness that is long compared to the film thickness, followed by a case study over the sinusoidal bottom. The long-wave instabilities of the ununiform film are used by deriving a nonlinear evolution equation in the classical long-wave expansion method framework. The one-equation model can track the free surface evolution and involve the bottom undulation, viscosity, gravity, surface tension, and phase change (evaporation/condensation) effects. Linear stability analysis shows that the bottom steepness ζ has a dual role. In the downhill region, increasing ζ destabilizes, whereas increasing ζ stabilizes in the uphill region. Weakly nonlinear waves are studied using the method of multiple scales to obtain the complex Ginzburg–Landau equation. The results show that both supercritical and subcritical solutions are possible for evaporating and condensate film. Interestingly, while one subcritical region is visible for an evaporating film, two subcritical unstable regions are found for condensate film. The numerical solution of the free-surface equation demonstrates the finite-amplitude behavior that tends to dry out for an evaporating film. For condensate film, the thickness increases rapidly. The rupture dynamics highly depend on the initial perturbation, and the bottom steepness has a negligible effect on it. Kutateladze number has a significant impact on the stability characteristic of the film flow as it represents a sort of efficiency of phase change that occurs at the interface.