Quintic Complex Ginzburg-Landau Equation Research Articles

Soliton-like nonlinear excitation in the FitzHugh–Nagumo cardiac model through the cubic–quintic complex Ginzburg–Landau equation

Soliton-like nonlinear excitation in the FitzHugh–Nagumo cardiac model through the cubic–quintic complex Ginzburg–Landau equation

Efficient techniques for nonlinear dynamics: a study of fractional generalized quintic Ginzburg-Landau equation

In this research, we explore the solution of the fractional Generalized Quintic Complex Ginzburg-Landau equation (GQCGLE) using the Controlled Picard Transform Method. We solve the equation by incorporating the Laplace transform (LT) and the Caputo-Fabrizio fractional derivative (CF). The fractional GQCGLE is a complex-valued equation that exhibits intricate dynamics and plays a significant role in various physical phenomena, particularly in the realm of nonlinear optics. The suggested method can efficiently address a wide variety of differential equations, encompassing both integer and fractional-order equations. This is achieved by integrating an extra parameter that enhances convergence, making it especially suitable for non-linear differential equations. To confirm the precision and convergence of our approach, we validate it by comparing it with other methods, presenting graphical representations to clarify the effects of different parameters on the solution behaviour, and confirming its precision and convergence. The existence and uniqueness of the solutions are also examined.

Impact of the Higher-Order Reactive Nonlinearity on High-Amplitude Dissipative Solitons

In this work, the impact of the higher-order reactive nonlinearity on very high-amplitude solitons of the cubic–quintic complex Ginzburg–Landau equation is investigated. These high amplitude pulses were found in a previous work in the normal and anomalous dispersion regimes, starting from a singularity found by Akhmediev et al. We focus mainly in the normal dispersion regime, where the energy of such pulses is particularly high. In the presence of the higher-order reactive nonlinearity effect, pulse formation are observed for much higher absolute values of dispersion. Under such effect, the amplitude and the energy of the VHA pulses decrease, while their spectral range shrinks. Numerical computations are in good agreement with the predictions based on the method of moments, in the absence of the higher-order reactive nonlinearity effect. However, in the presence of this effect such agreement becomes mainly qualitative. A region of existence of the very high-amplitude pulses was found in the semi-plane defined by the normal dispersion and nonlinear gain.

The Dynamics of Interacting Multi-pulses in the One-dimensional Quintic Complex Ginzburg–Landau Equation

The Dynamics of Interacting Multi-pulses in the One-dimensional Quintic Complex Ginzburg–Landau Equation

All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations

For both cubic and quintic nonlinearities of the one-dimensional complex Ginzburg-Landau evolution equation, we prove by a theorem of Eremenko the finiteness of the number of traveling waves whose squared modulus has only poles in the complex plane, and we provide all their closed form expressions. Among these eleven solutions, five are provided by the method used. This allows us to complete the list of solutions previously obtained by other authors.

Dynamics of the (2+1)$$ \left(2+1\right) $$‐dimensional coupled cubic–quintic complex Ginzburg–Landau model for convecting binary fluid: Analytical investigations and multiple‐soliton solutions

The ‐dimensional coupled cubic–quintic complex Ginzburg–Landau equations ( ‐DCC‐QCGLEs) can simulate a variety of binary fluid thermal convection characteristics, containing complex parameters. The analysis of pattern formation in chaotic and nonlinear dynamical systems can benefit greatly from the use of this thermal convection model. The primary goal of this study is to find analytical solutions for some recent advances that have been made for Rayleigh–Bénard convection by applying the ‐DCC‐QCGLEs model for slowly varying spatio‐temporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation are derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. Furthermore, we introduce the WTC‐Kruskal algorithm of the Painlevé methodology to examine the integrability of the ‐DCC‐QCGLEs and the truncated Painlevé expansion is used to extract the Bäcklund transform, from which new solitary solutions can be acquired. The results also demonstrated a good agreement with previous works and were more significant and accurate in two and three dimensions of the proposed model. Finally, the computational results indicate that the effects of the physical parameters of the considered equations can be demonstrated by utilizing 2D and 3D graphics for different values of these parameters.

The chirped wave propagation in modified cubic–quintic complex Ginzburg–Landau equation with parabolic law

This paper investigates the modified cubic–quintic complex Ginzburg–Landau equation with parabolic law that depictures the propagation of chirped wave. The used methodologies are the trial equation method and the complete discriminant system for polynomial method. As a result, nineteen exact chirped solutions are obtained, and several typical numerical simulations under specific parameter conditions are given.

Propagation characteristics of bright and mixed solitons based on the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation

In the study of telecommunication system, the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation is used as the optical solitons transmission model, which not only explains the physical meaning of the existing model with quintic terms, but also has more nonlinear dynamics characteristics of the higher dimensional system than the lower dimensional system. In this paper, the analytical soliton solutions of the (3+1)-dimensional cubic-quintic CGL equations with variable coefficients are obtained by using the modified Hirota method. By selecting certain parameters of the nonlinear coefficients and spectral filtering terms, a special kind of mixed soliton solution is obtained, which has the characteristics of bright soliton, dark soliton and kinked soliton at the same time. Subsequently, the influence of changing the nonlinear, spectral filtering, linear loss parameters and other parameters on the transmission characteristics of solitons is discussed respectively, so as to realize the control of optical solitons, which can not only control the propagation of optical solitons in different forms, but also can realize the adjustment of the amplitude and pulse width of the pulse and control the propagation direction and energy of the pulse for the mixed solitons of a particular form. The research results of high dimensional CGL system in this paper can be applied to nonlinear optical system, ultra-fast optical digital logic system and other different experiments and application fields.

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.

The essential spectrum of periodically stationary solutions of the complex Ginzburg–Landau equation

We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic–quintic complex Ginzburg–Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg–Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.

The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic–quintic complex Ginzburg–Landau equations on complicated geometries

Our motivation in this contribution is to propose a new numerical algorithm for solving cubic–quintic complex Ginzburg-Landau (CQCGL) equations. The developed technique is based on the following stages. At the first step, the nonlinear CQCGL equation is splitted in the three problems that two of them don’t have the space derivative e.g problems (I) and (III) and one of them has the space derivative e.g Problem (II). At the second stage, the Problems (I) and (III) can be considered as two ODEs and they are solved by using a fourth-order exponential time differencing Runge-Kutta (ETDRK4) method to get a high-order numerical approximation. Furthermore, the Problem (II) is solved by using direct meshless finite volume method. The proposed method is a new high-order numerical procedure based on a truly meshless method for solving the complex PDEs on non-rectangular computational domains. Moreover, various samples are investigated that verify the efficiency of the new numerical scheme.

Vector dissipative light bullets in optical laser beam

The dynamics of light bullets propagating in nonlinear media with linear/nonlinear, gain/loss and coupling described by the (2+1)-dimensional vectorial cubic–quintic complex Ginzburg–Landau (CGL) equations is considered. The evolution and the stability of the vector dissipative optical light bullets, generated from an asymmetric input with respect to two transverse coordinates x and y, are studied. We use the variational method to find a set of differential equations characterizing the variation of the light bullet parameters in the laser cavity. This approach allows us to analyze the influence of various physical parameters on the dynamics of the propagating beam and its relevant parameters. Then, we solve the original coupled (2+1)D cubic–quintic CGL equation using the split-step Fourier method. Numerical results and analytical predictions are confronted, and a good agreement between the two approaches is obtained.

Comments on “Application of the first integral method for solving (1 + 1) dimensional cubic–quintic complex Ginzburg–Landau equation, Optik 164 (2018) 210–217″

In their recently published article, G. Akram et al. investigated (1 + 1) dimensional cubic–quintic complex Ginzburg–Landau equation and constructed its exact solutions via the first integral method. We argue that all of solutions obtained by G. Akram et al. are incorrect and do not satisfy the considered equation. Furthermore, we present correct solutions as well as constraint conditions in this article.

Generation of ring-shaped optical vortices in dissipative media by inhomogeneous effective diffusion

By means of systematic simulations, we demonstrate generation of a variety of ring-shaped optical vortices (OVs) from a two-dimensional input with embedded vorticity, in a dissipative medium modeled by the cubic–quintic complex Ginzburg–Landau equation with an inhomogeneous effective diffusion (spatial filtering) term, which is anisotropic in the transverse plane and periodically modulated in the longitudinal direction. We show the generation of stable square- and gear-shaped OVs, as well as tilted oval-shaped vortex rings, and string-shaped bound states built of a central fundamental soliton and two vortex satellites, or of three fundamental solitons. Their shape can be adjusted by tuning the strength and modulation period of the inhomogeneous diffusion. Stability domains of the generated OVs are identified by varying the vorticity of the input and parameters of the inhomogeneous diffusion. The results suggest a method to generate new types of ring-shaped OVs with applications to the work with structured light.

Integrability of the coupled cubic–quintic complex Ginzburg–Landau equations and multiple-soliton solutions via mathematical methods

This paper is devoted to study the (1+1)-dimensional coupled cubic–quintic complex Ginzburg–Landau equations (cc–qcGLEs) with complex coefficients. This equation can be used to describe the nonlinear evolution of slowly varying envelopes of periodic spatial–temporal patterns in a convective binary fluid. Dispersion relation and properties of cc–qcGLEs are constructed. Painlevé analysis is used to check the integrability of cc–qcGLEs and to establish the Bäcklund transformation form. New traveling wave solutions and a general form of multiple-soliton solutions of cc–qcGLEs are obtained via the Bäcklund transformation and simplest equation method with Bernoulli, Riccati and Burgers’ equations as simplest equations.

Parametrically driven scalar field in an expanding background

We study the existence and dynamic behavior of localized and extended structures in a massive scalar inflaton field $\ensuremath{\phi}$ in $1+1$ dimensions in the framework of an expanding universe with constant Hubble parameter. We introduce a parametric forcing, produced by another quantum scalar field $\ensuremath{\psi}$, over the effective mass squared around the minimum of the inflaton potential. For this purpose, we study the system in the context of the cubic quintic complex Ginzburg-Landau equation and find the associated amplitude equation to the cosmological scalar field equation, which near the parametric resonance allows us to find the field amplitude. We find homogeneous null solutions, flat-top expanding solitons, and dark soliton patterns. No persistent non-null solutions are found in the absence of parametric forcing, and divergent solutions are obtained when the forcing amplitude is greater than $4/3$.

Numerical characterization of nonlinear oscillatory waves in a composite right‐ and left‐handed traveling‐wave field‐effect transistor

SummaryA specialized type of traveling‐wave field‐effect transistor, the gate and drain lines of which have composite right‐ and left‐handed structures, is considered as the platform to support nonlinear oscillatory waves. The cubic–quintic complex Ginzburg–Landau equation is obtained by application of the reductive perturbation method, by which we quantify the homogeneous oscillations including the property of the Andronov–Hopf bifurcation point, oscillation frequency, and amplitude. Several numerical calculations follow to validate the Ginzburg–Landau equation‐based analysis. Finally, the dynamics of numerically obtained stationary flat‐top pulses are discussed. Copyright © 2016 John Wiley & Sons, Ltd.

Spectra of Short Pulse Solutions of the Cubic–Quintic Complex Ginzburg–Landau Equation near Zero Dispersion

We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic–quintic complex Ginzburg–Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights the importance for analytical studies of perturbed nonlinear wave equations of using a pulse ansatz in which the phase is not constant, but rather depends on the perturbation parameter. In the presence of large dissipative effects, we discover variations in the structure of the spectrum as the dispersion crosses zero that are not predicted by the small dissipation theory. In particular, in the normal dispersion regime we observe a jump in the number of discrete eigenvalues when a pair of real eigenvalues merges with the intersection point of the two branches of the continuous spectrum. Finally, we contrast the method to computational Evans function methods.

Solving generalized quintic complex Ginzburg–Landau equation by homotopy analysis method

Abstract In this paper, the generalized quintic complex Ginzburg–Landau equation is considered to be solved, by means of the homotopy analysis method (HAM). Two examples are solved to illustrate the efficiency of the proposed method. By plotting the h-curve of the examples, the region of convergence is determined.

Parametric disorder effects on a subcritical stationary bifurcation

For a spatial modulation of the control parameter which describes, for instance, major effects of a rough container boundary in Rayleigh–Bénard convection, the threshold value of the bifurcation from a homogeneous basic state to a spatially periodic state is provided analytically and numerically, taking the one-dimensional cubic–quintic complex Ginzburg–Landau equation with real coefficients as an example. Above the threshold, using the Poincaré–Lindstedt expansion, we show that the quintic term affects both the stationary nonlinear solution and the Nusselt number.