Complex Ginzburg-Landau Equation Research Articles

Impact of higher-order effects on the dynamics of soliton solutions in the (3+1)D cubic-quintic-septic complex Ginzburg–Landau equation with higher-order dispersion terms

We investigate the influence of the higher-order correction term of the nonlinear refractive index and the nonlinear-gain absorption on the dynamics of soliton solutions in a three-dimensional (3D) dissipative medium described by the higher-order (3+1)-dimensional cubic-quintic-septic complex Ginzburg–Landau [(3+1)D CQS-CGL] equation with the viscous (spectral-filtering) term, self steepening, Raman effect, dispersion terms up to six, diffraction and cubic-quintic-septic nonlinearities. By mean of direct numerical simulation of the dynamical model, we show that these two parameters can have a significant impact on the dynamics of solitons govern by the above mentioned equation, since their formation deeply depends of the value of each of the above cited parameters. Some interesting particular cases have been detected concerning the combined action of the two parameters or the isolated action of each of them. Among those solutions, we have bell-shaped, snail and creeping dissipative solitons. Also, we have found that the propagation distance z is another important parameter which has allowed us to appreciate the impact of each parameter. At a given distance z or for different values of z, the formation of soliton depends on the value of the higher-order correction term and the nonlinear-gain absorption involved. We have also shown that the above solutions can be self-trapped over a huge propagation distance even in the presence of random perturbations. The discovered nonlinear dependence of the higher-order correction term and the nonlinear-gain absorption on the soliton solution can also be used for the better understanding of the (3+1)D dissipative light bullets in a doped nonlinear kerr medium.

Diverse optical solitons to the complex Ginzburg–Landau equation with Kerr law nonlinearity in the nonlinear optical fiber

Diverse optical solitons to the complex Ginzburg–Landau equation with Kerr law nonlinearity in the nonlinear optical fiber

Modulated blood waves in the coupled complex Ginzburg–Landau equations of Jeffrey fluids in arteries

Modulated blood waves in the coupled complex Ginzburg–Landau equations of Jeffrey fluids in arteries

A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity

In this short note, we consider the elliptic problem λϕ+Δϕ=η|ϕ|σϕ,ϕ|∂Ω=0,λ,η∈C,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lambda \\phi + \\Delta \\phi = \\eta |\\phi |^\\sigma \\phi ,\\quad \\phi \\big |_{\\partial \\Omega }=0,\\quad \\lambda , \\eta \\in {{\\mathbb {C}}}, \\end{aligned}$$\\end{document}on a smooth domain Omega subset {{mathbb {R}}}^N, Ngeqslant 1. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature of such bifurcations in some specific cases. We conclude with the stability analysis of these branches under the complex Ginzburg-Landau flow.

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions.

Highly Dispersive Optical Solitons in Birefringent Fibers of Complex Ginzburg–Landau Equation of Sixth Order with Kerr Law Nonlinear Refractive Index

In this paper, we derived optical soliton solutions with a highly dispersive nonlinear complex Ginzburg–Landau (CGL) equation in birefringent fibers that have Kerr law nonlinearity. We applied two mathematical methods, namely the addendum Kudryashov’s method and the unified Riccati equation expansion method. Straddled solitary solutions, bright soliton, dark soliton and singular soliton solutions were obtained.This model represents the propagation of a dispersive optical soliton through a birefringent fiber. This happens when pulses propagating through an optical fiber split into two pulses.

High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation

In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ∈(1,2]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order Oτ2+hx4+hy4, where τ denotes the time step size, hx and hy denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme.

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.

Obtaining the Soliton Type Solutions of the Conformable Time-Fractional Complex Ginzburg–Landau Equation with Kerr Law Nonlinearity by Using Two Kinds of Kudryashov Methods

The main idea of this study is to obtain the soliton-type solutions of the conformable time-fractional complex Ginzburg–Landau equation with Kerr law nonlinearity. For this aim, the generalized and modified Kudryashov methods are applied to the given model. The reason for using a conformable derivative is that the chain rule can be applied to this derivative. Thus, using the suitable wave transform, the given equation is converted into an ordinary differential equation. Then, the proposed methods are applied to the reduced equation. According to our results, both of the used methods are effective and powerful. Finally, 3D and contour plots are given for some results with suitable variables. Our findings in this paper are critical for explaining a wide range of scientific and physical applications. According to our knowledge, our results are new in the literature.

Dynamics of pulsating solitons derived from asymmetrical dispersive waves.

Pulsating soliton (PS) as a local structure of nonlinear systems has induced substantial interest in nonlinear photonics and ultrafast lasers. However, the interaction mechanism between PSs has not been fully studied. Here, the vital role of the asymmetric dispersive wave (DW) in PSs interaction is investigated for the first time. Based on the complex Ginzburg-Landau equation (CQGLE), we find that the asynchronous pulsating soliton molecule (PSM) composed of strong PSs and weak PSs will produce frequency shift due to the asymmetric DW, and the state of the PS can be transferred through the DW during the collision between PSs and PSM. Moreover, we firstly characterize the PS with asymmetric DW in experiment, and observe the drift of PSM, which agree with our simulation that the asymmetric DW can cause the frequency shift of PSMs. Our results provide new insights into the multi soliton interaction of nonlinear systems.

The modulation instability of shallow wake flows based on the higher-order generalized cubic-quintic complex Ginzburg–Landau equation

In the context of the parallel flow hypothesis, we derive a higher-order generalized cubic-quintic complex Ginzburg–Landau (GCQ-CGL) equation to describe the amplitude evolution of shallow wake flow from the dimensionless shallow water equations by using multi-scale analysis, perturbation expansion, and weak nonlinear theory. The evolution model includes not only the slowly changing envelope approximation but also the influence of higher-order dissipation, dispersion, and cubic and quintic nonlinear effects. We give the analytical solution of the higher-order GCQ-CGL equation based on the ansatz and coordinate transformation methods, and we discuss the influence of the higher-order dissipation coefficient on the amplitude and frequency of the wake flow by means of three-dimensional diagrams, contour maps, and plane graphs. The subsequent linear stability analysis gives a theoretical basis for the modulation instability (MI) of plane waves, and the linear theory predicts the instability of any amplitude of the main waves. Finally, we focus on the MI of shallow wake flows. Results show that the MI gain function is internally related to the background wave number, disturbance wave number, background amplitude, disturbance expansion parameter, and dissipation coefficient. The area of the MI decreases as the higher-order dissipation coefficient decreases.

Quantification of dissipative effects in a complex Ginzburg-Landau equation governed laser system by tracing soliton dynamics.

The concept of dissipative solitons has provided new insight into the complex pulse dynamics in mode-locked lasers and stimulated novel laser cavity designs. However, most of these studies are restricted to qualitative regimes, because it is difficult to quantify dissipative effects in a mode-locked laser. Meanwhile, the quantification of dissipative effects is a general problem that can be also encountered in other dissipative systems. In this paper, we demonstrate a method for quantifying dissipative effects in a mode-locked laser based on analyzing the soliton dynamics traced by time-stretch dispersive Fourier transform. As a result, we are able to quantitatively reproduce the evolution of the pulse that seeds mode-locking through simulations and gain a deeper understanding of the whole process. The obtained physical picture of mode-locking allows us to propose a simple method to quantify the energy threshold for mode-locking buildup and the stability of mode-locked states. A parameter is introduced to evaluate mode-locking conditions, which can serve as a criterion for designing mode-locked lasers. This work opens up new possibilities in the diagnosis and improvement of mode-locked lasers and studies of soliton physics.

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.

Stabilisation of spatially periodic states by non-Hermitian potentials

We uncover new families of stable periodic solutions by the introduction of non-Hermitian potentials in the universal complex Ginzburg–Landau equation. We perform a comprehensive analysis on the dynamics and stability of the system by determining and following these new solutions for a one-dimensional system, and demonstrate that the results hold for higher spatial dimensions and for the corresponding complex Ginzburg–Landau fractional order equation. We prove the robustness of the stabilisation within a broad range in parameter space. The universality of the CGLE allows extending these results to different actual systems described by other specific models. In particular, we provide results on the stabilisation for Vertical Cavity Surface Emitting Lasers.

New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature.

$G$-mean random attractors for complex Ginzburg–Landau equations with probability-uncertain initial data

$G$-mean random attractors for complex Ginzburg–Landau equations with probability-uncertain initial data

Inertial manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions

Inertial manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions

Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods

<abstract> <p>This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left({{{G'} / {{G^2}}}} \right)$ and $\left({{1 / {G'}}} \right) - $expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.</p> </abstract>

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.

Well-posedness of fractional stochastic complex Ginzburg-Landau equations driven by regular additive noise

This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landauequation driven by locally Lipschitz nonlinear diffusion terms defined on $ R^n $. We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation.