Fractional Complex Ginzburg Landau Equation Research Articles

Dispersive perturbations of solitons for conformable fractional complex Ginzburg–Landau equation with polynomial law of nonlinearity using improved modified extended tanh-function method

Dispersive perturbations of solitons for conformable fractional complex Ginzburg–Landau equation with polynomial law of nonlinearity using improved modified extended tanh-function method

Exact solutions to the fractional complex Ginzburg-Landau equation with cubic-quintic and Kerr law nonlinearities

The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved (G′/G2) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.

Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity

In this paper, the bifurcation theory of dynamical system is applied to investigate the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. We mainly consider the case of $\alpha\neq 2\beta$ which is not discussed in previous work. By overcoming some difficulties aroused by the singular traveling wave system, such as bifurcation analysis of nonanalytic vector field, tracking orbits near the full degenerate equilibrium and calculation of complicated elliptic integrals, we give a total of 20 explicit exact traveling wave solutions of the time-space fractional complex Ginzburg-Landau equation and classify them into 11 categories. Some new traveling wave solutions of this equation are obtained including the compactons and the bounded solutions corresponding to some bounded manifolds.

Analysis of lump solutions and modulation instability to fractional complex Ginzburg–Landau equation arise in optical fibers

In this paper, the fractional complex Ginzburg–Landau equation (CGLE) with Kerr law in nonlinear optics, which simulates soliton propagation in various waveguides in the presence of a detuning component which comes from the nonlinear Schrödinger equation (NLSE) with the inclusion of the growth and damping terms, is examined. The Hirota bilinear method is exercised to retrieve lump solitons such as the 1-kink wave solution, 2-kink wave solution, double exponential wave solution, and homoclinic breather wave solution to the model. We also scrutinize some M-shaped solutions in the forms of M-shaped rational solutions and the M-shaped interaction with rogue and kink waves. In addition, the instability modulation and gain spectra of the CGLE are examined. The originality of the study lies in the secured outcomes, which were never before produced and effectively balance the nonlinear physical aspects. To illustrate the dynamic of these waves, some of the solutions are sketched in three-dimensional, two-dimensional, contour, and density plots. The produced results are encouraging which can be used to describe the phenomena occurring in nonlinear optical or plasma physics. The computed solutions demonstrate that the suggested approaches are skillful, categorical, consistent, and effective in identifying exact solutions to a variety of complicated nonlinear problems that have recently arisen in nonlinear optics, applied sciences, and engineering.

Innovative solutions and sensitivity analysis of a fractional complex Ginzburg–Landau equation

Innovative solutions and sensitivity analysis of a fractional complex Ginzburg–Landau equation

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.

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.

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.

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>

A study of fractional complex Ginzburg–Landau model with three kinds of fractional operators

The fractional complex Ginzburg–Landau equation plays an important role in the field of optics, field theory and superconductivity. In this paper, two variable G′G,1G–expansion method has been used on the fractional complex Ginzburg–Landau equation. The governing model is studied with three kinds of nonlinearities, namely; Kerr, quadratic–cubic and parabolic laws. Each law applied on fractional complex Ginzburg–Landau model is further discussed along with three definitions of the derivative i.e., beta derivative, conformable derivative and M-truncated derivative. In order to observe the fractional effects, a graphical comparison is shown. The evolution of solutions of the governing model for increasing value of fractional parameter is depicted through the surface graphs and line plots.

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well.

Dynamics of Fractional Stochastic Ginzburg–Landau Equation Driven by Nonlinear Noise

In this work, we focus on the long-time behavior of the solutions of the stochastic fractional complex Ginzburg–Landau equation defined on Rn with polynomial drift terms of arbitrary order. The well-posedness of the equation based on pathwise uniform estimates and uniform estimates on average are proved. Following this, the existence and uniqueness of weak pullback random attractors are establsihed.

Exact solutions to the fractional complex Ginzburg–Landau equation with time-dependent coefficients under quadratic–cubic and power law nonlinearities

Exact solutions to the fractional complex Ginzburg–Landau equation with time-dependent coefficients under quadratic–cubic and power law nonlinearities

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.

An investigation of fractional complex Ginzburg–Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives

An investigation of fractional complex Ginzburg–Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives

A comparative study of the optical solitons for the fractional complex Ginzburg–Landau equation using different fractional differential operators

This paper presents a study of optical solitons for the fractional complex Ginzburg–Landau equation (CGLE) with Kerr law nonlinearity using different fractional differential operators. The soliton solutions of the equation have been investigated with conformable, β and M-truncated derivatives. A Jacobi elliptic function finder technique, namely, new extended ϕ6-model expansion method has been utilized for the construction of the solutions. A variety of soliton solutions to the complex Ginzburg–Landau equation have been obtained. The comparison of some of the retrieved solutions has also been graphically illustrated using the three fractional differential operators.

Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results

The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation.

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.

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.