Fractional Ginzburg-Landau Equation Research Articles

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.

Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on $ \mathbb{R}^n $

<abstract><p>This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.</p></abstract>

Dynamical behavior and multiple optical solitons for the fractional Ginzburg–Landau equation with $$\beta $$-derivative in optical fibers

Dynamical behavior and multiple optical solitons for the fractional Ginzburg–Landau equation with $$\beta $$-derivative in optical fibers

Unconditional long-time stability-preserving second-order BDF fully discrete method for fractional Ginzburg-Landau equation

Unconditional long-time stability-preserving second-order BDF fully discrete method for fractional Ginzburg-Landau equation

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

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.

Preconditioned fourth-order exponential integrator for two-dimensional nonlinear fractional Ginzburg-Landau equation

Preconditioned fourth-order exponential integrator for two-dimensional nonlinear fractional Ginzburg-Landau equation

Well-posedness of space fractional Ginzburg–Landau equations involving the fractional Laplacian arising in a Bose–Einstein condensation and its kernel based approximation

This study aims to investigate some theoretical results, numerical study and a real-word application of the SFGLE, involving the fractional Laplacian. First, we describe the application of SFGLE in the Bose–Einstein condensate, where the energy of bosons is transported over long distances. Then, the existence, uniqueness and regularity of the local weak solutions are briefly analyzed. These theoretical results are presented for the first time. Next, the new Green reproducing kernels are introduced for utilization in the numerical solution of SFGLE. The finite difference scheme, introduced by Meerschaer in 2004, is employed to estimate the fractional derivative. For the temporal discretization, an implicit midpoint difference method is applied. The stability, well-posedness of the numerical solution, and the doubled L2–convergence order are attended in details. Our numerical method has the ability to handle high dimensions for numerically solve the equations defined in the hypercube domain. The accuracy and efficiency of our method are evaluated by presenting various 2D and 3D examples and the results will be validated with the previous related works. Ultimately, the utilization of an artificial neural network and empirical outcomes is employed to anticipate the BEC dynamics of Strontium by SFGLE at a specific instance. Subsequently, a comparison between the obtained consequences and experimental data is carried out to demonstrate the authentic applicability of FGLE in precisely describing several physical phenomena.

Stability analysis and error estimates of local discontinuous Galerkin method for nonlinear fractional Ginzburg–Landau equation with the fractional Laplacian

Stability analysis and error estimates of local discontinuous Galerkin method for nonlinear fractional Ginzburg–Landau equation with the fractional Laplacian

A Space-Time Finite Element Method for the Fractional Ginzburg–Landau Equation

A fully discrete space-time finite element method for the fractional Ginzburg–Landau equation is developed, in which the discontinuous Galerkin finite element scheme is adopted in the temporal direction and the Galerkin finite element scheme is used in the spatial orientation. By taking advantage of the valuable properties of Radau numerical integration and Lagrange interpolation polynomials at the Radau points of each time subinterval In, the well-posedness of the discrete solution is proven. Moreover, the optimal order error estimate in L∞(L2) is also considered in detail. Some numerical examples are provided to evaluate the validity and effectiveness of the theoretical analysis.

Piecewise fractional Chebyshev cardinal functions: Application for time fractional Ginzburg–Landau equation with a non-smooth solution

In this paper, two important subjects are investigated. First, a fruitful family of functions with the interpolation property called the orthonormal piecewise fractional Chebyshev cardinal functions are generated, and a formulation for their fractional derivative matrix is provided. Second, these functions together with their expressed matrix are used to construct a numerical method for fractional Ginzburg–Landau equation with a non-smooth solution. In fact, the proposed method approximates the problem solution by the expressed fractional functions (in the temporal direction) and classical Chebyshev cardinal polynomials (in the space direction). This method is able to get highly accurate solutions for the problem under investigation if its solution is in the piecewise form or includes terms with fractional powers, or includes both of them. The ability and validity of the method are examined by solving some numerical examples.

The construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application

In this paper, a novel optimal fourth-order fractional-compact-type numerical differential formula for the Riesz derivatives is derived by constructing the appropriate generating function. Meanwhile, some interesting and important properties of the proposed formula are also presented and proved. Then, we take the nonlinear space fractional Ginzburg–Landau equations as an example and establish a high-order finite difference scheme by combining the compact integral factor method with Padé approximation. Furthermore, based on the discrete energy analysis method, the proposed difference scheme is proved to be uniquely solvable and convergent with order O(τ2+h4) under different norms, where τ and h denote the temporal and spatial mesh step sizes, respectively. Finally, some numerical results are provided to illustrate the accuracy of numerical differential formula and the effectiveness of numerical method, and further demonstrate the correctness of theoretical analysis.

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.

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>

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.

Non-Hermitian control of optical turbulence in systems with fractional dispersion

We show an efficient mechanism to control optical turbulence in systems with different dispersion laws, including parabolic, sub-diffractive, hyper-diffractive or general fractional dispersion. The method is based on the modification of the energy cascade through spatial scales leading to turbulence: a non-Hermitian spatio-temporal periodic potential allows unidirectional coupling between modes in the excitation process. We prove a significant increase and reduction of the energy flow in turbulent states, by either condensing the excitation towards small wave-numbers or affecting the energy transfer towards large wave-number. The study is based on the complex Fractional Ginzburg–Landau equation, a universal model for pattern formation and turbulence in a wide range of systems. The enhancement or reduction of turbulence is indeed dependent on the imposed direction of the energy flow, controlled by the phase shift between the real and imaginary parts of the temporal oscillation of the non-Hermitian potential.

The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I)

The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference operator for the approximation of the Riesz derivative with order α∈(1,2]. Moreover, based on the difference operator and the compact technique, a novel fourth-order fractional compact difference operator is also derived. Secondly, using the fourth-order difference operator in space and the Crank–Nicolson method in time, a high-order difference scheme is proposed for the nonlinear space fractional Ginzburg–Landau equations. Thirdly, besides the standard energy method, some new techniques and important lemmas are developed to prove the unique solvability, stability and convergence in the sense of different norms. It is proved that the difference scheme is unconditionally stable and convergent with order Oτ2+h4 for α∈(1,1.5), where τ and h are the temporal step size and spatial step size, respectively. Finally, some numerical examples are given to show the efficiency and accuracy of the numerical differential formulas and finite difference scheme.

A meshless method for solving nonlinear variable-order fractional Ginzburg–Landau equations on arbitrary domains

A meshless method for solving nonlinear variable-order fractional Ginzburg–Landau equations on arbitrary domains

Stochastic dynamics of non-autonomous fractional Ginzburg-Landau equations on $ \mathbb{R}^3 $

<p style='text-indent:20px;'>We investigate a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation with multiplicative white noise in three spatial dimensions. Of particular interest is the asymptotic behavior of its solutions. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in <inline-formula><tex-math id="M2">\begin{document}$ L^2( \mathbb{R}^3) $\end{document}</tex-math></inline-formula>. The existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions is then established. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in <inline-formula><tex-math id="M3">\begin{document}$ L^2( \mathbb{R}^3) $\end{document}</tex-math></inline-formula> is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.</p>

Numerical analysis of a fourth-order linearized difference method for nonlinear time-space fractional Ginzburg-Landau equation

<abstract><p>An efficient difference method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the $ L2 $-$ 1_{\sigma} $ scheme to handle Caputo fractional derivative, while a fourth-order difference method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the difference scheme is unconditionally convergent in pointwise sense with the rate of $ O(\tau^2+h^4) $, where $ \tau $ and $ h $ are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis.</p></abstract>