Stochastic Ginzburg-Landau Equation Research Articles

Analyzing the occurrence of bifurcation and chaotic behaviors in multi-fractional order stochastic Ginzburg-Landau equations

Analyzing the occurrence of bifurcation and chaotic behaviors in multi-fractional order stochastic Ginzburg-Landau equations

Numerical Method for Stochastic Nonlinear Schrödinger Equation Driven by Multivariate Gaussian Measure: Algorithms and Applications

In this paper, we present a novel Galerkin spectral method for numerically solving the stochastic nonlinear Schrödinger (NLS) equation driven by multivariate Gaussian random variables, including the nonlinear term. Our approach involves deriving the tensor product of triple random orthogonal basis and random functions, which enables us to effectively handle the stochasticity and nonlinear term in the equation. We apply this newly proposed method to solve both one- and two-dimensional stochastic NLS equations, providing detailed analysis and comparing the results with Monte Carlo simulation. In addition, the proposed method is applied to the stochastic Ginzburg–Landau equation. Our method exhibits excellent performance in both spatial and random spaces, achieving spectral accuracy in the numerical solutions.

The exact solutions of the variable-order fractional stochastic Ginzburg–Landau equation along with analysis of bifurcation and chaotic behaviors

This article explores the exact solutions of the variable order fractional derivative of the stochastic Ginzburg–Landau equation (GLE) using the G′G2-expansion method with the assistance of Matlab R2021a software. The paper presents three key aspects that contribute to its novelty: (1) Our study introduces and examines the variable order fractional derivative of the stochastic Ginzburg–Landau equation (VOFDSGLE) for the first time, demonstrating our best cognitive effort. We successfully obtain a significant number of exact solutions and provide illustrative examples and visual representations of the VOFDSGLE. These obtained solutions have the potential to offer enhanced availability and practicality in understanding the mechanisms behind the complex physical phenomena observed in various fields. (2) Additionally, we investigate the phase portraits of the variable-order fractional stochastic Ginzburg–Landau equation under a specific condition. We analyze the associated sensitivity and chaotic behaviors, which have not been examined in prior studies on stochastic Ginzburg–Landau equation (Mohammed et al., 2021; Alhojilan and Ahmed, 2023). This exploration adds a unique contribution to the existing literature and offers valuable insights into the dynamics of the system. (3) Our numerical observations underscore the significant impact of multiplicative noise on the solutions. The progressive degradation of patterns and the tendency towards planar surfaces indicate the disruption caused by increasing noise intensity. Moreover, the convergence of solution magnitudes to zero underlines the stabilizing influence of the multiplicative noise. Furthermore, we advance the existing knowledge regarding the response of the stochastic GLE model to small noise, offering valuable insights into the system’s transition behaviors in the energy landscape.

Novel analytical solutions of stochastic Ginzburg-Landau equation driven by Wiener process via the improved modified extended tanh function method

In this manuscript, the improved modified extended tanh integration technique is implemented to investigate the exact solutions of stochastic Ginzburg–Landau model driven by Wiener process which appeared in various fields of chemistry, mathematics and physics. Many types of stochastic wave solutions are raised such as hyperbolic stochastic solutions, trigonometric stochastic solutions, Weierstrass elliptic stochastic solutions and Jacobi elliptic stochastic solutions. Moreover, 3D and 2D graphical representations are depicted to show the effect of the multiplicative noise on the propagated waves. The results reveal that as the noise level increases, the wave degrades. The wave is distorted when σ⩾1. In addition, the 3D graphical representation show that the propagated wave is flattered when the noise strength is increased. These solutions may be applicable in various applications in applied science. The proposed method is efficient, direct and powerful.

Homogenization for stochastic Ginzburg-Landau equation on the half-line with fast boundary fluctuation

In this paper, we investigate the homogenization for complex stochastic Ginzburg-Landau equation on the half-line with fast boundary fluctuation. Precisely, we first verify the existence and uniqueness of the mild solution to the initial-boundary problem in a weighted space. Next, we discuss the tightness of the mild solution. Finally, we prove that the mild solution of the original system converges to the mild solution of the homogenized equation in probability, as ε goes to zero.

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.

Stability of the invariant measure for the 3D stochastic cubic Ginzburg–Landau systems

The current paper is devoted to 3D stochastic Ginzburg–Landau equations with degenerate random forcing. We establish the stability of stochastic systems by investigating the relationship between invariant measures under the action of transition semigroups corresponding to different sets of parameters. Towards this aim a new form of bound on the difference between solutions along with the spectral gap plays a significant role.

Dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise

This paper is concerned with the well-posedness as well as long-term dynamics of stochastic Ginzburg–Landau equations driven by nonlinear noise. We will apply a specific method to solve stochastic Ginzburg–Landau equations, known as the variational approach. We prove the existence and uniqueness of the solutions by assuming that the coefficients satisfy certain monotonicity assumptions. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. At the same time, the existence of invariant measures for the stochastic Ginzburg–Landau equations is also established.

Stochastic Ginzburg–Landau equation on a half-line driven by the multiplicative noise

We consider the stochastic nonlinear Ginzburg–Landau equations driven by the multiplicative noise on the half-line with Neumann brown-noise boundary conditions. We establish the existence and uniqueness of a solution to an initial-boundary value problem with values in Sobolev space H1. We are also interested in the regularity behavior of the solution, especially near the origin, where the boundary data are highly irregular.

Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces

<p style='text-indent:20px;'>In this paper we discuss the weak pullback mean random attractors for stochastic Ginzburg-Landau equations defined in Bochner spaces. We prove the existence and uniqueness of weak pullback mean random attractors for the stochastic Ginzburg-Landau equations with nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic Ginzburg-Landau equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.</p>

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>

Wong–Zakai approximations and limiting dynamics of stochastic Ginzburg–Landau equations

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.

Approximations of the solution of a stochastic Ginzburg-Landau equation

"This paper presents a method to approximate the solution of a stochas- tic Ginzburg-Landau equation with multiplicative noise term. Error estimates for the approximation of the solution are given."

Investigation on Ginzburg-Landau equation via a tested approach to benchmark stochastic Davis-Skodje system

We propose new numerical methods with adding a modified ordinary differential equation solver to the Milstein methods for solution of stiff stochastic systems. We study a general form of stochastic differential equations so that the Ginzburg-Landau equation and the Davis-Skodje model can be considered as special states of them. The efficiency of the method is experimented, in terms of the convergence rate and accuracy of approximate solution, employing some numerical examples, including stochastic Ginzburg-Landau equation and a paradigm of chemical reaction systems.

The exact solutions of the stochastic Ginzburg–Landau equation

The main goal of this paper is to obtain the exact solutions of the stochastic real-valued Ginzburg–Landau equation, which is forced by multiplicative noise in the Itô sense. It is necessary to get the exact solutions of this equation because it occurs in numerous fields of physics, mathematics and chemistry. We use three different methods such as the tanh-coth, the Riccati-Bernoulli sub-ODE and the generalized G′G-expansion methods in order to obtain a new trigonometric and hyperbolic stochastic solutions. The main advantage of these three methods is their applicability in solving similar models. The novelty of the present paper is that the results obtained here extend and improve some results that were previously obtained. Moreover, we plot 3D surfaces of analytical solutions obtained in this paper by using Matlab to illustrate the impact of multiplicative noise on the solutions of the stochastic real-valued Ginzburg–Landau equation.

Stochastic time-optimal control for time-fractional Ginzburg–Landau equation with mixed fractional Brownian motion

A theoretical approach for solving time-fractional stochastic Ginzburg–Landau equation with mixed fractional Brownian motion in Hilbert space is elaborated. Initially, the stochastic partial differential system is reformulated in the Hilbert space by using the properties of fractional order space and fractional Laplacian. We establish the existence of mild solutions by employing Mittag–Leffler functions, stochastic analysis, and Krasnoselskii’s fixed point theorem. A sufficient condition for the existence of a Lagrange optimal control problem is established via Balder’s theorem. Further, the existence of stochastic time-optimal control and stochastic optimal time are analyzed for the proposed control system. An example is given to illustrate the developed theory. Finally, an application to the stochastic optimal control of hydropower plant model is provided. The optimal control is termed as the amount of release of water through the reservoir and it is controlled with a suitable performance index.

LIMITING DYNAMICS OF NON-AUTONOMOUS STOCHASTIC GINZBURG-LANDAU EQUATIONS ON THIN DOMAINS

We examine the limiting dynamics of a class of non-autonomous stochastic Ginzburg-Landau equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. The existence and uniqueness of tempered pullback random attractors are established for the stochastic Ginzburg-Landau systems defined on $ (n+1) $-dimensional narrow domain. In addition, the upper semicontinuity of these attractors is obtained when a family of $ (n+1) $-dimensional thin domains collapses onto an $ n $-dimensional domain.

Asymptotic behavior for non-autonomous fractional stochastic Ginzburg–Landau equations on unbounded domains

In this paper, we first prove the existence and uniqueness of tempered pullback random attractors for a non-autonomous stochastic fractional Ginzburg–Landau equation driven by multiplicative noise with α ∈ (0, 1) in L2R3. Then, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. Due to the lack of the compactness of Sobolev embeddings on unbounded domains, we establish the pullback asymptotic compactness of solutions in L2(R3) by the tail-estimates of solutions.

The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation. Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern.This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple one-dimensional stochastic Ginzburg–Landau equation.

Regularity of random attractors for non-autonomous stochastic discrete complex Ginzburg-Landau equations

In this paper, we consider the asymptotic behaviour of non-autonomous stochastic discrete complex Ginzburg-Landau equations with additive noise in weighted space . The existences of tempered random attractors for this equation in spaces and are proved respectively by tail estimates, which implies that the obtained -random attractor is compact and attracting in the topology of space. The main difficulty here is the lack of compactness on infinite lattices. To deal with this, we introduce a common embedding space of and and derive some tail-estimates of solutions.