Multiplicative White Noise Research Articles

Bright soliton of Stochastic perturbed Biswas-Milovic equation with cubic-quintic-septic law having multiplicative white noise

For the first time, the adopted stochastic form of the perturbed Biswas-Milovic equation with cubic-quintic-septic law having spatio-temporal and chromatic dispersion in the presence of multiplicative white noise in Ito sense was presented and examined. The Biswas-Milovic equation ˆ models numerous physical phenomena occurring in optical fiber. We analyzed the optical soliton solutions of the stochastic model with the aid of a subversion of the new extended auxiliary equation method. Furthermore, we investigated the evaluation of the noise impacts and the effects of some model parameters on the dynamics of the generated soliton. Finally, graphical depictions of the derived soliton types were represented for some solution functions. The stochastic model and the derived results will contribute to the comprehension of the nonlinear dynamics of pulse propagation in optical fibers which has great importance for the advancement of optical communication engineering.

Structural damage identification with output-only strain measurements and swarm intelligence algorithms: a comparative study

The identification of structural damage with the unavailability of input excitations is highly desired but challenging since structural dynamic responses are affected by the coupling effect of structural parameters and external excitations. To deal with this issue, in this paper, an output-only damage identification strategy based on swarm intelligence algorithms and correlation functions of strain responses is proposed to identify structures subjected to single or multiple unknown white noise excitations. In the proposed strategy, four different population-based optimization algorithms—particle swarm optimization, the butterfly optimization algorithm, the tree seed algorithm, and a micro search Jaya (MS-Jaya)—are employed and compared. The micro search mechanism is integrated into a basic Jaya algorithm to improve its computational efficiency and accuracy by eliminating some damage variables with small values for the identified best solution after several iterations. The objective function is established based on the proposed auto/cross-correlation function of strain responses and a penalty function. The effectiveness of the proposed method is verified with numerical studies on a simply supported beam structure and a steel grid benchmark structure under ambient excitation. In addition, the effect of the reference point, number of sensors, and arrangement of strain gauges on the performance of the proposed method are discussed in detail. The investigated results demonstrate that the proposed approach can accurately detect, locate, and quantify structural damage with limited sensors and 20% noise-polluted strain responses. In particular, the proposed MS-Jaya algorithm presents a more superior capacity in solving the optimization-based damage identification problem than the other three algorithms.

Response analysis of the vibro-impact system under fractional-order joint random excitation

As a kind of good damping material, viscoelastic material is widely used in machinery, civil engineering, and other fields. In this paper, the viscoelasticity of the system is described by fractional differentiation. The dynamic response of a unilateral vibro-impact system with a viscoelastic oscillator under joint random excitation is studied, in which joint random excitation is composed of additive and multiplicative white noise. The fractional-order derivative was calculated based on Caputo’s definition, and the fractional derivative was equivalent to the corresponding linear damping force and linear restoring force. As a result, a new random system without fractional-order terms was obtained. A non-smooth transformation was introduced, which was equivalent to the original system to a new system without a velocity jump. The steady-state probability density functions of fractional-order vibro-impact systems under joint random excitation are solved by using the random average method and non-smooth transformation. In addition, the effects of parameters on the steady-state response of the system are analyzed.

High dispersion and cubic–quintic–septic–nonic nonlinearity effects on optical solitons in the complex Ginzburg–Landau equation of eighth-order with multiplicative white noise in the Itô sense

In this article, we present the complex Ginzburg–Landau equation involving eighth-order dispersion and multiplicative white noise effect in the context of Itô sense for the first time. In order to investigate and analyze the proposed model, two efficient and convenient techniques have been employed in this study. The results revealed the presence of various solitons, including bright, dark, and singular solitons. Additionally, the existence of straddled solitons was reported as well. These findings contribute to the current understanding of soliton dynamics and provide valuable insights for further research in this field. To better understand our findings, we have included several graphs that effectively illustrate the effect of the white noise. The white noise effect refers to a specific pattern of random fluctuations that can be observed in various systems and phenomena. It is characterized by a uniform distribution of frequencies across the entire spectrum, resulting in a constant and uncorrelated signal.

On strong convergence of two numerical methods for singular initial value problems with multiplicative white noise

Mean-square convergence of the Milstein method and the Euler–Maruyama (EM) method are investigated for stochastic singular initial value problems (SIVPs) driven by multiplicative white noise. For the first-order model equation, the existence, uniqueness, and moment boundedness of the exact solution are developed under some appropriate assumptions. On this basis, it is proved that the Milstein method is of 1/2−ϵ order convergence in the mean-square sense, and the convergence order can be increased to 1−ϵ when the diffusion coefficient vanishes at the origin, where ϵ is an arbitrarily small positive number. It is significantly different from the first-order mean-square convergence of the Milstein method to solve the classical stochastic ordinary differential equations. Meanwhile, whether the diffusion coefficient vanishes at the origin or not, the mean-square convergence order of the EM method is always 1/2−ϵ. Furthermore, a second-order stochastic SIVP with multiplicative noise is considered by converting it to a first-order system. When solving the system, it is shown that the EM method has the same form as the Milstein method, and thus enjoys the convergence order of the Milstein method. Numerical examples are provided to verify our theoretical prediction.

Stochastic Perturbation of Optical Solitons for the Concatenation Model with Power-Law of Self-Phase Modulation Having Multiplicative White Noise

Addressing the concatenation model, this paper explores the incorporation of power-law self-phase modulation and spatio-temporal dispersion, in addition to the chromatic dispersion. The inclusion of the white noise effect, along with deterministic Hamiltonian perturbation terms of arbitrary intensity, is also examined. Two integration approaches are applied to recover the soliton solutions, leading to the observation that the white noise effect remains confined to the phase component of these solutions.

The $$\Phi ^{6}$$-model expansion approach for solving the (2+1)-dimensional NLSE with Kerr law nonlinearity and STD with multiplicative white noise in birefringent fibers

The $$\Phi ^{6}$$-model expansion approach for solving the (2+1)-dimensional NLSE with Kerr law nonlinearity and STD with multiplicative white noise in birefringent fibers

Optical Solitons of the Generalized Stochastic Gerdjikov-Ivanov Equation in the Presence of Multiplicative White Noise

Optical Solitons of the Generalized Stochastic Gerdjikov-Ivanov Equation in the Presence of Multiplicative White Noise

Optical solitons for the dispersive concatenation model with spatio-temporal dispersion having multiplicative white noise

This investigation of the current work presents the dispersive concatenation model that integrates the spatio-temporal dispersion and the effect of multiplicative white noise in the Itô sense. The study aims to secure stochastic optical soliton solutions within this framework. To achieve this purpose, we implemented the enhanced Kudryashov technique and the new projective Riccati equations strategy. Initially, the traveling wave transform hypothesis was applied to convert the stochastic model into its nonlinear ordinary differential equation form, encompassing real and imaginary components with emerging the related parametric restrictions. Subsequently, we introduced the enhanced Kudryashov and the new projective Riccati equations techniques successfully employed in analyzing the generated nonlinear ordinary differential equation. By utilizing the appropriate solution sets, which encompass variables with unknown values and necessary transformations, we have derived many stochastic optical soliton-type solutions for the model. Detailed discussions for the dynamics of some selected soliton solutions are presented with the supporting of the 3D and 2D graphs that illustrate the white noise effect on these solitons.

Similarity reduction and new wave solutions for the 2D stochastic cubic Schrödinger equation with multiplicative white noise arising in optics

Similarity reduction and new wave solutions for the 2D stochastic cubic Schrödinger equation with multiplicative white noise arising in optics

The stochastic stability and H∞‐fuzzy control of stochastic bifurcation of a doubly‐fed induction generator

AbstractConsidering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

Density convergence of a fully discrete finite difference method for stochastic Cahn–Hilliard equation

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in L 1 ( R ) L^1(\mathbb {R}) of the numerical solution. Our results partially answer positively to the open problem posed by J. Cui and J. Hong [J. Differential Equations 269 (2020), pp. 10143–10180] on computing the density of the exact solution numerically.

Exact solutions of perturbed stochastic NLSE with generalized anti-cubic law nonlinearity and multiplicative white noise

The paper retrieves the perturbed stochastic nonlinear Schrödinger equation in magneto-optic waveguides with generalized anti-cubic law nonlinearity and multiplicative noise in the Itô sense. Exact solutions are obtained with the aid of the trial equation method and the complete discriminant system for a polynomial. Compared to previous solutions, this method has achieved a broader and more comprehensive set of exact solutions. The obtained exact solutions help explain light propagation, nonlinear effects, the impact of noise on the system, and various physical processes occurring within waveguides. Some graphical simulations have been provided to demonstrate the dynamical behavior of the solutions and the impact of noise term in Itô sense.

Optical solitons of stochastic perturbed Radhakrishnan–Kundu–Lakshmanan model with Kerr law of self-phase-modulation

This study aims to obtain the optical soliton solutions of the stochastic perturbed Radhakrishnan–Kundu–Lakshmanan equation which models the optical solitons in optical waveguides such as silica fibers with Kerr law in the presence of chromatic and third-order dispersions by multiplicative white noise in Itô calculus. The study constitutes one of the works aimed at adapting the model in accordance with the importance of the noise effect in nonlinear optics. In the first step, the nonlinear ordinary differential form of the investigated problem has been obtained by using an appropriate complex wave transformation. The enhanced Kudryashov technique, which is the combination of Kudryashov and the new Kudryashov methods, is applied to the nonlinear ordinary differential form. As the result, bright, dark and singular stochastic soliton solutions of the analyzed problem have been carried out by assigning appropriate parameter values and the impact of noise effect on the soliton behavior has been investigated. Lastly, the graphical representations and required comments have been presented in the related sections.

Impact of White Noise on the Exact Solutions of the Stochastic Riemann Wave Equation in Quantum Mechanics

In this article, the stochastic Riemann wave equation (SRWE) forced by white noise in the Itô sense is considered. The extended tanh function and mapping methods are applied to obtain new elliptic, rational, hyperbolic, and trigonometric stochastic solutions. Furthermore, we generalize some previous studies. The obtained solutions are important in explaining some exciting physical phenomena, since the SRWE is required for describing wave propagation. We plot numerous 3D and 2D graphical representations to explain how the multiplicative white noise influences the exact solutions of the SRWE. We can infer that the introduction of multiplicative white noise disrupts the symmetry of the solutions and serves to stabilize the solutions of the SRWE.

Highly dispersive optical solitons in fiber Bragg gratings for stochastic Lakshmanan–Porsezian–Daniel equation with spatio-temporal dispersion and multiplicative white noise

Nonlinear partial differential equations serve as key components for the mathematical representation of engineering phenomena across several domains within the known universe. In the field of engineering difficulties, the most significant considerations are invariably systems composition and physics of components. In this regard, the impact of noise emerges as a crucial aspect. Building on this foundational understanding, this study aims to present a novel version of the stochastic Lakshmanan–Porsezian–Daniel equation with higher order dispersion in fiber Bragg gratings. The equation incorporates multiplicative white noise in the Itô sense. To investigate this new model, we employ a couple of integration techniques which are the enhanced Kudryashov method and the generalized sine–Gordon equation method. Bright and singular solitons are achieved through the enhanced Kudryashov method, while straddled bright–dark solitons and straddled singular–singular solitons are obtained using the generalized sine–Gordon equation method. Solutions in the context of Jacobi’s elliptic functions are also considered. The emergence of all possible solutions using the proposed techniques has been noticed, each accompanied by specific constraints that guarantee their existence. Moreover, the effect of noise intensity on soliton dynamics is investigated and interpreted by simulating the graphical representations.

On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems

Stochastic infinite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are considered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure. This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures is again a (stochastic) Dirac structure under some assumptions. These interconnection results are applied on a system composed of a stochastic vibrating string actuated at the boundary by a mass–spring system with external input and output. This work is motivated by the problem of boundary control of SPHSs and will serve as a foundation to the development of stabilizing methods.

High-stochastic solitons for the eighth-order NLSE through Itô calculus and STD with higher polynomial nonlinearity and multiplicative white noise

High-stochastic solitons for the eighth-order NLSE through Itô calculus and STD with higher polynomial nonlinearity and multiplicative white noise

Applications of the Linear Quadratic Regulator Optimal Control With Completely and Partially Observed Markovian Switching

Abstract We study optimal control problems where the dynamic system evolves according to a linear stochastic differential equation with (a) multiplicative and additive white noise, (b) a mixture of white noise, and (c) white noise and colored noise. The drift rate and the diffusion matrix of the linear dynamic system depend on a continuous-time Markov chain with finite state space that is (I) partially observed and (II) completely observed. Using the results of Wonham filter theory, we reduce the partially observed problems to one with complete observation. We solve the control optimal problems explicitly with the help of dynamic programing technique, and three applications are presented to illustrate our theoretical results.

The traveling wave solution and dynamics analysis of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise

This article investigates the traveling wave solution of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise, and its conclusions can be applied to optical communication. Firstly, the equation is transformed into a nonlinear ordinary differential equation through traveling wave transformation, and its planar dynamic system is studied. The phase portraits of the equation in various cases are drawn by Maple software. Secondly, the traveling wave solutions of this equation are obtained by using the complete discriminant system theory of cubic polynomials. Finally, we draw two-dimensional graphics, three-dimensional graphics, density plots and contour plots for the module length of some solutions under specific parameters. In addition, the above research results can help readers have a more comprehensive and clear understanding of this equation and its application.