This work makes a significant contribution by providing a comprehensive study of exact solutions of the fractional complex Ginzburg-Landau equation based on the conformable fractional derivative. Fractional derivatives bring intrinsic non-locality to the theory that accommodates memory effects and complex temporal dynamics absent from the integer-order classical formulation. As a simple nonlinear equation, the complex Ginzburg-Landau equation plays a central role in describing many physical phenomena, including superconductivity, superfluidity, nonlinear optics, and pattern formation. The novelty in this work is to extend the modified direct algebraic method to the fraction form of the equation to generate a wide variety of exact solutions from dark, bright, and singular solitons to Jacobi elliptic, exponential, singular periodic, and rational ones. These diverse waveforms provide new analytical insight into the broad solution space of the fractional system. For a complete linear stability analysis, the stability of solutions so obtained is studied with respect to small amplitude perturbations to ensure their physical validity. Graphical representations are also given to present the dynamic development of these solutions for various orders of fractional derivative, revealing the wide influence of fractional effects on wave profiles. These findings not only make the list of solutions to known fractional Ginzburg-Landau systems richer but also pave the way for their application in nonlinear complex physics and engineering phenomena modeling in modern physics and engineering.

This study presents an in-depth investigation into the exact wave solutions arising within the framework of the Lord-Shulman (L-S) theory of thermoelasticity, with particular emphasis on the inclusion of gravitational effects and temperature-dependent material properties. Employing the Improved Simple Equation technique (ISET), the research yields precise analytical solutions to the governing field equations that capture the dynamic coupling between thermal and mechanical responses in deformable media. Special attention is given to the roles of gravitational forces and thermally varying material parameters, both of which critically influence the mechanical and thermal behavior of materials under a variety of stress and temperature conditions. The ISET facilitates the construction of more complex wave profiles by introducing flexible solution forms enriched with arbitrary constants. This methodological enhancement allows for the extraction of a broader spectrum of exact solutions, each representing distinct physical scenarios through tunable parameters. The analytical results derived in this work shed light on fundamental aspects of wave motion in thermoelastic environments, specifically revealing how gravitational loading and temperature-sensitive properties alter wave speed, amplitude, and energy distribution. Furthermore, the study offers a visual examination of the solutions that were produced by using graphical representations of important physical parameters such as stress tensors, displacement components, and temperature fields. These graphs not only confirm the theoretical results but also help clarify the complex interplay between thermal and mechanical influences when external forces are applied. This research provides meaningful understanding of how thermoelastic materials behave, underlining the significant impact of gravitational forces and temperature variations in the accurate simulation of wave phenomena in complex engineering and physical systems.

Today, cloud computing is a widely used technology that provides a wide range of services to numerous sectors around the world. This technology depends on the interaction and cooperation of virtual machines (VMs) to complete various computing tasks, propagating malware attacks quickly due to the complexity of cloud computing environments and users’ interfaces. As a result of the rising demand for cloud computing from multiple perspectives for complete analysis and decision-making across a range of life disciplines, multi-cloud environments (MCEs) are established. Therefore, in this work, we discuss impacted mathematical modeling for the MCEs’ network dynamics using two deterministic and stochastic approaches. In both approaches, appropriate assumptions are considered. Then, the proposed networks’ VMs are classified to have six different possible states covering media, healthcare, finance, and educational servers. After that, the two developed modeling approaches’ solution existence, uniqueness, equilibrium, and stability are carefully investigated. Using an optimal control strategy, both proposed models are tested for sustaining a certain level of security of the VMs’ states and reducing the propagation of malware within the networks. Finally, we verify the theoretical results by employing numerical simulations to track the malware’s propagation immunization. Results showed how the implemented control methods maintained the essential objectives of managing malware infections.

This study explores the sufficient conditions for the null controllability of noninstantaneous impulsive Atangana–Baleanu fractional stochastic differential equations with Poisson jumps in a Hilbert space. Leveraging fractional analysis, compact semigroup theory, fixed point theorems, and stochastic analysis, we establish fundamental results that deepen the understanding of null controllability in complex stochastic systems. To demonstrate the applicability of our findings, we present a detailed example.

This study investigates the nonlinear Fisher-Kolmogorov-Petrovsky-Piskunov (KPP) model, which is widely applied to phenomena such as population dynamics, combustion processes, wave propagation in excitable media, and other systems characterized by reaction-diffusion behavior. The primary objective is to examine the intricate interplay between diffusion and nonlinear growth, crucial for understanding processes where spatial spreading is coupled with local growth or decay, such as species invasion, disease transmission, and chemical reactions. The problem addressed is the challenge of obtaining accurate, analytical solutions to the nonlinear Fisher-KPP equation, which often resists conventional solution techniques due to its complex, coupled dynamics. This study employs two advanced analytical methods - the modified Khater (MKhat) and unified (UF) techniques - to derive new, exact solutions to the Fisher-KPP model. These solutions provide deeper insights into the underlying dynamics of the reaction-diffusion systems, showcasing the balance between diffusion-driven spreading and nonlinear growth effects. Numerical validation of the obtained solutions is carried out using He's variational iteration (HVI) scheme, ensuring the reliability and accuracy of the analytical results. This combination of exact and numerical solutions strengthens the study's findings and offers robust tools for future research. The expected results include a set of novel, exact analytical solutions that contribute significantly to the understanding of the diffusion and nonlinear growth interplay in reaction-diffusion systems. These solutions have practical applications in forecasting and modeling processes such as population growth, epidemic spread, combustion dynamics, and chemical reaction rates. The study's innovative use of MKhat and UF methods introduces new approaches for solving nonlinear differential equations, expanding the toolkit available to researchers in applied mathematics and related fields. In conclusion, this research provides a comprehensive exploration of the nonlinear Fisher-KPP model, offering both theoretical advancements and practical applications. The findings have wide-ranging interdisciplinary implications, benefiting fields such as biology, physics, chemistry, and engineering, where reaction-diffusion systems play a central role.

This paper investigates the nonlinear dynamics of solitary electrical signals propagating along a nonlinear transmission line governed by the modified complex Ginzburg-Landau equation. By applying Lie symmetry analysis, the governing partial differential equation is reduced to a system of ordinary differential equations, enabling the derivation of exact traveling wave solutions. Two advanced analytical techniques, the new Kudryashov method and the generalized Arnous method, are employed to construct bright, singular, and rational soliton solutions. A comprehensive bifurcation analysis is conducted to classify equilibrium points and explore their stability through phase portraits. Additionally, modulational instability analysis is used to identify the conditions under which continuous wave backgrounds become unstable, offering theoretical support for soliton formation. To examine chaotic behavior, external perturbations of various forms including trigonometric, Gaussian, and hyperbolic types are introduced into the reduced system. The presence of chaos is confirmed through the computation of Lyapunov exponents, analysis of phase portraits, and application of the Melnikov method. The study provides new insights into the interplay between soliton dynamics, bifurcation structures, and perturbation-induced chaos in nonlinear electrical transmission networks.

This study conducts an in-depth analysis of the Gerdjikov-Ivanov equation under the influence of multiplicative white noise, specifically within the context of birefringent fibers. By employing two advanced techniques-the enhanced direct algebraic method and the innovative projective Riccati equations method-the research uncovers a range of soliton behaviors. The results identify various soliton types, including bright, dark, singular, and straddled solitons. Additionally, the study presents solutions involving Jacobi and Weierstrass doubly periodic functions, which under certain conditions, transition into soliton solutions. This research introduces a novel model, with all solutions representing original contributions to the field. The influence of white noise on these soliton structures is vividly depicted through 3D, 2D, and contour plots, providing visual insights into the dynamics of solitons in the presence of noise disturbances. These graphical representations offer a deeper understanding of soliton behavior within birefringent fibers, thereby advancing the discourse on nonlinear dynamics in optical fibers.

This paper explores the dynamics of the generalized Chen-Lee-Liu equation, a fundamental model in nonlinear optics, extended to incorporate multiplicative white noise. By employing Ito calculus, the stochastic behavior of the system was is rigorously analyzed, providing insights into the effects of perturbations on soliton dynamics. The improved extended modified tanh-function approach was utilized to derive a variety of soliton solutions, including singular, dark, and bright solitons, as well as newly identified straddled solitons. This analytical approach highlights the transformative relationships between soliton types under specific conditions, expanding the spectrum of known solutions. The incorporation of multiplicative white noise reveals intricate changes in soliton stability, amplitude, and velocity, illustrating the interplay between deterministic and stochastic influences. These findings offer theoretical advancements in the understanding of soliton behavior in noisy environments and have practical implications for optical communication systems and nonlinear wave modeling. This study enriches the theoretical landscape of soliton dynamics and sets the stage for future research into stochastic soliton systems.

We consider a coupled nonlinear system modeling light propagation in nematic liquid crystal media, comprising a nonlinear Schrödinger-type equation and a nonlocal molecular reorientation equation. To incorporate temporal stochastic effects, we introduce multiplicative noise within the Stratonovich framework, ensuring consistency with physical and mathematical principles. The stochastic forcing is taken to be homogeneous in space and random only in time, providing an analytically tractable mean-field description of uniform fluctuations acting on the beam, while not capturing spatially localized or fully spatiotemporal noise effects. A traveling wave reduction combined with stochastic averaging transforms the system into a deterministic framework that retains the essential features of the underlying randomness. A detailed modulational instability analysis is carried out, yielding precise conditions for the growth or suppression of perturbations under stochastic influences. To construct explicit analytical solutions, we employ two advanced techniques: the enhanced direct algebraic method and a projective Riccati equation method adapted to the present nonlocal nematicon system. This yields a broad class of exact solutions, including bright and dark solitons, singular structures, Jacobi and Weierstrass elliptic functions, and rational composite forms. We further analyze the influence of stochastic perturbations on the amplitude, symmetry, and stability of nonlinear modes, and our results indicate that noise can suppress, deform, or stabilize localized structures. The results contribute to the mathematical theory of nonlinear stochastic systems and provide analytical insights into wave phenomena in nonlocal media.

This paper introduces the technique of the improved modified extended tanh function method to examine the effects of laser pulse phenomena on a thermo-elastic material with temperature dependence within a coupled theory. Nonlinear thermo-elasticity is considered here due to its relevance in scenarios where a material’s response to varying thermal loads results in significant alterations to both its shape and intrinsic properties. This area of study is essential for accurately capturing real-world behaviors, such as thermal stress distributions in large-scale structures, material performance at different temperatures, and the complex interplay between mechanical and thermal effects. Using the proposed method, we have derived a range of exact solutions with distinct free parameters. These include bright soliton, rational, exponential, and hyperbolic solutions. Additionally, some of these findings, covering temperature, displacement, and stress tensor components, are illustrated graphically to enhance clarity and the interpretation of the results.

This paper introduces the modified extended (ME) direct algebraic approach, which is used to evaluate equations based on the Lord-Shulman (L-S) theory of thermoelasticity. Nonlinear thermoelasticity investigates situations in which a material's properties and shape change significantly as a result of fluctuating thermal loads. This study topic is critical for understanding real-world phenomena such as material behavior at high temperatures, thermal stresses in large structures, and the complex interplay between mechanical and thermal forces. The proposed approach generates a wide range of precise solutions, including exponential, polynomial, singular soliton, and hyperbolic solutions, each with unique free parameters that have yet to be published. Furthermore, the research includes graphical representations of various displacement components, temperature fluctuations, and stress tensors.

This work examines the dynamics of solitons traveling through birefringent fibers, regulated by the coupled Kaup-Newell equations and influenced by stochastic disturbances. The analysis utilizes Stratonovich calculus to describe physically consistent multiplicative noise, concentrating on soliton stability, bifurcation processes, and the possible formation of chaotic behavior. An extensive analysis of equilibrium points, stability criteria, and the Hamiltonian energy landscape elucidates the effects of stochastic perturbations on soliton dynamics. Moreover, stochastic traveling wave solutions are formulated using hyperbolic, Jacobi, and Weierstrass elliptic functions. The findings indicate that stochastic perturbations can dramatically influence soliton behavior, possibly instigating transitions among stable, periodic, and chaotic regimes. These findings provide significant insights into soliton dynamics inside optical communication systems, especially during random environmental perturbations.

This study presents the improved Modified Extended (IME) tanh function method, utilized to analyze equations based on the Green-Naghdi (G-N II) theory thermo-elasticity. Unlike conventional methods, the proposed technique enhances the ability to derive a wide spectrum of exact solutions, making it particularly effective for capturing the complex interplay between mechanical and thermal effects in nonlinear thermoelastic systems. Nonlinear thermo-elasticity examines scenarios when a material's characteristics and form experience substantial changes due to varying thermal stresses. This research domain is essential for understanding real-world phenomena, including material behavior at elevated temperatures, thermal stresses in extensive structures, and intricate interactions between mechanical and thermal forces. The suggested method produces a diverse array of exact solutions, encompassing exponential, polynomial, Jacobi elliptic (JE), singular soliton, and hyperbolic solutions, with distinct free parameters that have not been documented. Additionally, the study incorporates graphical depictions of diverse displacement components, temperature variations, and stress tensors.

In this paper, the averaging result for impulsive (delta, psi)-Hilfer fractional stochastic delayed differential equations (FSDDEs) caused by the Levy process was derived. In the sense of mean square, the relationship between the equivalent solutions of the original equations and the averaged equation solutions was demonstrated. Our findings allowed us to shift our attention from the original, more complicated system to the averaged system. Additionally, to demonstrate the relevance and practicality

Modern technologies networks due to the increasing demand for high-speed data transfer. In this work, higher-order dispersion and nonlinear effects are included into a high-order nonlinear Schrödinger equation (NLSE) formulated in an inhomogeneous optical fiber medium in the presence of conformable fractional derivative (CFD). The improved modified extended tanh-function method is used to construct a range of new exact analytical solutions, such as Weierstrass elliptic- and exponential-type solutions, Jacobi elliptic function (JEF) solutions, bright, dark, and singular solitons, rational solutions, and singular periodic waves. These solutions’ uniqueness is confirmed using the chosen analytical framework. To show how the solutions behave in different scenarios, parametric studies are performed. The method’s validity is supported by visualization in the form of 2D/3D surface plots produced with Wolfram Mathematica. The outcomes validate the method’s accuracy, suitability, and resilience in encapsulating the intricate dynamics of nonlinear wave propagation. These results provide a deeper comprehension of high-order NLSEs and pave the way for applying these techniques to increasingly complex nonlinear physical models.

This study delves into utilizing the Kudryashov auxiliary equation method on the time-fractional (3+1)-dimensional Sasa–Satsuma equation, pivotal for transmitting ultra-fast pulses in optical fibers. The main aim is to generate a variety of optical solitons for the time-fractional nonlinear (3+1)-dimensional Sasa–Satsuma model, including dark, bright, singular, and straddled solitons. These solitons are derived from exponential and hyperbolic function-type solutions, presenting novel findings not previously reported. The study demonstrates the effectiveness of the implemented method in deriving analytical solutions for fractional differential equations. The (3+1)-dimensional Sasa–Satsuma model’s potential in optical fiber transmission is highlighted, offering significant insights into generating optical solitons and their applications.

The rise in everyday internet users worldwide can be ascribed to advancements made to the internet worldwide. Attacks by malware cause data loss, hardware destruction, and large financial losses. As a result, this research creates a novel mathematical compartmental dynamical model that, under both fixed and dynamic size network assumptions, can be utilized to precisely predict malware outbreaks. The modeling processes divided the network into seven distinct states and considered the diverse ways that users of the internet interacted with potentially hazardous links. The Hilfer-Katugampola fractional operator was employed to obtain the intended varied states of networks. The existence, uniqueness, equilibrium, and stability of the provided models are thoroughly examined. Then the best control plan is implemented for every network topology. The controllers' objectives are to minimize the costs of data loss due to infections, malware tracing, and public awareness improvements. Ultimately, we confirm the theoretical results by observing the spread of malware using numerical simulations. Results demonstrated the effectiveness of the enforced control measures in maintaining the necessary goals of malware infection management.

In this study, we introduce the new (3+1)-dimensional $ \beta $-fractional Boussinseq-Kadomtsev-Petviashvili (KP) equation that describes the wave propagation in fluid dynamics and other physical contexts. By using the modified extended direct algebraic method, we investigate diverse wave solutions for the proposed fractional model. The acquired solutions, include (dark, bright) soliton, hyperbolic, rational, exponential, Jacobi elliptic function, and Weierstrass elliptic doubly periodic solutions. The primary objective is to investigate the influence of fractional derivatives on the characteristics and dynamics of wave solutions. Graphical illustrations are presented to demonstrate the distinct changes in the amplitude, shape, and propagation patterns of the soliton solutions as the fractional derivative parameters are varied.

The primary aim of this study is to solve the complex Akbota equation using the Khater II (Khat II) and unified (UF) methods as analytical techniques, while validating the accuracy of the constructed solutions through the Adomian decomposition method as a numerical scheme. The complex Akbota equation is a significant nonlinear evolution equation with crucial applications in fluid dynamics, optical fibers, and quantum field theory. It shares characteristics with other well-known nonlinear equations such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, highlighting its importance in modeling diverse physical phenomena. The Khater II and UF methods enable the construction of exact analytical solutions, while the Adomian decomposition method ensures the numerical accuracy of these solutions, demonstrating their applicability to real-world scenarios. The study’s findings reveal precise solutions that align closely with numerical results, underscoring the robustness of the proposed methods. This research contributes to the field by providing novel analytical solutions and verifying their accuracy, thereby enhancing our understanding of the complex Akbota equation’s physical behavior. The significance lies in the potential applications of these solutions in various scientific and engineering domains. Conclusively, the study offers new insights and methodologies for tackling complex nonlinear equations, advancing both theoretical and practical knowledge in applied mathematics and physics.