In this paper, we explore the approximative controllability of fractional stochastic differential inclusions (SDIs) of Sobolev-type with fractional derivatives in Atangana-Baleanu (AB) sense and Poisson jumps. Our findings are supported by the fixed point theorem, multi-valued map theory, compact semigroup theory and stochastic analysis principles. In the later part, an illustration is provided to clarify the established outcomes.

This study sought to improve the comprehension of wave propagation in thermo-elastic materials according to Lord-Shulman (L-S) theory by developing precise wave solutions for the governing equations that take into consideration temperature-dependent material features. The research utilized the improved simple equation method (ISEM) to analyze the interrelated thermal and mechanical properties of these materials, allowing the creation of analytical solutions that exactly characterize intricate wave processes. The ISEM facilitates the development of various wave shapes. These solutions, defined by configurable free parameters, offer a flexible framework for examining diverse physical circumstances in thermo-elasticity. The work includes detailed graphical representations of crucial discoveries such as temperature distributions, stress tensors, and displacement which provide amazing visual insights into the complex interactions that occur within thermo-elastic systems.

This work investigated analytical solutions for a coupled system of nonlinear perturbed Schrödinger equations in fiber Bragg gratings (FBGs), characterized by sixth-order dispersion and a combination of Kerr and parabolic nonlocal nonlinear refractive indices. Chromatic dispersion, which restricts wave propagation in standard optical fibers, was effectively compensated using FBGs, making them indispensable in modern optical networks. In this study, the modified Sardar sub-equation technique (MSSE) was applied to the system for the first time. This method was chosen for its advantages, including low computational cost, high consistency, and simplicity in calculations. The novelty of this work lied in the derivation of new analytical solutions, such as exponential, singular periodic, hyperbolic, and rational solutions, which have not been previously reported in the literature. Additionally, bright gap solitons and singular gap solitons, previously studied, were also obtained. All solutions were rigorously verified by direct substitution into the system. Another significant contribution of this work was the derivation of modulation instability (MI) analysis using linear stability analysis. For the first time in the literature, an analytical expression for the MI gain spectrum was derived. This gain spectrum depended on key parameters such as normalized power, perturbation wave number, dispersion coefficients, phase modulation coefficients, and nonlinearity coefficients. The study also included 2D and 3D graphical representations of selected exact solutions, with parameters chosen to satisfy specific limiting conditions, as well as visual illustrations of the MI gain spectrum. The solutions derived in this work have profound implications for optical communication systems. Exponential and hyperbolic solutions can model pulse propagation in FBGs with high accuracy, enabling better design of dispersion-compensating devices and improving signal integrity over long distances. Singular periodic and rational solutions provided insights into the behavior of nonlinear waves in FBGs, which can be exploited for advanced signal processing applications, such as pulse shaping and wavelength conversion. Bright and singular-gap solitons were crucial for maintaining stable signal transmission in FBG-based systems, particularly in high-power scenarios where nonlinear effects were significant. The MI analysis further enhanced the practical relevance of this work. By understanding the conditions under which MI occured, engineers can design FBG systems that minimize signal degradation and optimize performance. The MI gain spectrum provided a tool to predict and control instability, ensuring robust and efficient optical communication networks. This work not only advanced the theoretical understanding of nonlinear wave dynamics in FBGs but also offered practical tools and solutions for improving optical communication systems. The derived solutions and MI analysis have direct applications in enhancing signal stability, dispersion management, and overall system performance, making this research highly relevant to the field of photonics and optical engineering.

Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.

We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter ( $\alpha$) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems.

We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.

This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.

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.