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The fractional nonlinear generalized (3 + 1) -dimensional Chaffee–Infante (FGCI) equation is a nonlinear evolution equation that governs the propagation of waves in dispersive media, exhibiting both dissipative and dispersive effects. It finds applications in various fields such as fluid mechanics, plasma physics, and nonlinear optics, where the interplay between nonlinearity, dispersion, and dissipation plays a crucial role in determining the dynamics of the system. This study aims to obtain analytical and numerical solutions for the FGCI equation, which is a generalization of the Chaffee–Infante equation. The analytical approach employed is the Khater II (Khat II) method, an efficient technique for constructing approximate analytical solutions to nonlinear fractional differential equations. Additionally, He's variational iteration (HVI) method is implemented as a numerical scheme to validate the accuracy of the analytical solutions obtained from the Khat II method. The results obtained through these methods demonstrate the effectiveness of the proposed techniques in solving the FGCI equation and provide valuable insights into the dynamics of wave propagation phenomena governed by the equation. The significance of this study lies in its contribution to the understanding of nonlinear wave propagation phenomena and the development of analytical and numerical techniques for solving fractional differential equations. The novelty of the research lies in the application of the Khat II method to the FGCI equation, which has not been previously explored, and the validation of the analytical solutions through the HVI method. The study is conducted in the field of applied mathematics, with a focus on nonlinear fractional differential equations and their applications in various physical systems. [ABSTRACT FROM AUTHOR]

This study investigates the nonlinear time-fractional Harry Dym ( ℍ ) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym (ℍ ) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the ℍ equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He's variational iteration (ℍ ) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the ℍ equation's behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the ℍ equation, this study paves the way for further investigations into its potential applications in diverse physical settings. [ABSTRACT FROM AUTHOR]

This study focuses on employing recent and accurate computational techniques, specifically the Sardar-sub equation () method, to explore novel solitary wave solutions of the Gilson–Pickering (ℙ) equation. The GP equation is a mathematical model with implications in fluid dynamics and wave phenomena. It describes the behavior of solitary waves, which are localized disturbances propagating through a medium without changing shape. The physical significance of the ℙ equation lies in its ability to capture the dynamics of solitary waves in various systems, including water waves, optical fibers, and nonlinear acoustic waves. The study's findings contribute to the advancement of mathematical modeling approaches and offer valuable insights into solitary wave phenomena. The stability of the constructed solutions is investigated using the properties of the Hamiltonian system. The accuracy of the computational solutions is demonstrated by comparing them with approximate solutions obtained through He's variational iteration (ℍ ) method. Furthermore, the effectiveness of the employed computational techniques is validated through comparisons with other existing methods. [ABSTRACT FROM AUTHOR]

This study employs three advanced computational and numerical techniques to solve the nonlinear fractional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation in magnetized plasma. The reductive perturbation approach is utilized to investigate the dynamics of various components, namely isothermal species, immobile background species, and warm adiabatic fluid, in magnetized plasma. Emphasis is placed on unraveling the asymmetrical propagation characteristics of nonlinear electrostatic waves. The model's solutions encompass diverse types of solitons, including ion-acoustic, dust acoustic, and electron acoustic solitons. Analytical solutions are obtained using a variety of mathematical functions, such as exponents, trigonometry, and hyperbolas. Two- and three-dimensional density graphs illustrate the practical behavior of a single soliton. The primary objective of employing numerical schemes is to assess the accuracy of the derived solutions, and the outcomes demonstrate the efficacy of the analytical method in solving nonlinear mathematical and physical problems. Several techniques are employed to validate the consistency between calculated and estimated results, ensuring the study's accuracy and reliability. Overall, this investigation underscores the effectiveness of numerical and analytical techniques in tackling complex mathematical models, offering a promising avenue for future research in the field. The findings carry significant implications for comprehending nonlinear phenomena in magnetized plasma and contribute to advancing the field. [ABSTRACT FROM AUTHOR]

The soliton wave solutions to the (2+1)-dimensional Heisenberg ferromagnetic spin chain (ℋ ℱ ) problem are explored in this paper. The (2+1)-dimensional ℋ ℱ equation describes the nonlinear spin dynamics of Heisenberg ferromagnetic spin chains in the semiclassical limit. This formula is based on the spin-chain of the Heisenberg field. The bilinear and anisotropic interactions are accounted for in this equation. Even ferromagnetism may be explained by this paradigm, since it describes the propagation of waves inside a substance (the magnetic ordering in ferromagnetic materials). To construct a new soliton wave solution and validate its correctness, modern analytical and approximative schemes are utilized such as the Khater II and Adomian decomposition methods. Numerous interpretations for solitary waves are shown by various plots such as polar plots, contour plots, two-dimensional plots, and three-dimensional plots. The study's novelty may be appreciated by contrasting our findings with those presented in previously published research publications. Most of the many computational simulations used are performed with the aid of a program called Mathematica 13.1. [ABSTRACT FROM AUTHOR]