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Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ( ℂ ) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ( ℂ ) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics. [ABSTRACT FROM AUTHOR]

This study comprehensively explores the (3 + 1) -dimensional Mikhailov–Novikov–Wang (ℕ) integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified () , Khater II ( hat.II) method, and He's variational iteration (ℍ ) method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation's pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics. [ABSTRACT FROM AUTHOR]

This study introduces innovative analytical solutions for the (2 + 1) -dimensional nonlinear Jaulent–Miodek () equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of waves. Within the realm of nonlinear fluid dynamics, the equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the (2 + 1) -dimensional equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field. [ABSTRACT FROM AUTHOR]

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He's Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]