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This study delves into investigating the stability of solutions pertaining to the generalized Davey–Stewartson ( D S ) equation by employing the generalized rational method and projective Riccati equations method within the Hamiltonian system framework. The central focus lies in unraveling the physical significance of this model within the realm of plasma physics, shedding light on its intricate connections with analogous equations. The generalized D S equation, a pivotal nonlinear evolution model, serves as a vital tool in comprehending the evolution of coupled waves across various physical systems, notably within the domains of fluid dynamics and plasma physics. Its significance lies in its capacity to elucidate the intricate interplay among different wave components, offering a holistic perspective on wave interactions within plasma mediums. Through the meticulous application of rigorous analytical methods, this investigation uncovers pivotal insights into the stability of solutions, thereby enriching our understanding of mathematical models in plasma physics. The findings presented in this research contribute significantly to advancing the understanding of nonlinear dynamics, with profound implications for future exploration in plasma physics and related fields. In essence, this study underscores the analytical prowess of the generalized rational and projective Riccati equations methods, accentuating their relevance and applicability to the generalized D S equation and, by extension, their role in enhancing our comprehension of coupled wave phenomena within diverse plasma systems. [ABSTRACT FROM AUTHOR]

This investigation employs advanced computational techniques to ascertain novel and precise solitary wave solutions of the Camassa–Holm ( ℋ) equation, a partial differential equation governing wave phenomena in one-dimensional media. Originally designed for the representation of shallow water waves, the ℋ equation has exhibited versatility across various disciplines, including nonlinear optics and elasticity theory. It intricately delineates the interplay between nonlinear and dispersive effects in wave systems, with nonlinearity arising from component interactions and dispersion rooted in the temporal spreading of waves. Furthermore, the ℋ equation governs the spatiotemporal evolution of wave profiles, encompassing both nonlinear and dispersive influences. Notably, the equation allows for soliton solutions — localized wave packets sustaining their form over extended distances. The identification of precise solitary wave solutions holds paramount significance for comprehending the ℋ equation's behavior in diverse physical contexts, such as fluid dynamics and nonlinear optics. Moreover, this study establishes a correlation between the investigated model and plasma physics, demonstrating the efficacy and efficiency of the employed computational techniques through benchmarking against alternative computational methods. This augmentation underscores the broader relevance of the ℋ equation, extending its applicability to provide insights into wave phenomena analogous to those encountered in plasma physics. [ABSTRACT FROM AUTHOR]

This paper explains the importance and benefits of using AI in fluid mechanics, emphasizing its capability of finding the solution of fluid flow systems through iterative optimization techniques. It explores how AI can enhance the understanding of fluid dynamics to improve engineering processes. In the presented studies, the heat and mass transport of the Casson nanofluidic flow model on a nonlinear slanted extending sheet (HMT‐CNFM) via AI‐based Levenberg Marquard methodology with backpropagated trained neural networks (TNN‐BLMM) is studied. By applying an appropriate transformation, the governing PDEs representing HMT‐CNFM are converted into a system of nonlinear ODEs. The dataset for Levenberg Marquard methodology with backpropagated trained neural network (TNN‐BLMM) for all six scenarios of this proposed model via computational power of the Lobatto IIIA scheme using the "bvp4c" package in MATLAB and then graphically illustrate all these six scenarios through nftool to attain mean square error, regression, error histogram, performance, and fit curve. Training, testing, and validation processes of NN‐BLMM are organized for the investigation of the HMT‐CNFM model. [ABSTRACT FROM AUTHOR]