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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]