This investigation employs advanced computational techniques to ascertain novel and precise solitary wave solutions of the Camassa–Holm (${𝒞}{\mathscr{H}}$) equation, a partial differential equation governing wave phenomena in one-dimensional media. Originally designed for the representation of shallow water waves, the ${𝒞}{\mathscr{H}}$ 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 ${𝒞}{\mathscr{H}}$ 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 ${𝒞}{\mathscr{H}}$ 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 ${𝒞}{\mathscr{H}}$ equation, extending its applicability to provide insights into wave phenomena analogous to those encountered in plasma physics.

这项研究采用先进的计算技术来确定卡马萨-霍尔姆 ( ${𝒞}{\mathscr{H}}$ ) 方程的新颖且精确的孤立波解，该方程是控制一维介质中波动现象的偏微分方程。 ${𝒞}{\mathscr{H}}$ 方程最初是为表示浅水波而设计的，现已展现出跨多个学科的多功能性，包括非线性光学和弹性理论。它复杂地描述了波系统中非线性和色散效应之间的相互作用，其中非线性是由成分相互作用和色散产生的，而色散根源于波的时间传播。此外， ${𝒞}{\mathscr{H}}$ 方程控制着波浪剖面的时空演化，包括非线性和色散影响。值得注意的是，该方程允许孤子解——局域波包在很长的距离内保持其形式。精确的孤立波解的识别对于理解 ${𝒞}{\mathscr{H}}$ 方程在不同物理环境（例如流体动力学和非线性光学）中的行为具有至关重要的意义。此外，这项研究建立了所研究的模型和等离子体物理学之间的相关性，通过对替代计算方法进行基准测试，证明了所采用的计算技术的功效和效率。这种增强强调了 ${𝒞}{\mathscr{H}}$ 方程更广泛的相关性，扩展了其适用性，以提供对类似于等离子体物理学中遇到的波现象的见解。