Exact Solitary-Wave Solutions of the Camassa–Holm Hierarchy via the Sine–Cosine Method

The Sine–Cosine method has emerged as a robust analytical approach to derive solitary wave solutions for nonlinear dispersive partial differential equations. In this work, we systematically employ this technique to the Camassa–Holm hierarchy, encompassing the classical Camassa–Holm, Degasperis–Procesi, Fornberg–Whitham, and Fuchssteiner–Fokas–Camassa–Holm equations. Each member of the hierarchy models shallow-water wave propagation under specific integrability conditions, exhibiting rich dynamical behavior. By applying an appropriate wave transformation, we reduce the governing equations to ordinary differential equations and construct exact travelling-wave solutions in terms of trigonometric and hyperbolic functions. The solutions obtained include compacton like profiles and classical sech² and cosh² structures, with explicit expressions for wave speed and amplitude as functions of model parameters. Comparative analysis highlights the effectiveness and simplicity of the Sine–Cosine method relative to more elaborate techniques such as the Hirota bilinear formalism and the inverse scattering transform. Our contributions lie in the unified application of this method across the entire hierarchy and the presentation of a comprehensive classification of solitary wave families. These results extend existing literature and provide valuable benchmarks for numerical simulations and theoretical investigations of nonlinear wave dynamics in fluid mechanics and related fields. Studies illustrate the influence of nonlinearity and dispersion coefficients on wave morphology.