Dark, bright, and peaked solitons for Camassa–Holm nonlinear Schrödinger equation

Soliton dynamics in water waves are crucial for understanding and mitigating their impacts in coastal engineering, oceanography, and climate studies. This research investigates the soliton solutions of the Camassa–Holm nonlinear Schrödinger equation, a model suitable for studying the interaction between shallow and deep water waves. By applying a traveling wave transformation and the extended sinh-Gordon equation expansion method, the novel exact wave solutions are derived. These solutions, expressed in trigonometric and hyperbolic functions, reveal a variety of patterns, including dark solitons, bright singular solitons, two-bright singular solitons, periodic anti-peakons, V-shaped, and W-shaped periodic waves. The dominance of the Camassa–Holm equation component is evident in the peakon solutions, while the dark and bright solitons highlight the influence of the nonlinear Schrödinger equation component. Furthermore, the attained solutions are compared with existing results obtained using alternative techniques for this model.