Construction of Exact Solutions for Gilson–Pickering Model Using Two Different Approaches

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences.