Abstract
The purpose of this article is to achieve new soliton solutions of the Gilson-Pickering equation (GPE) with the assistance of Sardar's subequation method (SSM) and Jacobi elliptic function method (JEFM). The applications of the GPE is wider because we study some valuable and vital equations such as Fornberg-Whitham equation (FWE), Rosenau-Hyman equation (RHE) and Fuchssteiner-Fokas-Camassa-Holm equation (FFCHE) obtained by particular choices of parameters involved in the GPE. Many techniques are available to convert PDEs into ODEs for extracting wave solutions. Most of these techniques are a case of symmetry reduction, known as nonclassical symmetry. In our work, this approach is used to convert a PDE to an ODE and obtain the exact solutions of the NLPDE. The solutions obtained are unique, remarkable, and significant for readers. Mathematica 11 software is used to derive the solutions of the presented model. Moreover, the diagrams of the acquired solutions for distinct values of parameters were demonstrated in two and three dimensions along with contour plots.