In this paper, we investigated the Gilson-Pickering (GP) equation and many new solutions are obtained with the aid of two different approaches, namely Jacobi elliptic functions and exponential rational function approach. Different choices of the parameters in obtained results lead to the solutions of some well known models, which are Camassa-Holm equation, the Fornberg-Whitham equation and the Rosenau-Hyman equation. The methods considered here can also help to have a panoply of new wave surfaces concerning other related partial differential equations. Further more, 2D and 3D graphical presentations of these surfaces are presented for the various parameters. Moreover, bifurcation behavior of nonlinear travelling waves of GP equation is discussed. Bifurcation theory of planer dynamical system is utilized to observe that considered model contains nonlinear periodic wave, bell shaped solitary wave and shock wave.
