| Summary: | In this article, we study the fully non-linear third-order partial differential equation, namely the Gilson-Pickering equation. The (1/G′)-expansion method, and the generalized exponential rational function method are used to construct various exact solitary wave solutions for a given equation. These methods are based on a homogeneous balance technique that provides an order for the estimation of a polynomial-type solution. In order to convert the governing equation into a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. As a result, we have constructed a variety of solitary wave solutions, such as singular solutions, compound singular solutions, complex solutions, and topological and non-topological solutions. Besides, the 2D, 3D, and contour surfaces are plotted for all obtained solutions by choosing appropriate parameter values.
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