This article possesses new exact wave structures to the Gilson–Pickering equation (GPE) that describes the prorogation of waves in plasma physics. The solutions are achieved in single and combined behavior like shock, singular, shock-singular, singular periodic waves and periodic as well rational function by utilizing innovative integration norms namely (G′ G 2)-expansion method and expansion function method (EFM). Moreover, under the suitable choice of involved parameters 3-, 2-dimensional, and their corresponding contour plots are also sketched. The obtained results show that the applied computational schemes are straightforward, efficient, concise and can be utilized for more complex physical phenomena in various fields of sciences. The reported results are helpful to understand the studying of wave propagation and are also vital for numerical and experimental verification in plasma physics.