In this research article, we explore new generalized perturbed or geophysical Korteweg–de Vries (gpKdV) equation under the time–space conformable operators. Using the modified tanh \((\frac{\phi }{2})\) -expansion method, we obtain a large variety of exact wave solutions such as bright, dark, kink, concave, convex, solitary, bell-shaped, line, and lump solitons. The obtained solutions are shown through 2D, 3D, and contour plots, which reveal their unique structural features to demonstrate their physical behavior. Furthermore, the role of the Coriolis effect in creating and determining the evolution of these solutions is explored in a systematic manner. The analysis verifies that the theoretical computational approach is practical, reliable, and adaptable to solve nonlinear applied mathematics and geophysical fluid dynamics models. In addition to providing additional understanding of nonlinear wave processes, the findings can assist scientists and engineers in advancing mathematical physics applications and theoretical study.
Graphical abstract
