This study establishes a unified approach to utilising symmetry methods to uncover diverse analytical solutions and provides deeper insights into the nonlinear phenomena governed by the Kadomtsev Petviashvili (KP) framework. The (3+1) dimensional generalized Kadomtsev-Petviashvili (gKP) equation with arbitrary nonlinearity f(u) is examined for its underlying symmetry properties, optimal systems, and exact solutions. We employ Lie symmetry methods to classify the algebraic structure and develop one-dimensional optimal systems for different nonlinearities, including power and exponential forms of f(u). Additionally, using square-law nonlinearity, we derive the modified Korteweg–de Vries-Kadomtsev Petviashvili (mKdV-KP) equation, which is then subjected to symmetry analysis. By utilising the invariants, we systematically reduce the (3+1) gKP equation to lower-dimensional forms, facilitating the discovery of static and travelling wave solutions. Notably, the square-law nonlinearity reveals rich solution behaviour, which includes elliptic and Jacobi function representations. Furthermore, graphical illustrations provide a better understanding of underscoring their relevance to soliton theory and nonlinear wave dynamics.

A detailed analysis of the invariant point transformations for the first four partial differential equations which belong to the complex Burgers’ hierarchy is performed. Moreover, a detailed application of the reduction process through the Lie-point symmetries is presented while we construct the similarity solutions. We conclude that the differential equations of our consideration are reduced to first-order equations such as the Abel, Riccati, and to a linearisable second-order differential equation by using similarity transformations.