Exploring the (3+1)-generalized Kadomtsev-Petviashvili equation: symmetries, reductions, and exact solutions

Abstract 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.