New Exact Solutions of a Variable-Coefficient (2+1)-Dimensional Kadomtsev- Petviashvili Equation Using a Combination of Lie Symmetry Analysis and the Generalized Expansion Method

Abstract In this paper, we apply the variable-coefficient (2 + 1)-dimensional Kadomtsev-Petviashvili (vcKP) equation by using a combination of Lie symmetry analysis and the generalized expansion method. First, by using the method of Lie symmetry analysis, three similarity transformations associated with three Lie symmetry generators are obtained, which reduce the original equation into three different (1 + 1)-dimensional nonlinear evolution equations. These reduced equations are then solved analytically by the Generalized Expansion Method (GEM). During the solution process, different admissible classes of the variable coefficients are obtained in a natural way without any a priori assumptions about their forms. The resulting classification establishes the correspondence between the chosen Lie symmetry generators, their similarity transformations, and the corresponding classes of variable coefficients. Based on this classification, several families of exact solutions are derived, including bright solitons, dark solitons, kink-like solitary waves, double-dark solitons, M-shaped bright solitons, and singular periodic solitons. The influence of the variable coefficients on the wave structures is analyzed, and the dynamical behavior of the obtained solutions is illustrated through two-dimensional, three-dimensional, and contour plots.