Dynamical analysis of soliton wave solutions of $(2+1)$-Dimensional Kadomtsev-Petviashvili (KP) Equation through two efficient analytical techniques

In this paper, we consider generalized Kadomtsev-Petviashvili (KP) equation and presents new closed-form solutions through an analytical technique and Lie symmetry analysis. The KP equation, a key tool in modeling long waves and frequency dispersion, is relevant in various nonlinear physical systems. New soliton solutions—including single, multi, elastic, kink, bright, and dark types—are derived using the (2+1)-dimensional KP equation. The approach employs both a novel analytical method and Lie symmetry transformations to generate invariant solutions, reducing the governing equation to various ordinary differential equations (ODEs). These solutions are analyzed for dynamic behavior with detailed 2-D, 3-D, contour, and density plots created via Mathematica, providing insights into complex phenomena in fields like geophysics, optics, and atmospheric sciences.