Analyzing Soliton Solutions of the (n + 1)-dimensional generalized Kadomtsev-Petviashvili equation: Comprehensive study of dark, bright, and periodic dynamics

This paper investigates the (n + 1) dimensional integrable extension of the Kadomtsev-Petviashvili (KP) equation. The study focuses on extracting Auto-Backlund transformations for the given model using the extended homogeneous balance (HB) method in conjunction with Maple. These transformations are then employed to obtain explicit analytic solutions for the equation. Additionally, a Bilinear Backlund transformation is constructed based on the Hirota bilinear form, leading to the acquisition of exponential function solutions. Furthermore, complexiton solutions for the KP equation are obtained utilizing the bilinear form and the extended transformed rational function technique. The physical properties of the obtained solutions are explored through visualization techniques, including 3D, 2D, and contour plots. The analysis reveals the presence of dark, bright, and periodic solitons. The obtained solutions of the KP equation have practical applications in nonlinear optics, plasma physics, and wave propagation analysis.