Painleve Analysis and Analytic Solutions of a Variable-Coefficient (3 + 1)-Dimensional KP-Type Equation in Shallow Water Waves, Ion-Acoustic Waves and Fluid Flow Dynamics

This paper investigates a variable-coefficient (3+1)-dimensional Kadomtsev–Petviashvili (KP)-type equation with time-dependent drift terms to determine how coefficient-dependent propagation can be retained within an analytically tractable nonlinear dispersive model. The Weiss–Tabor–Carnevale method with the simplified Kruskal ansatz establishes resonance compatibility in the Laurent-series framework, while the dependent-variable transformation u = 2(ln f) xx yields a Hirota bilinear equation and its general N-soliton tau function. A conjugate-pair long-wave limit of that tau function produces a multi-lump family, whose first-order member is characterized by explicit extrema, algebraic localization and a computable center trajectory, whereas complex-conjugate reductions generate breather and general multi-breather–stripe states. Because the time-dependent functions enter the phase variables through integral terms, they modulate propagation trajectories and fixed-point phase rates without altering the amplitude, characteristic width or pairwise interaction factors determined by the constant spectral parameters. Analytical characterizations and graphical illustrations are supplied for the soliton, lump, breather and interaction structures.