Novel (3 + 1)-dimensional variable-coefficient B-type KP equation: exploring integrability, soliton solutions and interaction behaviours

This paper investigates the integrability and localized wave dynamics of a generalized (3 + 1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili (BKP) equation. Starting from a broad non-autonomous model, the Weiss–Tabor–Carnevale (WTC) Painlevé analysis is employed to identify the Painlevé-admissible variable-coefficient BKP subclass. The compatibility conditions show that the higher-order dispersive and nonlinear coefficients are constrained, while several lower-order coefficients may remain arbitrary differentiable functions of time. For this selected subclass, the Hirota bilinear form is derived through a dependent-variable transformation, and multi-soliton solutions are constructed. Breather solutions are obtained by imposing complex conjugate wave parameters, and algebraic lump solutions are generated through the long-wave limit method. The limiting procedure, nonsingularity condition and algebraic localization of the lump solutions are discussed explicitly. In addition, hybrid interactions between lumps and line solitons or breathers are analyzed by asymptotic sector decompositions. Explicit phase-shift formulas show that the lump component preserves its algebraic profile and extremal amplitude relative to the soliton background, while acquiring a permanent coordinate shift. The roles of the variable coefficients in modulating phase accumulation, instantaneous velocity, breather oscillation, lump trajectory and interaction location are also clarified. These results provide a systematic description of parameter-dependent localized wave dynamics in a Painlevé-admissible non-autonomous BKP framework.