On the multiple soliton and lump solutions to the (3+1)-dimensional Painlevé integrable Boussinesq-type and KP-type equations

This study aims to construct and investigate a novel (3+1)-dimensional model that combines both the Boussinesq-type and the Kadomtsev-Petviashvili (KP) equations, which is called a (3+1)-dimensional Boussinesq-KP-type (B-KP-type) equation. We demonstrate that the combined model does not exhibit Painleve integrability. However, it does provide two separate integrable equations: one of the (3+1)-dimensional Boussinesq-type equation and the other of the (3+1)-dimensional KP-type equation. The simplified Hirota’s direct approach can formally admit multiple soliton solutions for each integrable model. In addition, we employ Maple symbolic computation with the Hirota bilinear form to derive a class of lump solutions for each equation. Moreover, the (3+1)-dimensional B-KP-type equation is analyzed using the families of tanh and tan methods for deriving traveling wave solutions, including shock wave solutions, periodic solutions, and others. Due to the significant correlation and wide range of applications of the Boussinesq-type and KP equations, the derived equations will play a crucial role in elucidating and interpreting various nonlinear phenomena observed in fluid mechanics and other fields of nonlinear physics and engineering issues.