Painlevé analysis and Hirota direct method for analyzing three novel physical fluid extended KP, Boussinesq, and KP-Boussinesq equations: Multi-solitons/shocks and lumps

Due to the extensive and diverse connections of the Kadomtsev-Petviashvili (KP) equation with various physical phenomena, such as the propagation of waves in sea and ocean water and the propagation of nonlinear waves in plasmas when two-dimensional perturbations are considered, thus in this study and based on the original KP equation, we construct and investigate three extended models, which are called the extended KP (eKP) equation, the extended Boussinesq (eBO) equation, and the extended KP-Boussinesq (eKP-BO) equation. These equations are commonly observed in many physical contexts involving nonlinear and dissipative media. Our research begins with thoroughly verifying the complete integrability of the three extended models. Once this crucial step is accomplished, we will examine these three extended models using the simplified Hirota approach (SHA), leading us to derive multiple soliton/shock solutions. Furthermore, the Hirota bilinear form of each model will be employed to derive a set of lump solutions for these three extended models, utilizing different parameter values. We also analyze some derived solutions graphically by presenting some two- and three-dimensional graphics to understand the dynamic behavior of the waves described by these solutions. The obtained results are anticipated to benefit a broad spectrum of academics interested in studying fluid mechanics, water wave characterization, and other interdisciplinary domains. Additionally, these models are anticipated to be essential in describing and understanding the dynamics of several nonlinear phenomena that arise and propagate in different plasma systems when perturbations are considered in multidimensions.