A GKM-Auxiliary Framework for the (2+1)-Dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation Arising in Fluid Dynamics and Nonlinear Optics

Abstract

The (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation is a significant model for studying weakly dispersive waves in fluid dynamics and nonlinear optics. However, current algorithms are insufficient to fully understand it across the full range of its complex, nonlinear lattices. In this work, we propose a novel GKM-Auxiliary approach that combines the Generalized Kudryashov Method and the Extended Auxiliary Equation Method to obtain broad families of solitary wave solutions and multi-soliton solutions. This unified treatment yields a wide range of closed-form solutions, including bright, kink, dark, and compactons, as well as periodic solutions and multilayer solitons. By employing the traveling wave ansatz approach and the homogeneous balance principle technique, we obtain exact solutions in hyperbolic, trigonometric, exponential, and rational functions. Graphical simulations through Maple and Mathematica confirm not only the richness of shapes but also the physical relevance of the framework used in this work. Furthermore, this combined approach could contribute to the development of analytical methods applied to high-order NLPDEs on a physical level, as well as to potential applications in (2+1)-dimensional hydrodynamics, plasma oscillations, and optical soliton theory.