This study investigates the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony (KP-BBM) equation using two effective methods: the unified scheme and the advanced auxiliary equation (AAE) scheme. The goal is to derive precise wave solutions, which are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through various figures, highlighting the significant impact of nonlinearity. Additionally, the Hamiltonian function and stability of the model are analyzed using a planar dynamical system approach, examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results show that these methods are reliable, straightforward, and powerful for analyzing nonlinear evolution equations in physics, applied mathematics, and engineering. The KP-BBM equation is a mathematical model that combines features of the KP and BBM equations. It describes the evolution of two-dimensional, weakly nonlinear, and weakly dispersive water waves. The equation is used to study wave patterns in coastal regions and harbors, providing insights into nonlinear phenomena in fluid dynamics. The study presents a comprehensive analysis of the KP-BBM model, including the derivation of numerous wave solutions using the unified and AAE methods. These solutions are compared with existing literature, revealing a diverse range of solutions with distinct behaviors. The study also conducts a bifurcation analysis of the model, assessing the stability of equilibrium points. The resulting phase portraits of the model are illustrated in figures, providing visual representations of the solutions. The research highlights the significant role of the nonlinear parameter p in influencing soliton formations and the dynamics of wave solutions within the KP-BBM model. The findings demonstrate that variations in parameter values can lead to changes in the dynamics of soliton solutions. The study concludes that the AAE and unified methods are more effective than the sine-cosine and tanh methods for solving the KP-BBM equation, yielding a significantly larger number of wave solutions.This study investigates the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony (KP-BBM) equation using two effective methods: the unified scheme and the advanced auxiliary equation (AAE) scheme. The goal is to derive precise wave solutions, which are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through various figures, highlighting the significant impact of nonlinearity. Additionally, the Hamiltonian function and stability of the model are analyzed using a planar dynamical system approach, examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results show that these methods are reliable, straightforward, and powerful for analyzing nonlinear evolution equations in physics, applied mathematics, and engineering. The KP-BBM equation is a mathematical model that combines features of the KP and BBM equations. It describes the evolution of two-dimensional, weakly nonlinear, and weakly dispersive water waves. The equation is used to study wave patterns in coastal regions and harbors, providing insights into nonlinear phenomena in fluid dynamics. The study presents a comprehensive analysis of the KP-BBM model, including the derivation of numerous wave solutions using the unified and AAE methods. These solutions are compared with existing literature, revealing a diverse range of solutions with distinct behaviors. The study also conducts a bifurcation analysis of the model, assessing the stability of equilibrium points. The resulting phase portraits of the model are illustrated in figures, providing visual representations of the solutions. The research highlights the significant role of the nonlinear parameter p in influencing soliton formations and the dynamics of wave solutions within the KP-BBM model. The findings demonstrate that variations in parameter values can lead to changes in the dynamics of soliton solutions. The study concludes that the AAE and unified methods are more effective than the sine-cosine and tanh methods for solving the KP-BBM equation, yielding a significantly larger number of wave solutions.