Energy-driven soliton dynamics in the (2+1) KP-BBM model: Exact solutions and numerical insights

In this article, we explore the energy-guided evolution of the (2+1)-dimensional Kadomtsev–Petviashvili– Benjamin–Bona–Mahony (KP-BBM) equation — an important nonlinear evolution equation that describes bidirectional traveling water surface waves and simulates intricate fluid energy flows. To reveal the abundant energy structures underlying this system, we use the modified Sardar sub-equation technique, a powerful and effective analytical tool for producing multifamily solutions with ease. By following this method, we obtain a range of new and physically relevant exact energy solutions, such as dark solitons, bright solitons, singular solitons, exponential solitons, and singular periodic waveforms. To better expose wave energy propagation and interaction in the KP-BBM model, we offer extensive graphical visualizations: 2D profiles, 3D surface plots, and contour maps show in detail the complex energy landscapes and wave dynamics. To justify the accuracy and energetic stability of such solutions, we use the differential transform method (DTM) for numerical approximations. The good agreement between analytical and numerical solutions provides justification for the efficiency and reliability of the suggested method in modeling nonlinear wave energy dynamics. In general, this study not only shows the versatility of the upgraded Sardar sub-equation technique for the solution of energy-abundant nonlinear wave equations but also broadens the theoretical horizon on wave energy transport in fluid mechanics. To ensure the robustness of our results, we performed a stability analysis on the derived solutions, confirming their reliability and physical applicability.