Invariance Analysis, Optimal System, and Wave Propagation of Solutions of (3+1)‐Dimensional Broer–Kaup–Kupershmidt (BKK) Model in Shallow Water Dynamics

The paper investigates a (3+1)‐dimensional (D) system of Broer–Kaup–Kupershmidt (BKK) equations using the method of Lie group analysis to understand the dynamics of nonlinear gravity and long waves in fluid phenomena. Using symbolic computation, we derive infinitesimal generators, vector fields, and the optimal system of the subalgebras. For different cases, we perform the similarity reductions utilizing the optimal system. The similarity reduction reduces the nonlinear partial differential equations (PDEs) into the nonlinear ordinary differential equations (ODEs), which provides closed‐form group invariant solutions. As a result, a plethora of exact solutions, including solitary and periodic wave type solutions, are obtained for the considered model. The obtained solutions contain the arbitrary parameters and functions, offering significant potential for modeling shallow water wave propagation and complex geophysical fluid dynamics. To further analyze the results, we plot the obtained closed‐form solutions into three‐dimension, which shows how the solutions evolve across the varying parameters. All the solutions derived are new and distinct from the earlier findings. We compare our results with the existing solutions presented in the literature highlights similarities, differences, and potential advancements. The figures plotted in this paper demonstrates the variation effects of the nonlinear parameters to the BKK system on the evolution of solitons. The physical structure of the solutions varies significantly with changes in parameters. The physical significance of the obtained solutions is discussed in detail, which demonstrates the model's capacity to describe multidimensional nonlinear wave interactions. The obtained solutions have potential applications in nonlinear physics, quantum mechanics, and optical fibers. The symbolic computations involved in this study were carried out utilizing the software Mathematica and Maple.