Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq–Burgers system in ocean waves

Water waves, a common natural phenomenon, have been influential in various fields, such as energy development, offshore engineering, mechanical engineering, and hydraulic engineering. To describe the shallow water waves near an ocean coast or in a lake, we use the (1 + 1)-dimensions Boussinesq–Burgers system. By means of Lie symmetry analysis, symmetry groups and infinitesimal generators are obtained for the (1 + 1)-dimension Boussinesq–Burgers system. For the sake of finding the invariant solutions of the Boussinesq–Burgers system, the optimal one-dimensional subalgebra system is computed. Furthermore, using similarity reduction and the generalized Kudryashov method, we attain the abundant wave solutions of the Boussinesq–Burgers system presented in this research paper. Additionally, the exact solutions, which illustrate the effectiveness of the proposed method, also reveal the physical interpretation of the nonlinear models. To demonstrate the significance of interaction phenomena, dynamical behaviors of some attained solutions are depicted geometrically and theoretically through suitable parameter values. Consequently, kink, singular, periodic, solitary wave solutions, and their elastic nature have been shown to validate these solutions with physical phenomena. With the aid of the obtained results, the researchers could gain an understanding of the different modes of shallow water waves nearby an ocean beach. The computational work ascertained that the imposed methods are sturdy, precise, modest, and widely applicable.