By employing the one-parameter Lie group of transformations method, abundant exact invariant solutions are obtained for a (2+1)-dimensional dissipative long wave (DLW) system, which describes the water wave model of hydrodynamics with wide channels or open seas of finite depth and can also be used to illustrate nonlinear wave propagation in the dissipative medium. Initially, we derived the Lie symmetries, geometric vector fields, and commutative table, and adjoint relations of the examined vectors for the system. A one-dimensional optimal system of symmetry subalgebras is well-structured. It leads to the reduction of independent variables of the considered DLW system and leaves the system invariant. The (2+1)-dimensional DLW system is reduced into nonlinear ordinary differential equations (NLODEs) for the various subalgebras obtained through two phases of similarity reductions. Some novel closed-form solutions are obtained, including arbitrary independent functional parameters, and have never been reported in the literature. Moreover, we depict the dynamical behaviors of obtained closed-form invariant solutions like multi-wave solitons, single solitons, breather-waves, rogue waves, bell-shaped structures, Kink-waves, and Lump-waves, wave-wave interactions, and annihilation of Kink-wave profiles through three-dimensional graphics. Eventually, a comparison with other results is discussed.

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.