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.

This paper investigates the new KP equation with variable coefficients of time ‘ t ’, broadly used to elucidate shallow water waves that arise in plasma physics, marine engineering, ocean physics, nonlinear sciences, and fluid dynamics. In 2020, Wazwaz [1] proposed two extensive KP equations with time-variable coefficients to obtain several soliton solutions and used Painlevé test to verify their integrability. In light of the research described above, we chose one of the integrated KP equations with time-variable coefficients to obtain multiple solitons, rogue waves, breather waves, lumps, and their interaction solutions relating to the suitable choice of time-dependent coefficients. For this KP equation, the multiple solitons and rogue waves up to fourth-order solutions, breather waves, and lump waves along with their interactions are achieved by employing Hirota's method. By taking advantage of Wolfram Mathematica , the time-dependent variable coefficient's effect on the newly established solutions can be observed through the three-dimensional wave profiles, corresponding contour plots. Some newly formed mathematical results and evolutionary dynamical behaviors of wave-wave interactions are shown in this work. The obtained results are often more advantageous for the analysis of shallow water waves in marine engineering, fluid dynamics, and dusty plasma, nonlinear sciences, and this approach has opened up a new way to explain the dynamical structures and properties of complex physical models. This study examines to be applicable in its influence on a wide-ranging class of nonlinear KP equations.