In this paper, we investigate the Grammian determinant solutions of a (2+1)-dimensional generalized nonlinear system in fluid mechanics. The solutions of this system are expressed in terms of Grammian determinants, with distinct classes of solutions obtained by selecting different functional forms for the determinant elements, including Airy functions, exponential functions, and semi-rational functions. Additionally, via modifying the Grammian determinant form and its elements using differential operators \(P_{\iota }\) and \(Q_{j}\) , a series of distinct solutions are obtained. The propagation and interaction dynamics of these nonlinear waves are analyzed. When the function m is represented by the Airy function, the distinct nonlinear wave propagation phenomena are observed. By utilizing the exponential functions, the lump chains and soliton-like waves are obtained. Moreover, asymptotic analysis of these solutions is performed, and the propagation velocities and periods of lump chains are explicitly calculated. The interaction dynamics between soliton-like waves and lump chains are systematically discussed. Employing semi-rational functions allows for a detailed investigation of the interaction between lump waves and soliton-like waves. Finally, increasingly complex lump wave patterns are demonstrated, with their interaction mechanisms are thoroughly analyzed. It is revealed by our findings that the properties of the solutions are fundamentally governed by both the determinant structure and the specific functional forms adopted for its elements.
