We have developed a theoretical analysis to systematically study the early and late-time evolution of the Rayleigh-Taylor instability in a finite-sized spatial domain. The nonlinear dynamics of fluids with similar and contrasting densities are considered for two-dimensional flows driven by sustained acceleration. The flows are periodic in the plane normal to the direction of acceleration and have no external mass sources. Group theory analysis is applied to accurately account for the mode coupling. Asymptotic linear and nonlinear solutions are found to describe the interfacial dynamics far from and near the boundaries. The influence of the size of the domain on the diagnostic parameters of the flow is identified. In particular, it is shown that in a finite-sized domain the flow is slower compared to the spatially extended case. The direct link between the multiplicity of solutions and the interfacial shear is explored. It is suggested that the interfacial shear function acts as a natural parameter to the family of nonlinear asymptotic solutions.