Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity

Abstract This paper studies the asymptotic behavior of a solution of dispersive shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation. The energy decay rates of the solution to the model are examined through the Fourier transform method. Moreover, we develop a pseudo-compact finite difference scheme for solving the model, study solution behavior, and confirm our theoretical results. The fundamental energy-decreasing property, which is obtained from the model of coupling the Rosenau-RLW equation with the Rosenau-Burgers equation, is derived and preserved by the present numerical scheme. The existence, uniqueness, convergence, and stability of the numerical solution are theoretically analyzed. Some numerical experiments are also conducted to demonstrate the accuracy and robustness of the present method. Finally, to verify the optimal decay rates, the numerical results are carried out at variant time scales by applying the moving boundary technique. The simulations are successfully constructed to support the theoretical results.