Stability and Well-Posedness Analysis of the Fractional Shallow Water Wave Models with Applications

Abstract

In this work, we present a comprehensive analytical investigation of the fractional Whitham-Broer-Kaup Equations (WBKEs), which model nonlinear oscillatory behavior in shallow water wave phenomena. The equations are formulated using the Atangana-Baleanu-Caputo (ABC) fractional derivative to capture nonlinear oscillations and the dynamics of tsunami-induced shallow water waves. Within the framework of Banach spaces endowed with the compact-open topology, the existence, uniqueness, and Hyers-Ulam stability of the fractional WBKEs are rigorously established via fixed point theorems. Furthermore, a hybrid analytical technique combining the double Laplace transform and the Adomian decomposition method, referred to as the Double Laplace Transform Adomian Decomposition Method (DLADM), is developed to obtain approximate series solutions. This method effectively handles nonlinearities and fractional-order effects, yielding rapidly convergent solutions that agree well with exact analytical results. The findings also demonstrate the robustness of the fractional model under small perturbations and its effectiveness in describing nonlinear shallow water wave dynamics.