Exploring the shallow water waves near ocean beaches and lake shores via a Boussinesq–Burgers system

Water waves shape coastlines, influence marine engineering, and drive energy transport across ocean beaches and lake shores. For such shallow-water dynamics, we investigate a Boussinesq–Burgers system through computerized symbolic computation. With respect to the horizontal velocity and surface elevation fields, we construct a hybrid Bäcklund transformation coupling the system to both the heat equation and the Kaup–Boussinesq integrable system, alongside a generalized set of similarity reductions yielding a novel fourth-order ordinary differential equation. Our fourth-order framework captures higher-order dispersion effects and multi-speed propagation phenomena inaccessible to previous second-order reductions. A comprehensive dynamical analysis reveals modulational instability (MI) for 1 < β < 3 , homoclinic bifurcations near β ≈ 2.0 , and the onset of chaos under periodic perturbation, with the Melnikov threshold derived explicitly in terms of β . Numerical simulations confirm the analytical predictions and establish positive Lyapunov exponents in the chaotic regime. Our results recover previously known reductions as special cases while introducing significant new physics, including the first demonstration of deterministic chaos in this system.