Oceanic shallow-water investigations on a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system

To date, with respect to water waves, researchers have studied certain systems, including a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system that describes, e.g., the dispersive long waves in the oceanic shallow water, which we study here. With respect to, e.g., the horizontal velocity of the water wave as well as the height of the deviation from the equilibrium position of the water, with symbolic computation, on the one hand, the system is found to pass the Painlevé test under some coefficient constraints, while on the other hand, two families of the bilinear forms and two families of the N-soliton solutions are constructed, with N as a positive integer. Related constraints are shown. Our bilinear forms and N-soliton solutions depend on the coefficients in the system.