Lax pair,infinitely-many conservation laws and soliton solutions for a set of the time-dependent Whitham-Broer-Kaup equations for the shallow water

Under investigation in this paper is a set of the time-dependent Whitham–Broer–Kaup equations, which is used for the shallow water under the Boussinesq approximation. The equations can be transformed into generalized time-dependent coefficient Ablowitz–Kaup–Newell–Segur system via the variable transformation. Lax pair, infinitely-many conservation laws and bilinear forms of the Ablowitz–Kaup–Newell–Segur system are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. The solitons are physically related to the horizontal velocity field and height that deviates from equilibrium position of the water. Features of the solitons are studied: Soliton amplitude is related to the wave number parameters, while the soliton velocity is related to the wave number parameters and variable coefficient. Interactions between/among the solitons could be elastic or inelastic, determined by the wave number parameters. Interaction property could not be affected by the variable coefficient. Soliton stability is studied via the numerical calculation, which indicates that the solitons could only propagate steadily in a limited time.