Extended double Wronskian solutions to the Whitham-Broer-Kaup equations in shallow water

Whitham-Broer-Kaup (WBK) equations describing the propagation of shallow-water waves, with a variable transformation, are transformed into a generalized Ablowitz-Kaup-Newell-Segur system, the bilinear forms of which are obtained via the rational transformations. Employing the matrix extension and symbolic computation, we derive types of solutions of the WBK equations through the selection of different canonical matrices, including solitons, rational solutions, and complexitons. Furthermore, dynamic properties of the solutions are discussed graphically and a novel phenomenon is observed, i.e., the coexistence of the elastic-inelastic interactions without disturbing each other.