Elastic–inelastic-interaction coexistence and double Wronskian solutions for the Whitham–Broer–Kaup shallow-water-wave model

By virtue of a variable transformation, Whitham–Broer–Kaup (WBK) model describing the propagation of the shallow water waves is transformed into the generalized Ablowitz–Kaup–Newell–Segur (AKNS) systems, the bilinear forms of which are derived with a rational transformation. Explicit multi-soliton solutions are obtained subsequently by means of the Wronskian technique and symbolic computation. Furthermore, interactions of the solitons are investigated graphically for the WBK model and a phenomenon is revealed that the elastic and inelastic interactions occur simultaneously without affecting each other, i.e., the elastic and fission interactions coexist at the same time, and they do not disturb mutually during the collision. Our results could be useful to explain certain physical phenomena in the shallow water models.