Gauge transformation, elastic and inelastic interactions for the Whitham–Broer–Kaup shallow-water model

Whitham–Broer–Kaup (WBK) model is a model for the dispersive long wave in shallow water. With symbolic computation, gauge transformation between the WBK model and a parameter Ablowitz–Kaup–Newell–Segur (AKNS) system is hereby constructed. By selecting seeds, we derive two sorts of multi-soliton solutions for the WBK model via a N-fold Darboux transformation (DT) of the parameter AKNS system, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Different from the bilinear way, the double Wronskian solutions can be obtained via the N-fold DT with a linear algebraic system and matrix differential equation solved. A novel inelastic interaction is graphically discussed, in which the soliton complexes are formed after the collision. Our results could be helpful for interpreting certain shallow-water-wave phenomena.