Vadermonde-type odd-soliton solutions for the Whitham-Broer-Kaup model in the shallow water small-amplitude regime

Under investigation in this paper, with symbolic computation, is the Whitham-Broer-Kaup (WBK) system for the dispersive long waves in the shallow water small-amplitude regime. N-fold Darboux transformation (DT) for a spectral problem associated with the WBK system is constructed. Odd-soliton solutions in terms of the Vandermonde-like determinant for the WBK system are presented via the N-fold DT and evolution of the three-soliton solutions is graphically studied. Our results could be used to illustrate the bidirectional propagation of the waves in the shallow water small-amplitude regime.