N-Fold Darboux Transformation and Bidirectional Solitons for Whitham–Broer–Kaup Model in Shallow Water

Abstract

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) system, which describes the dispersive long wave in shallow water. Through a variable transformation, the WBK system is casted into a general Broer–Kaup system whose Lax pair can be derived by the Ablowitz–Kaup–Newell–Segur technology. With symbolic computation, based on the aforementioned Lax pair, the N-fold Darboux transformation is constructed with a gauge transformation and the multi-soliton solutions are obtained. Finally, the elastic interactions of the two-soliton solutions (including the head-on and overtaking collisions) for the WBK system are graphically studied. Those multi-soliton collisions can be used to illustrate the bidirectional propagation of the waves in shallow water.