Exact travelling wave solutions of the Whitham–Broer–Kaup and Broer–Kaup–Kupershmidt equations

In this paper, an interesting fact is found that the auxiliary equation method is also applicable to a coupled system of two different equations involving both even-order and odd-order partial derivative terms. Furthermore, singular travelling wave solutions can also be obtained by considering other types of exact solutions of auxiliary equation. The Whitham–Broer–Kaup and the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations are chosen as examples to illustrate the effectiveness of the auxiliary equation method.