Abstract
Water waves, which are essentially complex waves controlled by gravity fields and surface movements, have been studied actively. In this article, first, the Bäcklund transformation of Whitham–Broer–Kaup (WBK) equations is studied using the homogeneous balance method. Second, the solitary wave solutions and similar wave solutions of WBK equations are constructed using the obtained Bäcklund transformation, and the conclusions obtained from the homogeneous balance method and the Painlevé analysis method are compared. Then, based on the auxiliary equation method and the Bäcklund transformation obtained previously, the Weierstrass elliptic function solutions and degenerate solutions of WBK equations are attained. Finally, using the symbolic calculation system Mathematica, the dynamic characteristics of solutions are analyzed through images, which helps us increase the diversity of solutions and get more new phenomena. It is worth mentioning that by comparing the Bäcklund transformation and soliton solutions obtained by the two methods, we obtain the same and different contents, respectively. The waves in the ocean are complex and diverse. By studying the dynamic characteristics of waves, theoretical basis is provided for the motion of ships in the ocean. Furthermore, the results derived from this work have not been presented before.
Funder
the Graduate Students' Research and Innovation Fund of Inner Mongolia Normal University, China
the Fundamental Research Funds for the Inner Mongolia Normal University
the National Natural Science Foundation of China
the Natural Science Foundation of Inner Mongolia Autonomous Region, China
the Graduate Students's Scientific Research Innovation Fund Program of Inner Mongolia Normal University, China