Nonlocal Symmetry, CTE Solvability and Interaction Solutions of Whitham–Broer–Kaup Equations**Supported by the Key Foundation of Anhui Education Bureau under Grant No. KJ2013A028, the 211 Project of Anhhui University under Grant No. J18520104, Scientific Training Project for University Students, National Natural Science Foundation of China under Grant Nos. 11471015, 11571016, and Natural Science Foundation of Anhui Province under Grant No. 1408085MA02

Abstract

Whitham–Broer–Kaup (WBK) equations in the shallow water small-amplitude regime is hereby under investigation. Nonlocal symmetry and Bäcklund transformation are presented via the truncated Painlevé expansion. This residual symmetry is localised to Lie point symmetry by the properly enlarged system. The finite symmetry transformation of the prolonged system is computed. Based on the CTE method, WBK equations are linearized and new analytic interaction solutions between solitary waves and cnoidal waves are given with the aid of solutions for the linear equation.