Residual symmetries and interaction solutions for the Whitham–Broer–Kaup equation

The nonlocal residual symmetry for the Whitham–Broer–Kaup (WBK) equation is derived by the truncated Painleve analysis. The nonlocal residual symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Based on the consistent tanh expansion method (CTE), some exact interaction solutions among different nonlinear excitations are explicitly presented. Some special interaction solutions are investigated in both analytical and graphical ways.