Analytical Study on the Generalized Fifth-Order Kaup–Kupershmidt Equation from the Shallow Water Wave

In this paper, the generalized fifth-order Kaup–Kupershmidt (KK) equation from the shallow water wave, have been investigated. With the help of bilinear method and auxiliary function, the multi-soliton solutions of the generalized fifth-order KK equation have been obtained, and those solutions have not been given before. From the analytical view, the interactions of the solitons have been presented. Through the discussion, we have found that the velocity of the soliton is related to the sign of the physical parameters $$b$$ and $${{c}_{j}}$$ . Also, the (bright or dark-like) types of the soliton interaction are related to the physical parameters $$a$$ . Hence, the velocities and bright-dark types of the soliton interactions could be controlled by adjusting the physical parameters.