A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup-Newell equation

Due to the fact that the higher-order Kaup-Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann-Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single-double zeros, and double-double zeros are especially shown.