  Due to higher-order Kaup-Newell (KN) system has more complex and diverse
solutions than classical second-order flow KN system, the research on it has
attracted more and more attention. In this paper, we consider a higher-order KN
equation with third order dispersion and quintic 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.
