Several categories of exact solutions of the third-order flow equation of the Kaup–Newell system

In this paper, we introduce the third-order flow equation of the Kaup–Newell (KN) system. We study this equation, and we obtain different types of solutions by using the Darboux transformation (DT) and the extended DT of the KN system, such as solitons, positons, breathers, and rogue waves. The extended DT is obtained by taking the degenerate eigenvalues $$ \lambda _{i} \rightarrow \lambda _{1} (i=3,5,7,\ldots ,2k-1)$$ and by performing the Taylor expansion near $$\lambda _{1}$$ of the determinants of DT. Some analytic expressions are explicitly given for the first-order solutions. We study the unique waveforms of both the first-order and higher-order rogue-wave solutions for special choices of parameters, and we find different types of such wave structures: fundamental pattern, triangular, modified-triangular, pentagram, ring, ring-triangular, and multi-ring wave patterns. We conclude that the third-order dispersion and quintic nonlinear term of the KN system modify both the trajectories and speeds of the solutions as compared with those corresponding to the second-order flow equation of the KN system.