∂̄-dressing method for the coupled third-order flow equation of the Kaup–Newell system

In this article, we investigate the nonlinear spectral properties of the coupled third-order flow of the Kaup–Newell (TOFKN) system by formulating a local matrix $\bar{\partial }$-equation with non-normalized boundary conditions and two linear constraint equations. Furthermore, we derive a coupled Kaup–Newell hierarchy with sources using recursive operator. By employing a specially designed spectral transformation matrix and the $\bar{\partial }$-dressing method, we construct the $N$-soliton solutions of the coupled TOFKN system, providing explicit expressions for the one- and two-soliton solutions. These findings demonstrate the effectiveness of the $\bar{\partial }$-dressing method in capturing the complex nonlinear dynamics of the TOFKN system, offering new insights into its soliton interactions and stability properties.