Noncommutative AKNS Equation Hierarchy and Its Integrable Couplings with Kronecker Product

We present a noncommutative version of the Ablowitz--Kaup--Newell--Segur (AKNS) equation hierarchy, which possesses the zero curvature representation. Furthermore, we derive the noncommutative AKNS equation from the noncommutative (anti-)self-dual Yang--Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, theintegrable coupling system of the noncommutative AKNS equation hierarchy is constructed by using the Kronecker product.