Non-isospectral integrable couplings of Ablowitz--Kaup--Newell--Segur (AKNS) hierarchy with self-consistent sources

A hierarchy of non-isospectral Ablowitz--Kaup--Newell--Segur (AKNS) equations with self-consistent sources is derived. As a general reduction case, a hierarchy of non-isospectral nonlinear Schr\"odinger equations (NLSE) with self-consistent sources is obtained. Moreover, a new non-isospectral integrable coupling of the AKNS soliton hierarchy with self-consistent sources is constructed by using the Kronecker product.