Darboux Transformation for the Non-isospectral AKNS Hierarchy and Its Asymptotic Property

In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-B\"acklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.