Non-isospectral Integrable System And The Corresponding Soliton Equations The Results

In this article, the non-isospectral polynomial spectral problem and the corresponding soliton equations will be discussed, especially the N×N AKNS system. Non-isospectral AKNS system is more complicated than the isospectral case. The soliton equations corresponding to the isospectral AKNS system are usually the differential equations, while the equations corresponding to the non-isospectral case are the integral equations and the integral-differential equations. We prove that the non-isospectral AKNS system and the reductive AKNS system are well-defined with some asymptotic conditions. We construct the universal Darboux transformation for this system. We also prove that the Darboux transformation of this type conserves the asymptotic property and only the integral constants are changed. By using this type of Darboux transformation, the soliton solutions of the corresponding equations can be obtained.