Kaup-Newell Hierarchy With Self-Consistent Sources And Their Reduction

B?cklund transformations (BTs) are an important tool in the studies of integrable systems. In recent years some new properties of BTs for finite-dimensional integrable Hamiltonian systems have attracted some attention. These explicit BTs are shown to be canonical transformations including B?cklund parameter and to satisfy a spectrality property with respect to and the "conjugated" variable for which the pair or for some function lies on the spectral curve. As model we construct the B?cklund transformation for the high-order constrained flows of Kaup-Newell (KN) hierarchy via the Darboux transformations of their Lax representation. We show that these BTs are canonical transformations by presenting their generating functions and possess the spectrality property with respect to and conjugate variable . As the expansion of normal soliton equations, the soliton equations with self-consistent sources (SESCS) appear widely in the models of hydrodynamics, plasma physics, solid-state physics and have important significance. In recent years people have found that the high-order constrained flows are exactly the case of "" for the SESCS, which makes us obtain the Lax pair of SESCS naturally. Then the inverse scattering method and Darboux transformations (DTs), which are related to the Lax representations of SESCS, have new development. This paper is mainly on the DTs of SESCS. First, we get rid of the restriction of the spectrality parameter in a known DT of the KN equation. It was pointed out that the normal DTs for SESCS which provides auto-BTs can not be used to construct solution of SESCE from the trivial solutions. In this paper, we present binary Darboux transformations (BDTs) with arbitrary -dependent functions for KN hierarchy. This BDTs offers a non-auto BTs between KN equations with different degrees of source and enables us to obtain non-trivial solutions from trivial one. The derivative nonlinear Schr?dinger equation (DNLS+ or DNLS-) and the DNLS with self-consistent cources can be reduced from KN equation and the KN equation with self-consistent sources. The paper also present the BDTs with arbitrary -dependent functions for DNLS with self-consistent sources. And we get some solutions for the DNLS with self-consistent sources, such as soliton solutions, algebraic soliton solutions and positon solutions. Under a proper choice of the scattering data, the positons are slowly decreasing, oscillating solutions whose transmission coefficient is unity, and it has no pole in this DNLS with self-consistent sources model.