Darboux Transformation Of The Family Of Unified Form For The Nonlinear Dirac Equation And Applications

In this paper,we focus on the Darboux transformation of the Dirac hierarchy and its application,in the first chapter,we simply introduce the soliton theory and the beginning and development process of the Darboux transformation related to this paper,in the second chapter,we start from the Dirac spectral problem,then derive a Dirac hierarchy by using Lenard operator pairs and the zero-curved equation. in the third chapter,an explicit and universal Darboux transformation for the whole hierarchy is constructed based on its Lax pair in theory.In the end,exact solutions of the Dirac hierarchy are constructed explicitly as an application of Darboux transformation.