The Darboux Transformations Of KP And Its Related Integrable Hierarchies

KP hierarchy is one of the important research objects in the field of integrable systems,which has been widely used in mathematical physics.In recent years,KP and its related integrable hierarchies have attracted the interest of many researchers,and many remarkable achievements have been achieved.The Darboux transformation is an efficient way to solve integrable systems,various types of solutions of integrable systems can be obtained by Darboux transformation.In this thesis,under the framework of Sato theory,combined with the highest weight representation of infinite dimensional Lie algebra,the Darboux transformations of KP and its related integrable hierarchies will be studied and the bilinear equation and various applications of Darboux transformations will be dicussed,including the bilinear equation,the squared eigenfunction symmetry and the additional symmetry,by using the language of the free fermion and the free boson.This thesis is divided into six chapters as follows:In Chapter 1,starting from the research background of KP and its related integrable hierarchies,including modified KP hierarchy(mKP for short),BKP hierarchy and CKP hierarchy,we review some basic properties of these integrable hierarchies,as well as the research of Darboux transformation,the bilinear equation,the squared eigenfunction symmetry and additional symmetry.Finally,the main contents,research methods and ideas,as well as the innovation of the research are introduced.In Chapter 2,the KP hierarchy and its Darboux transformation are mainly studied by using free fermions.Firstly,the properties of free fermions are reviewed,and some important relations of free fermions are given.Then the Lax structure,dressing operator,tau function are constructed.Secondly,the spectral representation of KP hierarchy is given.Then,the fermionic tau functions of the KP hierarchy under various Darboux transformations are studied by using the properties of the free fermion and the spectral representation,and then the bilinear equations satisfied by the transformed tau function are given.Finally,we conclude this chapter with some examples of bilinear equations.In Chapter 3,the mKP hierarchy and its Darboux transformation are mainly studied by using free fermions.Firstly,the bilinear equations of the 1st mKP hierarchy,that is,the Kupershmidt Kiso version of the mKP hierarchy,are reviewed,and the corresponding Lax structure is written.Then,the properties of Miura transformation are reviewed.Through Miura transformation,the spectral representation of mKP hierarchy is obtained from the spectral representation of KP hierarchy.Then the Darboux transformation in fermionic picture of mKP hierarchy is discussed.The fermionic tau function under the successive application of Darboux transform is given.On this basis,the bilinear equation satisfied by the transformed tau function is obtained.Some examples are also listed in the last section.In Chapter 4,the BKP hierarchy and its Darboux transformation are mainly studied using neutral free fermions.Firstly,the BKP hierarchy is reviewed from two methods.On the one hand,the BKP hierarchy can be constructed from neutral free fermions.At the same time,some relations of neutral free fermions are also given.On the other hand,the BKP hierarchy can be regarded as the mKP hierarchy under Kupershmidt reduction.Secondly,the squared eigenfunction potential of BKP hierarchy can be obtained by its relationship with mKP hierarchy.Then,on this basis,the Darboux transformation in fermionic picture of BKP hierarchy is given,and then the bilinear equation satisfied by tau function under the action of Darboux transformation is given.Finally,some examples of bilinear equations are also given.In Chapter 5,the CKP hierarchy and its Darboux transformation are mainly studied by using free bosons.Firstly,the construction of CKP hierarchy is reviewed by using free bosons,and then the bilinear equation and Lax equation of modified CKP hierarchy are obtained.At the same time,the solutions of constrained CKP hierarchy are obtained.Next,through the relationship between the Darboux transformation of KP hierarchy and the squared eigenfunction symmetry,the Darboux transformation of CKP hierarchy can be expressed in the form of free boson.Finally,on this basis the actions of additional symmetry on the tau function of CKP hierarchy are studied.In Chapter 6,we summarize the main contents of this thesis,and then discuss possible research directions in the future.