The Research On The Bilinear Equations And Symmetries For The DKP And Its Related Hierarchies

The highest weight representations of infinite-dimensional Lie algebras play an important role in integrable systems.The investigation of linking infinite-dimensional Lie algebras d∞ with integrable systems has always attracted the attention of many scholars.In this paper we mainly investigate some topics involving the D-type Kadomtsev-Petviashvili(DKP)and modified DKP(mDKP)hierarchies.Firstly,the representation of an infinite-dimensional Lie algebra d∞ on a fermionic Fock space is used,then we establish boson-fermion correspondence,and further we investigate the bilinear equations of the DKP and mDKP hierarchies in different bosonizations.Also define the corresponding wave functions and tau functions,and investigate the Lax equations,Zakhalov-Shabat equations,additional symmetries,Darboux transformations of the DKP and mDKP hierarchies.The specific conclusions are as follows:In Chapter 2,we firstly introduce the infinite-dimensional Lie algebra d∞ and the free fermions.And then the representation on the fermions Fock space of the infinitedimensional Lie algebras d∞ is given.Then the Casimir operator SD of the infinitedimensional Lie algebra d∞ is given.And SD commutes with the action of d∞.Finally,bilinear equations of the fermions DKP and mDKP hierarchies are given.In Chapter 3,starting from the bilinear equations of the fermionic form of the DKP and mDKP hierarchies and using the first boson-fermion correspondence,the first bosonic forms of bilinear equations of the DKP and mDKP hierarchies is given.It is found that the corresponding bilinear equations are the same as the bilinear equations of the B-type Kadomtsev-Petviashvili(BKP)and modified BKP(mBKP)hierarchies.Then,based on the above conclusions,we prove that the conjecture raised by Y.You in the literature[Y.You,Physica D 50(1991)429-462]is correct,that’s:the product of two tau functions in the mDKP hierarchy is one of the tau functions in the KP hierarchy.Finally,we investigate the Darboux transformation in the free fermionic forms of the DKP and mDKP hierarchies,and the solutions of the constrainted BKP hierarchy in the context of d∞ is given.In Chapter 4,we firstly rearrange the labels of free fermions and obtain the free charged fermions.Then,by establishing the second boson-fermion correspondence,the second bosonic bilinear equations of the DKP and mDKP hierarchies are given.It is found that this kind of bosonic bilinear equations of the DKP and mDKP hierarchies is just the bilinear equation of the large BKP hierarchy.Therefore,if the related properties are investigated from the second bosonic bilinear equations of the DKP and mDKP hierarchies,we only need to investigate the bilinear equation of the large BKP hierarchy.In Chapter 5,we mainly investigate the second bosonic bilinear equations of the DKP and mDKP hierarchies.Since the combination of DKP and mDKP hierarchies is exactly the large BKP hierarchy,therefore,it is only necessary to investigate the bilinear equation of the large BKP hierarchy.Firstly,starting from the bilinear equation of the large BKP hierarchy,we construct the corresponding Lax structure,and give the corresponding Lax equation and specific examples.Then,from the Lax structure of the large BKP hierarchy,the bilinear equation of the DKP and mDKP hierarchies are given.Finally,the existence of the tau functions of the DKP and mDKP hierarchies are proved.In Chapter 6,we investigate the additional symmetries in the second bosonic DKP and mDKP hierarchies.Since the large BKP hierarchy consists of the DKP and mDKP hierarchies,we only need to discuss the additional symmetries of the large BKP hierarchy.Firstly,according to the wave functions and the Lax structure,the OrlovSchulman operator M± is defined and the relations between them are given.Then,the related properties of the additional symmetries of the large BKP hierarchy are investigated.Finally,the Adler-Shiota-van Moerbeke formulas for the large BKP hierarchy and the action of the additional symmetries on the tau functions are given.In Chapter 7,the content of this paper is summarized,and the relevant discussion is given,and the subsequent research directions on DKP and mDKP hierarchies are proposed.