The Bilinear Equations,Darboux Transformations And Additional Symmetry Of Modified BKP And Its Related Hierarchies

The integrable system is one of the important research fields in mathematical physics.Sato theory on KP and its related hierarchies has significant place in theory of integrable system.It is known that the BKP hierarchy is an important sub-hierarchy of the KP hierarchy.Just like the modified KP(mKP)and KP hierarchies,the modified BKP(mBKP)hierarchy has close connection with the BKP hierarchy.We mainly study the bilinear equations,Darboux transformation,additional symmetries of the mBKP hierarchy.Then the Lax formulation and bilinear equation of the constrained mBKP hierarchy are discussed.The relations of the two-component mBKP and B-type Toda(B-Toda)hierarchies with mBKP hierarchy are discussed at last.The main content of this thesis is as follow:In Chapter 1,we firstly introduce the theoretical background of this thesis,including Sato theory,BKP hierarchy,mBKP hierarchy,Miura transformations of the BKP and mBKP hierarchies.Then we introduce the bilinear equations,Darboux transformation,additional symmetry and squared eigenfunction symmetry.At last,the research ideas,research significance and research content innovation are showed.In Chapter 2,the basic knowledge is given.Firstly,we simply introduce the BKP hierarchy and its squared eigenfunction potential.Next we introduce the neutral free fermion,fermionic field and their properties.Then we introduce the boson-fermion correspondence.The last is B-type squared eigenfunction potential which is prepared for the calculation about symmetry and Darboux transformation of the mBKP hierarchy.In Chapter 3,we mainly discuss the bilinear equation of the mBKP hierarchy.We firstly study the connection between bilinear equation and Lax structure of the mBKP hierarchy.We discuss the Lax operator and dressing operator from the bilinear equation.Then the bilinear equation in terms of wave functions of the mBKP hierarchy is obtained from the corresponding Lax structure.Based upon above,the existence of tau functions of the mBKP hierarchy is discussed.In Chapter 4,we mainly study the Darboux transformations of the mBKP hierarchy.Similar to the mKP hierarchy,the Darboux transformation of the mBKP hierarchy has its own properties.Because of the complicated Lax constraint of the mBKP hierarchy,it is not easy to construct its Darboux transformation.In this chapter,the Darboux transformations can be computed according to boson-fermion correspondence and Btype squared eigenfunction potential.Then the relations among these Darboux transformations are discussed.Finally,we calculate the bosonic forms of the transformed tau functions of the mBKP hierarchy under the Darboux transformations.In Chapter 5,the additional symmetry and squared eigenfunction symmetry of the mBKP hierarchy are discussed.Due to the special constraint of the mBKP hierarchy,the corresponding symmetry is difficult to construct.The additional symmetry generator can be calculated by introducing additional symmetry flow.Then the action of additional symmetry on the tau functions of the mBKP hierarchy can be obtained.While the Adler-Shiota-van Moerbeke formula can also be calculated.In Chapter 6,we simply discuss a class of related integrable hierarchy with the mBKP hierarchy,that is the constrained mBKP hierarchy.Firstly,the constrained mBKP hierarchy can be defined by putting a constraint on the Lax operator of the mBKP hierarchy.And we prove this definition is reasonable.Then the bilinear equation of the constrained mBKP hierarchy can be obtained.In Chapter 7,we summarize the main contents of this thesis.At first,we briefly discuss the relations of the 2-component mBKP and B-Toda hierarchies with mBKP hierarchy.While we sum up the main research of this thesis,and then discuss possible research directions in the future.The research of the mBKP and its related hierarchies in this thesis is of great significance for the study of integrable system.It is an important application of the Sato theory.