The Integrabibity Of The BC_r-KP Hierarchy

The BCr-KP hierarchy is a generalization of BKP hierarchy and CKP hierarchy,which is an important sub-hierarchy of the KP hierarchy.We study the gauge transfor-mation,the bilinear identities,the squared eigenfunction symmetries for the BCr-KP hierarchy and its constrained case in this paper.Then,we investigate the compatibility of the additional symmetries and the gauge transformation for the BCr-KP hierarchy.The main contents of this paper is as follows:In Chapter 1,we introduce the background knowledge of the KP hierarchy,BKP hierarchy,the CKP hierarchy,the BCr-KP hierarchy and the constrained BCr-KP hier-archy.Next,we introduce the gauge transformation,the bilinear identities,the squared eigenfunction symmetries and the additional symmetries.Lastly,we introduce the re-search significance and content.In Chapter 2,we firstly introduce the relevant properties of pseudo-differential operators.Then,the KP hierarchy and the definition of the gauge transformation are introduced.Next we introduce the BKP hierarchy and the CKP hierarchy.Lastly,we introduce the definition of the BCr-KP hierarchy and its the constrained case.In Chapter 3,the gauge transformation for the BCr-KP hierarchy is investigat-ed.Different from the KP hierarchy,the gauge transformation not only keep the Lax equation,but also keep the constraint of the BCr-KP hierarchy.In addition to the pre-vious two conditions,we also need to maintain the form of the Lax operator for the constrained BCr-KP hierarchy.In Chapter 4,the bilinear identities for the BCr-KP hierarchy and its constrained case are studied.We not only obtain the bilinear identities,but also give another de-scription of the bilinear identities for the BCr-KP hierarchy.For the bilinear identi-ties of the constrained BCr-KP hierarchy,we need to keep the Lax operator for the constrained BCr-KP hierarchy.By studying the bilinear identities for the BCr-KP hierarchy,the BCr-KP hierarchy can be better understood.In Chapter 5,we investigate the squared eigenfunction symmetries for the BCr-KP hierarchy and its constrained case.For the squared eigenfunction symmetries for the BCr-KP hierarchy,the key point is that the definition of the(?)? needs to keep the BCr-constraint.It is necessary that we need to poof the rationality of the definition of(?)?.Finally,we found that we can define the constrained BCr-KP hierarchy by identifying the time flow with the squared eigenfunction symmetries.In Chapter 6,we explore the compatibility between the additional symmetry and the gauge transformation for the BCr-KP hierarchy.Discussing the changes of the additional symmetries under the gauge transformations is very interesting.In Chapter 7,we summarize the main results of this paper and discuss possible research directions in future.