BKP hierarchy and modified BKP hierarchy

The highest weight representation of the infinite Lie algebras is used to construct various soliton equations by the famous Kyoto school. In this chapter, we firstly review the construction of the BKP hierarchy corresponding to o(∞) by using the neutral free Fermions. As an important reduction of the BKP hierarchy, the constrained BKP hierarchy is further investigated in the aspect of the bilinear equations. It is shown that the bilinear equations in the form of tau functions can fully determine the constrained BKP hierarchy. Then the bilinear equations of the constrained BKP hierarchy are given in the Fermionic picture, and further interpretations of these bilinear equations are obtained. Particularly, there is the structure of the modified BKP hierarchy in the constrained BKP hierarchy, which is presented in the forms of the bilinear equations. Last, we investigate the dressing structure and the Lax structure of the modified BKP hierarchy. Note that the soliton equations obtained by the highest weight representation are usually given in the forms of the bilinear equations; therefore, the work here may be one try in further study of the integrable properties for these soliton equations.