Polynomial tau-functions of the orthogonal KP, multicomponent orthogonal KP and BUC hierarchies

Abstract

This paper is concerned with the construction of the polynomial tau-functions (PTFs) of the orthogonal KP (OKP) hierarchy, multicomponent OKP hierarchy and universal character hierarchy of B-type (BUC hierarchy), which are proved as zero modes of certain combinations of the generating functions. By carrying out the action of the (multicomponent) quantum fields on vacuum vector 1, the generating functions for the orthogonal Schur function, multicomponent orthogonal Schur function and generalized Q-function have been presented. The remarkable feature is that PTFs are the coefficients of certain family of generating functions. Furthermore, in terms of the Vandermonde-like identity and properties of Pfaffian, it is showed that the PTFs of the OKP and BUC hierarchies can be written as determinant and Pfaffian forms, respectively. Meanwhile, the PTFs of the multicomponent OKP hierarchy can be expressed as the product of determinants. In addition, the soliton solutions of the OKP hierarchy have also been discussed.