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

Denghui Li and Zhaowen Yan∗ School of Mathematical Sciences, Inner Mongolia
University,
Hohhot, Inner Mongolia 010021, P. R. China

###### Abstract.

This paper is concerned with the construction of the polynomial tau-functions
of the symplectic KP (SKP), orthogonal KP (OKP) hierarchies and universal
character hierarchy of B-type (BUC hierarchy), which are proved as zero modes
of certain combinations of the generating functions. By applying the strategy
of carrying out the action of the quantum fields on vacuum vector, the
generating functions for symplectic Schur function, orthogonal Schur function
and generalized $Q$-function have been presented. The remarkable feature is
that polynomial tau-functions 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 polynomial tau-functions of
the SKP, OKP and BUC hierarchies can be written as determinant and Pfaffian
forms, respectively. In addition, the soliton solutions of the SKP and OKP
hierarchies have been discussed.

Keywords: polynomial tau-functions; symplectic/orthogonal Schur function;
generalized $Q$-function; generating functions; soliton solutions
Mathematics Subject Classifications (2000): 17B80, 35Q55, 37K10

*Corresponding author. Email<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on symmetric functions
  3. 3 Polynomial tau-functions and $n$-soliton solutions of the SKP hierarchy
    1. 3.1 Quantum fields presentation of symplectic Schur functions and the SKP hierarchy
    2. 3.2 Generating functions and the polynomial tau-functions of the SKP hierarchy
    3. 3.3 $N$-soliton solutions of the SKP hierarchy
  4. 4 Polynomial tau-functions and $n$-soliton solutions of the OKP hierarchy
    1. 4.1 Quantum fields presentation of orthogonal Schur functions and the OKP hierarchy
    2. 4.2 Generating functions and the polynomial tau-functions of the OKP hierarchy
    3. 4.3 $N$-soliton solutions of the OKP hierarchy
  5. 5 Polynomial tau-functions of the BUC hierarchy
    1. 5.1 Quantum fields presentation of the generalized $Q$-functions and the BUC hierarchy
    2. 5.2 Generating functions and polynomial tau-functions of the BUC hierarchy
  6. 6 Conclusions and discussions
  7. 7 Acknowledgements

##  1\. Introduction

Symmetric functions are the characters of the irreducible highest weight
representation of the classical groups [1]-[3], which play a significant role
in mathematical physics especially in the theory of integrable systems
[4]-[6]. Schur functions and Schur $Q$-functions are the basic symmetric
functions which are the solutions of differential equations in Kadomtsev-
Petviashvili (KP) and Kadomtsev-Petviashvili sub-hierarchy of B-type (BKP)
hierarchies, respectively. In the famous work of the Kyoto School, the authors
[7]-[13] investigated the core connection of the infinite dimensional Lie
algebra and their highest weight vectors to the integrable hierarchies
involving KP, BKP, discrete KP (DKP), modified KP (MKP) and $s$-component KP
hierarchies.

Koike [14] introduced a polynomial with a pair of partitions called the
universal character, which is a generalization of Schur function. The
universal character (UC) hierarchy, proposed by Tsuda [15], as an infinite-
dimensional integrable system satisfied by the universal character. It can be
regarded as a extension of the KP hierarchy. Furthermore, in the subsequent
paper [16]-[19], Tsuda presented the relations between the $q$-Painlevé
equations and the lattice $q$-UC hierarchy which are the extended $q$-KP and
$q$-UC hierarchies. Based upon this, the structure and properties of the
Painlevé equations and their higher order analogues have been developed, such
as rational solutions, Lax formalism, bilinear relations for $\tau$-functions
and Weyl group symmetry. Wang et.al [20] discussed the algebra of universal
characters and the phase model of strongly correlated bosons. Recently, by
means of the vertex operator realization of symplectic and orthogonal Schur
functions [21], the authors [22, 23] established generalizations of symplectic
KP (SKP) and orthogonal KP (OKP) hierarchies called the symplectic and
orthogonal universal character hierarchies corresponding to symplectic and
orthogonal universal characters, respectively. Ogawa [24] defined a
generalized $Q$-function expressed by the Pfaffian and constructed an
integrable UC hierarchy of B-type (BUC hierarchy) characterized by the
generalized $Q$-function. Lately, in the paper [25, 26], the author
generalized the theory of BUC hierarchy to a coupled and plethystic cases,
which can derive coupled and plethystic infinite order nonlinear PDEs.

All the polynomial tau-functions of the KP hierarchy can be expressed as a
disjoint union of Schubert cells and the Schur polynomial is the center of the
Schubert cell. You [27, 28] showed polynomial tau-functions of the BKP, DKP
and modified DKP (MDKP) hierarchies all include the $Q$-Schur polynomials
which are the centers of Schubert cells of the infinite-dimensional orthogonal
Grassmann manifold. Kac et al. [29] constructed all the polynomial tau-
functions of the KP and MKP hierarchies from Schur polynomials by some shift
of arguments. Moreover, the polynomial tau-functions of the BKP, DKP, and MDKP
hierarchies have been well discussed in terms of boson-fermion correspondence
[30]. Then by means of the $s$-comonent boson-fermion correspondence, Kac et
al. [31] have studied the polynomial tau-functions of the multi-component KP
hierarchy. Based on the quantum fields, they also develop the twisted quantum
fields presentation of Hall-Littlewood polynomials and derived a novel
deformed boson-fermion correspondence [32]. Rozhkovskaya [33] proved
multiparameter Schur $Q$-functions are tau-functions of the BKP hierarchy.
Besides, in the frame work of quantum fields presentation and generating
functions of the symmetric functions, recent study has shown that the
polynomial tau-functions of the KP, BKP and $s$-component KP hierarchies can
be expressed as the zero-modes of certain combinations of generating functions
[34]. Recently, the polynomial tau-functions of the UC and multi-component UC
hierarchies have also been analyzed [35]. To our best knowledge, there is no
existing references on the study on the polynomial tau-functions of the SKP,
OKP and BUC hierarchies. Based upon the facts, in the view point of the
quantum field presentation of the symmetric functions, we will concentrate on
the construction of the exact solutions of these integrable hierarchies
including polynomial-type and soliton-type solutions. We shall prove the
polynomial tau-functions of SKP, OKP and BUC hierarchies can be regarded as
zero modes of certain combinatorial generating functions.

The present paper is organized as follows. In Section 2, we begin with a
review of the elementary, complete symmetric functions, power sums and Schur
polynomial. Section 3 is devoted to construction of quantum fields of
symplectic Schur functions and the polynomial tau-functions of the SKP
hierarchy. Meanwhile, the $n$-soliton solutions of this integrable system are
derived. In section 4, the generating functions for the orthogonal Schur
functions are investigated by the action of the operators on the vacuum vector
$1$. In terms of quantum fields presentation, we also study the polynomial
tau-functions and $n$-soliton solutions of the SKP hierarchy. The fact that
the polynomial tau-functions of the BUC hierarchy are the coefficients of
certain family generating functions is described in Section 5. The last
Section are conclusions and discussions.

##  2\. Preliminaries on symmetric functions

In this section, we mainly retrospect some basic facts and properties about
symmetric functions.

Let $\Lambda(\mathbf{x})$ be the ring of symmetric functions in variables
$\mathbf{x}=(x_{1},x_{2}\ldots)$. The $r$th elementary symmetric function
$e_{r}$, complete symmetric function $h_{r}$ and power sum $p_{r}$ are defined
by (cf. [3])

$\displaystyle
e_{r}(\mathbf{x})=\sum_{i_{1}<i_{2}<\ldots<i_{r}<\infty}x_{i_{1}}x_{i_{2}}\ldots
x_{i_{r}},\quad for\quad r\geq 1,$ (2.1) $\displaystyle
h_{r}(\mathbf{x})=\sum_{i_{1}\leq i_{2}\leq\ldots\leq
i_{r}\leq\infty}x_{i_{1}}x_{i_{2}}\ldots x_{i_{r}},\quad for\quad r\geq 1,$
$\displaystyle p_{r}(\mathbf{x})=\sum_{i}x_{i}^{r}.$

It is universally known that
$h_{r}(\mathbf{x})=e_{r}(\mathbf{x})=p_{r}(\mathbf{x})=0$ for $r<0$ and
$h_{0}=e_{0}=p_{0}=1$. The generating functions for these symmetric functions
are

$\displaystyle E(u)$ $\displaystyle=$ $\displaystyle\sum_{k\geq
0}e_{k}(\mathbf{x})u^{k}=\prod_{i\geq
1}\left(1+x_{i}u\right)=\exp\left(-\sum_{n\geq
1}\frac{(-1)^{n}p_{n}}{n}u^{n}\right),$ $\displaystyle H(u)$ $\displaystyle=$
$\displaystyle\sum_{k\geq 0}h_{k}(\mathbf{x})u^{k}=\prod_{i\geq
1}\frac{1}{1-x_{i}u}=\exp\left(\sum_{n\geq 1}\frac{p_{n}}{n}u^{n}\right),$
$\displaystyle P(u)$ $\displaystyle=$ $\displaystyle\sum_{k\geq
1}p_{k}(\mathbf{x})u^{k-1}=\frac{H^{\prime}(u)}{H(u)}.$ (2.2)

It is easy to obtain that $H(u)E(-u)=1$.

The polynomials $S_{k}(t_{1},t_{2},\ldots)$, $k\in\mathbb{Z}$, is determined
by the generating function

$\
\sum_{k\in\mathbb{Z}}S_{k}(t_{1},t_{2},\ldots)u^{k}=\exp\left(\sum_{j=1}^{\infty}t_{j}u^{j}\right).$
(2.3)

The Schur polynomial $S_{\lambda}(t_{1},t_{2},\ldots)$ can be expressed as
(cf. [3])

$\
S_{\lambda}(t_{1},t_{2},\ldots)=\det[S_{\lambda_{i}-i+j}(t_{1},t_{2},\ldots)]_{1\leq
i,j\leq l},$ (2.4)

where $\lambda=(\lambda_{1},\ldots,\lambda_{l})$ is a partition.

For each symmetric function $f\in\Lambda$, let
$f^{\bot}:\Lambda\rightarrow\Lambda$ be the adjoint of multiplication by $f$

$\ \langle f^{\bot}g,\omega\rangle=\langle g,f\omega\rangle,\quad
g,f,\omega\in\Lambda.$ (2.5)

Let us consider generating functions of the adjoint operators

$\displaystyle E^{\bot}(u)=\sum_{k\geq 0}\frac{e^{\bot}_{k}}{u^{k}},\quad
H^{\bot}(u)=\sum_{k\geq 0}\frac{h^{\bot}_{k}}{u^{k}}.$ (2.6)

It is straightforward to show that

$\displaystyle E^{\bot}(u)=\exp\left(-\sum_{k\geq
1}(-1)^{k}\frac{\partial}{\partial{p_{k}}}\frac{1}{u^{k}}\right),\quad
H^{\bot}(u)=\exp\left(\sum_{k\geq
1}\frac{\partial}{\partial{p_{k}}}\frac{1}{u^{k}}\right).$ (2.7)

######  Proposition 2.1.

The generating functions satisfy the following relations (cf. [3])

$\displaystyle\left(1-\frac{v}{u}\right)E^{\bot}(u)E(v)=E(v)E^{\bot}(u),$
$\displaystyle\left(1-\frac{v}{u}\right)H^{\bot}(u)H(v)=H(v)H^{\bot}(u),$
$\displaystyle H^{\bot}(u)E(v)=\left(1+\frac{v}{u}\right)E(v)H^{\bot}(u),$
$\displaystyle E^{\bot}(u)H(v)=\left(1+\frac{v}{u}\right)H(v)E^{\bot}(u).$
(2.8)

##  3\. Polynomial tau-functions and $n$-soliton solutions of the SKP
hierarchy

By means of charged free fermions, we devote to discussing structures and
properties of polynomial tau-functions for the SKP hierarchy. Furthermore, the
generating functions for polynomial tau-functions of the SKP hierarchy can be
obtained by acting the quantum fields of symplectic Schur functions on the
bosonic Fock space $\mathcal{B}^{m}$. There is an interesting conclusion that
the polynomial tau-functions of the SKP hierarchy are the coefficients of
certain family of generating functions. Finally, the soliton-type solutions of
the SKP hierarchy have been derived.

###  3.1. Quantum fields presentation of symplectic Schur functions and the
SKP hierarchy

The symmetric polynomial ring $\Lambda$ :
$\Lambda=\mathbb{C}[e_{1},e_{2},\ldots]=\mathbb{C}[h_{1},h_{2},\ldots]=\mathbb{C}[p_{1},p_{2},\ldots]$
can be generated by elementary, complete symmetric functions and power sums,
respectively. Introduce the bosonic Fock space
$\mathcal{B}=\mathbb{C}[z,z^{-1}]\bigotimes\Lambda$, it is decomposed to
obtain the charged graded space

$\displaystyle\mathcal{B}=\bigoplus\limits_{m\in\mathbb{Z}}\mathcal{B}^{m},\ \
\text{where}\ \
\mathcal{B}^{m}=z^{m}\cdot\mathbb{C}[p_{1},p_{2},\ldots]=z^{m}\Lambda.$ (3.1)

Let $R(u)$ act on the elements of the form $z^{m}f,\ f\in\Lambda,\
m\in\mathbb{Z}$, $R(u):\mathcal{B}\rightarrow\mathcal{B}$ is defined as (cf.
[36])

$\displaystyle R(u)\left(z^{m}f\mathbf{(x)}\right)=z^{m+1}u^{m+1}f.$ (3.2)

Then it leads to

$\displaystyle R^{-1}(u)\left(z^{m}f\mathbf{(x)}\right)=z^{m-1}u^{-m}f.$ (3.3)

Operators $R^{\pm 1}(u)$ map the grading of the boson Fock space
$\mathcal{B}^{(m)}$ into $\mathcal{B}^{(m\pm 1)}$.

Define the quantum fields $\psi^{Sp,\pm}(u)$ [37]

$\displaystyle\psi^{Sp,+}(u)=u^{-1}R(u)H(u)E^{\bot}(-u)E^{\bot}(-\frac{1}{u})=\sum\limits_{k\in\mathbb{Z}+\frac{1}{2}}\psi^{Sp,+}_{k}u^{-k-\frac{1}{2}},$
(3.4)
$\displaystyle\psi^{Sp,-}(u)=(1-u^{2})R^{-1}(u)E(-u)H^{\bot}(u)H^{\bot}(\frac{1}{u})=\sum\limits_{k\in\mathbb{Z}+\frac{1}{2}}\psi^{Sp,-}_{k}u^{-k-\frac{1}{2}}.$

######  Proposition 3.1.

Quantum fields $\psi^{Sp,+}(u),\psi^{Sp,-}(u)$ satisfy the anticommutation
relations

$\displaystyle\psi^{Sp,\pm}(u)\psi^{Sp,\pm}(v)+\psi^{Sp,\pm}(v)\psi^{Sp,\pm}(u)=0,$
(3.5)
$\displaystyle\psi^{Sp,+}(u)\psi^{Sp,-}(v)+\psi^{Sp,-}(v)\psi^{Sp,+}(u)=\delta(u,v),$

where $\delta(u,v)=\sum\limits_{{k,m\in\mathbb{Z}}\atop{k+m=-1}}u^{k}v^{m}$ is
the delta-distribution.

From Eq.(3.4), Eq.(3.5) is equivalent to the relations with charged free
fermions

$\displaystyle\psi^{Sp,\pm}_{k}\psi^{Sp,\pm}_{l}+\psi^{Sp,\pm}_{l}\psi^{Sp,\pm}_{k}=0,$
(3.6)
$\displaystyle\psi^{Sp,+}_{k}\psi^{Sp,-}_{l}+\psi^{Sp,-}_{l}\psi^{Sp,+}_{k}=\delta_{k,-l}.$

######  Remark 3.2.

From Eqs. (2) and (2.7), we easily get the bosonic form of the quantum fields
$\psi^{Sp,\pm}(u)$:

$\displaystyle\psi^{Sp,+}(u)=u^{-1}R(u)\exp\left(\sum\limits_{n\geq
1}\frac{p_{n}}{n}u^{n}\right)\exp\left(-\sum\limits_{n\geq
1}\frac{\partial}{\partial
p_{n}}\frac{1}{u^{n}}\right)\exp\left(-\sum\limits_{n\geq
1}\frac{\partial}{\partial p_{n}}u^{n}\right),$ (3.7)
$\displaystyle\psi^{Sp,-}(u)=(1-u^{2})R^{-1}(u)\exp\left(-\sum\limits_{n\geq
1}\frac{p_{n}}{n}u^{n}\right)\exp\left(\sum\limits_{n\geq
1}\frac{\partial}{\partial
p_{n}}\frac{1}{u^{n}}\right)\exp\left(\sum\limits_{n\geq
1}\frac{\partial}{\partial p_{n}}u^{n}\right).$

Hence one has $\psi^{Sp,+}_{i}(z^{m})=0$ if $i>-m-\frac{1}{2}$ and
$\psi^{Sp,-}_{i}(z^{m})=0$ if $i>m-\frac{1}{2}$.

######  Definition 3.3.

For an unknown function $\tau=\tau(x)$, the bilinear equation

$\displaystyle\widehat{\Omega}(\tau\otimes\tau)=0,$ (3.8)

is called the SKP hierarchy, where

$\displaystyle\widehat{\Omega}=\sum\limits_{l\in\mathbb{Z}+\frac{1}{2}}\psi_{l}^{Sp,+}\otimes\psi_{-l}^{Sp,-}.$
(3.9)

######  Lemma 3.4.

Let $\widehat{X}=\sum\limits_{i>N}C_{i}\psi_{i}^{Sp,+}$, where
$C_{i}\in\mathbb{C},N\in\mathbb{Z}$. Then $\widehat{X}^{2}=0$.

###### Proof.

Due to $\psi^{Sp,+}_{k}\psi^{Sp,+}_{l}+\psi^{Sp,+}_{l}\psi^{Sp,+}_{k}=0$,
$i,k\in\mathbb{Z}+\frac{1}{2}$, one immediately has

$\displaystyle\
\widehat{X}^{2}=\widehat{X}\cdot\widehat{X}=\sum\limits_{l>N}C_{l}\psi_{l}^{Sp,+}\cdot\sum\limits_{k>N}C_{k}\psi_{k}^{Sp,+}=\sum\limits_{l>N}\sum\limits_{k>N}C_{l}C_{k}\psi_{l}^{Sp,+}\psi_{k}^{Sp,+}=0.$
(3.10)

∎

######  Lemma 3.5.

Let $\widehat{X}=\sum\limits_{i>N}C_{i}\psi_{i}^{Sp,+}$, where
$C_{i}\in\mathbb{C},N\in\mathbb{Z}$. Then
$\widehat{\Omega}(\widehat{X}\otimes\widehat{X})=(\widehat{X}\otimes\widehat{X})\widehat{\Omega}$.

###### Proof.

Based on $\psi_{-l}^{Sp,-}\widehat{X}=-\widehat{X}\psi_{-l}^{Sp,-}+C_{l}$, we
have

$\displaystyle\ \widehat{\Omega}(\widehat{X}\otimes\widehat{X})$
$\displaystyle=$
$\displaystyle\sum\limits_{l\in\mathbb{Z}+\frac{1}{2}}\psi_{l}^{Sp,+}\widehat{X}\otimes\psi_{-l}^{Sp,-}\widehat{X}=\sum\limits_{l\in\mathbb{Z}+\frac{1}{2}}(-\widehat{X}\psi_{l}^{Sp,+})\otimes(-\widehat{X}\psi_{-l}^{Sp,-}+C_{l})$
(3.11) $\displaystyle=$
$\displaystyle(\widehat{X}\otimes\widehat{X})\widehat{\Omega}-\widehat{X}\sum\limits_{l\in\mathbb{Z}+\frac{1}{2}}C_{l}\psi_{l}^{Sp,+}\otimes
1=(\widehat{X}\otimes\widehat{X})\widehat{\Omega}-\widehat{X}^{2}\otimes
1=(\widehat{X}\otimes\widehat{X})\widehat{\Omega}.$

∎

######  Corollary 3.6.

Let $\tau\in\mathcal{B}^{m}$ be a tau-function of the SKP hierarchy, and let
$\widehat{X}=\sum\limits_{i>N}C_{i}\psi_{i}^{Sp,+}$, where
$C_{i}\in\mathbb{C},N\in\mathbb{Z}$. Then
$\widehat{\tau}=\widehat{X}\tau\in\mathcal{B}^{m+1}$ is also a tau-functions
of the SKP hierarchy.

###### Proof.

Multiplying $\widehat{X}\otimes\widehat{X}$ left on both sides of
$\widehat{\Omega}(\tau\otimes\tau)=0$, we get
$(\widehat{X}\otimes\widehat{X})\widehat{\Omega}(\tau\otimes\tau)=0$.
According to
$\widehat{\Omega}(\widehat{X}\otimes\widehat{X})=(\widehat{X}\otimes\widehat{X})\widehat{\Omega}$,
it follows that
$(\widehat{X}\otimes\widehat{X})\widehat{\Omega}(\tau\otimes\tau)=\widehat{\Omega}(\widehat{X}\otimes\widehat{X})(\tau\otimes\tau)=\widehat{\Omega}(\widehat{X}\tau\otimes\widehat{X}\tau)=0$.
Therefore, $\widehat{X}\tau$ is the solution of the SKP hierarchy. ∎

It is known that symplectic Schur functions are tau-functions of the SKP
hierarchy [22]. If the non-zero solution of (3.8) is a polynomial function of
variables $(p_{1},p_{2},\ldots)$, we call the non-zero solution a polynomial
tau-function. It follows from Remark 3.2 that $z^{m}$ is a solution of the SKP
hierarchy. We now turn our attention to the polynomial tau-function of the SKP
hierarchy.

###  3.2. Generating functions and the polynomial tau-functions of the SKP
hierarchy

Let $\widehat{G}(u_{1},\ldots,u_{l})$ be a generating function of the
symplectic Schur function in $\mathbf{u}=(u_{1},\ldots,u_{l})$ defined by

$\displaystyle\ \widehat{G}(u_{1},\ldots,u_{l})=\prod\limits_{1\leq i<j\leq
l}\left(u_{i}-u_{j}\right)\left(1-u_{i}u_{j}\right)\prod\limits_{i=1}^{l}H(u_{i}).$
(3.12)

From Proposition 2.1, we have

$\displaystyle\psi^{Sp,+}(u_{1})\psi^{Sp,+}(u_{2})\cdots\psi^{Sp,+}(u_{l})(z^{k}f)$
(3.13) $\displaystyle=$ $\displaystyle z^{k+l}u_{1}^{l+k-1}\cdots
u_{l-1}^{k+1}u_{l}^{k}\prod\limits_{1\leq i<j\leq
l}\left(1-u_{i}u_{j}\right)\left(1-\frac{u_{j}}{u_{i}}\right)\prod\limits_{i=1}^{l}H(u_{i})E^{\bot}(-u_{i})E^{\bot}(-\frac{1}{u_{i}})(f)$
$\displaystyle=$ $\displaystyle z^{k+l}u_{1}^{k}\cdots
u_{l}^{k}\widehat{G}(u_{1},\ldots,u_{l}).$

Let $\widehat{A}_{1}(u),\ldots,\widehat{A}_{l}(u)$ be the set of formal
Laurent series, define the formal Laurent series
$\widehat{T}_{i}(u)=\widehat{A}_{i}(u)H(u)=\sum\limits_{p\in\mathbb{Z}}\widehat{T}_{i,p}u^{p},\
i=1,\ldots,l$. Besides, let $\widehat{T}(u_{1},\ldots,u_{l})$ be a formal
Laurent series in $(u_{1},\ldots,u_{l})$ defined by

$\displaystyle\ \widehat{T}(u_{1},\ldots,u_{l})=\prod\limits_{1\leq i<j\leq
l}\left(u_{i}-u_{j}\right)\left(1-u_{i}u_{j}\right)\prod\limits_{i=1}^{l}\widehat{A}_{i}(u_{i})H(u_{i}).$
(3.14)

For any vector $\boldsymbol{\xi}=(\xi_{1},\ldots,\xi_{l})\in\mathbb{Z}^{l}$,
$\widehat{T}_{\boldsymbol{\xi}}$ is the coefficient of the following expansion

$\displaystyle\
\widehat{T}(u_{1},\ldots,u_{l})=\sum\limits_{\boldsymbol{\xi}\in\mathbb{Z}^{l}}\widehat{T}_{\boldsymbol{\xi}}u_{1}^{\xi_{1}}\cdots
u_{l}^{\xi_{l}}.$ (3.15)

######  Theorem 3.7.

  * 1)

Formal Laurent series $\widehat{T}(u_{1},\ldots,u_{l})$ can be expressed as

$\displaystyle\widehat{T}(u_{1},\ldots,u_{l})=\frac{1}{2}\det\left[\left(u_{i}^{l-j}+u_{i}^{l+j-2}\right)\widehat{T}_{i}(u_{i})\right]_{1\leq
i,j\leq l}.$ (3.16)

  * 2)

The coefficient $\widehat{T}_{\boldsymbol{\xi}}$ of $u_{1}^{\xi_{1}}\cdots
u_{l}^{\xi_{l}}$ in (3.15) can be written as follows

$\displaystyle\widehat{T}_{\boldsymbol{\xi}}=\frac{1}{2}\det\left[\widehat{T}_{i,\xi_{i}-j}+\widehat{T}_{i,\xi_{i}+j-2}\right]_{1\leq
i,j\leq l},$ (3.17)

where $\boldsymbol{\xi}=(\xi_{1},\ldots,\xi_{l})$.

  * 3)

$\widehat{T}_{\boldsymbol{\xi}}$ is a polynomial tau-function of the SKP
hierarchy.

###### Proof.

  * 1)

According to Vandermonde-like identity [38]

$\displaystyle\det\left[u_{i}^{k-j}+u_{i}^{k+j-2}\right]$ $\displaystyle=$
$\displaystyle 2\prod\limits_{1\leq i<j\leq
k}\left(u_{i}-u_{j}\right)\left(1-u_{i}u_{j}\right)$ (3.18) $\displaystyle=$
$\displaystyle\sum\limits_{{\sigma\in S_{k}}\atop{\varepsilon_{i}=\pm
1}}sgn(\sigma)(u_{1}\cdots
u_{k})^{k-1}u_{1}^{\varepsilon_{1}(\sigma(1)-1)}\cdots
u_{k}^{\varepsilon_{k}(\sigma(k)-1)},$

it is easy to verify that

$\displaystyle\widehat{T}(u_{1},\ldots,u_{l})$ $\displaystyle=$
$\displaystyle\prod\limits_{1\leq i<j\leq
l}\left(u_{i}-u_{j}\right)\left(1-u_{i}u_{j}\right)\prod\limits_{i=1}^{l}\widehat{A}_{i}(u_{i})H(u_{i})$
(3.19) $\displaystyle=$
$\displaystyle\frac{1}{2}\det\left[u_{i}^{l-j}+u_{i}^{l+j-2}\right]\prod\limits_{i=1}^{l}\widehat{T}_{i}(u_{i})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\det\left[\left(u_{i}^{l-j}+u_{i}^{l+j-2}\right)\widehat{T}_{i}(u_{i})\right]_{1\leq
i,j\leq l}.$

  * 2)

Observe that

$\displaystyle\widehat{T}(u_{1},\ldots,u_{l})$
$\displaystyle=\frac{1}{2}\det\left[\sum\limits_{p_{i}\in\mathbb{Z}}\widehat{T}_{i,p_{i}}\left(u_{i}^{l+p_{i}-j}+u_{i}^{l+p_{i}+j-2}\right)\right]$
$\displaystyle=\frac{1}{2}\sum\limits_{p_{i}\in\mathbb{Z}}\sum\limits_{{\sigma\in
S_{l}}\atop{\varepsilon_{i}=\pm 1}}sgn(\sigma)u_{1}^{l+p_{1}-1}\cdots
u_{l}^{l+p_{l}-1}\widehat{T}_{1,p_{1}}u_{1}^{\varepsilon_{1}(\sigma(1)-1)}\cdots\widehat{T}_{l,p_{l}}u_{l}^{\varepsilon_{l}(\sigma(l)-1)}$
$\displaystyle=\sum\limits_{\xi_{i}\in\mathbb{Z}}\frac{1}{2}\sum\limits_{{\sigma\in
S_{l}}\atop{\varepsilon_{i}=\pm
1}}sgn(\sigma)\widehat{T}_{1,\xi_{1}-l+1-\varepsilon_{1}(\sigma(1)-1)}\cdots\widehat{T}_{l,\xi_{l}-l+1-\varepsilon_{l}(\sigma(l)-1)}u_{1}^{\xi_{1}}\cdots
u_{l}^{\xi_{l}}$
$\displaystyle=\sum\limits_{\xi_{i}\in\mathbb{Z}}\frac{1}{2}\det\left[\widehat{T}_{i,\xi_{i}-j}+\widehat{T}_{i,\xi_{i}+j-2}\right]_{1\leq
i,j\leq l}u_{1}^{\xi_{1}}\cdots u_{l}^{\xi_{l}},$ (3.20)

therefore, the coefficient $\widehat{T}_{\boldsymbol{\xi}}$ of
$u_{1}^{\xi_{1}}\cdots u_{l}^{\xi_{l}}$ is
$\frac{1}{2}\det\left[\widehat{T}_{i,\xi_{i}-j}+\widehat{T}_{i,\xi_{i}+j-2}\right]_{1\leq
i,j\leq l}$.

  * 3)

It is apparent from (3.12) that

$\displaystyle\widehat{A}_{1}(u_{1})\cdots\widehat{A}_{l}(u_{l})\psi^{Sp,+}(u_{1})\cdots\psi^{Sp,+}(u_{l})(z^{k}\cdot
1)=z^{l+k}u_{1}^{k}\cdots u_{l}^{k}\widehat{T}(u_{1},\ldots,u_{l}).$ (3.21)

Let $\widehat{A}_{j}(u)=\sum\limits_{M_{j}\leq r\leq
N_{j}}\widehat{A}_{j,r-\frac{1}{2}}u^{r}(A_{j,r-\frac{1}{2}}\in\mathbb{C},M_{j},N_{j},r\in\mathbb{Z},j=1,\ldots,l)$
be a power series expansion of the variable $u$. Therefore,
$\widehat{T}_{\boldsymbol{\xi}}$ can be written as

$\displaystyle\widehat{T}_{\boldsymbol{\xi}}=z^{-l-k}\widehat{X}_{1}\cdots\widehat{X}_{l}(z^{k}\cdot
1),$ (3.22)

where

$\displaystyle\widehat{X}_{j}=\sum\limits_{M_{j}-\xi_{j}-k-\frac{1}{2}\leq
i_{j}\leq
N_{j}-\xi_{j}-k-\frac{1}{2}}\widehat{A}_{j,\xi_{j}+k+i_{j}}\psi^{Sp,+}_{i_{j}},\quad
j=1,\cdots,l.$ (3.23)

Particularly, by Remark 3.2 and Corollary 3.6, the coefficient
$\widehat{T}_{\boldsymbol{\xi}}$ is a tau-function of the SKP hierarchy with
$k=0$. Since $\widehat{T}_{\boldsymbol{\xi}}$ is a finite linear combination
of $\psi_{i_{1}}^{Sp,+}\cdots\psi_{i_{l}}^{Sp,+}(1)$, it is a polynomial tau-
function.

∎

By replacing $\widehat{A}_{j}(u)$ with $u^{\xi_{j}}\widehat{A}_{j}(u)$, we
have

$\displaystyle\widehat{A}_{i}(u)=u^{N_{i}}\widehat{h}_{i}\sum\limits_{k=0}^{\infty}a_{i,k}u^{k},\quad\quad
N_{i}\in\mathbb{Z},\widehat{h}_{i},a_{i,k}\in\mathbb{C},a_{i,0}=1,\widehat{h}_{i}\neq
0,i=1,\ldots,l,$ (3.24)

where $\widehat{A}_{j}(u)$ are non-zero Laurent series defined in the
$\widehat{T}(u_{1},\ldots,u_{l})$.

According to (2.3), $\sum\limits_{k=0}^{\infty}a_{i,k}u^{k}$ can be written as

$\displaystyle\sum\limits_{k=0}^{\infty}a_{i,k}u^{k}=\exp\left(\sum\limits_{l=1}^{\infty}\widehat{c}_{i,l}u^{l}\right),\quad\text{and}\quad
a_{i,k}=S_{k}(\widehat{c}_{i,1},\widehat{c}_{i,2},\ldots),$ (3.25)

where $\\{\widehat{c}_{i,l}\\}$ is a set of constants in $\mathbb{C}$.

From (2) and setting $t_{l}=\frac{p_{l}}{l}$, we obtain

$\displaystyle\widehat{T}_{i}(u)$ $\displaystyle=$
$\displaystyle\widehat{A}_{i}(u)H(u)=u^{N_{i}}\widehat{h}_{i}\exp\left(\sum\limits_{l=1}^{\infty}\widehat{c}_{i,l}u^{l}\right)\exp\left(\sum\limits_{l=1}^{\infty}\frac{p_{l}}{l}u^{l}\right)$
(3.26) $\displaystyle=$ $\displaystyle
u^{N_{i}}\widehat{h}_{i}\exp\left(\sum\limits_{l=1}^{\infty}(\widehat{c}_{i,l}+t_{l})u^{l}\right)=u^{N_{i}}\widehat{h}_{i}\sum\limits_{l=0}^{\infty}S_{l}(t_{1}+\widehat{c}_{i,1},t_{2}+\widehat{c}_{i,2},\ldots)u^{l}.$

Hence
$\widehat{T}_{i,p}=\widehat{h}_{i}S_{p-N_{i}}(t_{1}+\widehat{c}_{i,1},t_{2}+\widehat{c}_{i,2},\ldots),\
i=1,\ldots,l$. From Theorem 3.7, polynomial tau-functions of the SKP hierarchy
are given by

$\displaystyle\widehat{T}_{\boldsymbol{\xi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\det\left[\widehat{T}_{i,\xi_{i}-j}+\widehat{T}_{i,\xi_{i}+j-2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\det\left[\widehat{h}_{i}S_{\xi_{i}-j-N_{i}}(t_{1}+\widehat{c}_{i,1},t_{2}+\widehat{c}_{i,2},\ldots)+\widehat{h}_{i}S_{\xi_{i}+j-2-N_{i}}(t_{1}+\widehat{c}_{i,1},t_{2}+\widehat{c}_{i,2},\ldots)\right]$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{l}\widehat{h}_{i}\frac{1}{2}\det\left[S_{\xi_{i}-j-N_{i}}(t_{1}+\widehat{c}_{i,1},t_{2}+\widehat{c}_{i,2},\ldots)+S_{\xi_{i}+j-2-N_{i}}(t_{1}+\widehat{c}_{i,1},t_{2}+\widehat{c}_{i,2},\ldots)\right]_{i,j=1,\ldots,l}.$

When $\widehat{c}_{i,l}=0$, $\widehat{h}_{i}=1$ and $N_{i}+2=i$,
$\widehat{T}_{\boldsymbol{\xi}}$ reduces to the symplectic Schur functions
[22]. The polynomial tau-functions (3.2) of the SKP hierarchy are the
generalization of the solution of the SKP hierarchy in [22], which are the
zero mode of an appropriate combinatorial generating functions.

###  3.3. $N$-soliton solutions of the SKP hierarchy

Now let us consider another extremely important exact solution of SKP
hierarchy called the soliton solution.

Let

$\displaystyle\Gamma^{Sp}(p,q)=p^{-1}(1-q^{2})R(p)R^{-1}(q)H(p)E(-q)E^{\bot}(-p)H^{\bot}(q)E^{\bot}(-\frac{1}{p})H^{\bot}(\frac{1}{q}).$
(3.28)

From Proposition 2.1, it is easy to check that

$\displaystyle\Gamma^{Sp}(p_{i},q_{i})\Gamma^{Sp}(p_{j},q_{j})=A_{ij}:\Gamma^{Sp}(p_{i},q_{i})\Gamma^{Sp}(p_{j},q_{j}):,$
(3.29)

where

$\displaystyle
A_{ij}=\frac{(1-p_{i}p_{j})(1-q_{i}q_{j})(p_{i}-p_{j})(q_{i}-q_{j})}{(1-q_{i}p_{j})(1-p_{i}q_{j})(q_{i}-p_{j})(p_{i}-q_{j})}.$
(3.30)

In particular $\Gamma^{Sp}(p,q)^{2}=0$; therefore
$e^{pc\Gamma^{Sp}(p,q)}=1+pc\Gamma^{Sp}(p,q)$.

######  Lemma 3.8.

If $\tau$ is a solution of the SKP hierarchy, then $\Gamma^{Sp}(u,v)\tau$ is
also a solution.

###### Proof.

Using Eq. (3.6), we obtain

$\displaystyle\widehat{\Omega}\left(\psi^{Sp,+}(u)\psi^{Sp,-}(v)\otimes\psi^{Sp,+}(u)\psi^{Sp,-}(v)\right)$
(3.31) $\displaystyle=$
$\displaystyle(\sum\limits_{l\in\mathbb{Z}+\frac{1}{2}}\psi_{l}^{Sp,+}\otimes\psi_{-l}^{Sp,-})(\sum\limits_{m,n\in\mathbb{Z}+\frac{1}{2}}\psi_{m}^{Sp,+}u^{-m-\frac{1}{2}}\psi_{n}^{Sp,-}v^{-n-\frac{1}{2}}\otimes\sum\limits_{m,n\in\mathbb{Z}+\frac{1}{2}}\psi_{m}^{Sp,+}u^{-m-\frac{1}{2}}\psi_{n}^{Sp,-}v^{-n-\frac{1}{2}})$
$\displaystyle=$
$\displaystyle\sum\limits_{l\in\mathbb{Z}+\frac{1}{2}}(\sum\limits_{m,n\in\mathbb{Z}+\frac{1}{2}}\psi_{l}^{Sp,+}\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}\otimes\sum\limits_{m,n\in\mathbb{Z}+\frac{1}{2}}\psi_{-l}^{Sp,-}\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}})$
$\displaystyle=$
$\displaystyle\sum\limits_{l,m,n\in\mathbb{Z}+\frac{1}{2}}-\psi_{m}^{Sp,+}\delta_{l,-n}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}\otimes\delta_{m,l}\psi_{n}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}+\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}\psi_{l}^{Sp,+}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}$
$\displaystyle\otimes\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}\psi_{-l}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}-\psi_{m}^{Sp,+}\delta_{l,-n}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}\otimes\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}\psi_{-l}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}$
$\displaystyle+\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}\psi_{l}^{Sp,+}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}\otimes\delta_{m,l}\psi_{n}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\sum\limits_{l,m,n\in\mathbb{Z}+\frac{1}{2}}\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}\psi_{l}^{Sp,+}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}\otimes\psi_{m}^{Sp,+}\psi_{n}^{Sp,-}\psi_{-l}^{Sp,-}u^{-m-\frac{1}{2}}v^{-n-\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left(\psi^{Sp,+}(u)\psi^{Sp,-}(v)\otimes\psi^{Sp,+}(u)\psi^{Sp,-}(v)\right)\widehat{\Omega}.$

It can easily be checked that

$\displaystyle\left(\psi^{Sp,+}(u)\psi^{Sp,-}(v)\otimes\psi^{Sp,+}(u)\psi^{Sp,-}(v)\right)\widehat{\Omega}(\tau\otimes\tau)$
(3.32) $\displaystyle=$
$\displaystyle\widehat{\Omega}\left(\psi^{Sp,+}(u)\psi^{Sp,-}(v)\otimes\psi^{Sp,+}(u)\psi^{Sp,-}(v)\right)(\tau\otimes\tau)$
$\displaystyle=$
$\displaystyle\widehat{\Omega}\left(\psi^{Sp,+}(u)\psi^{Sp,-}(v)\tau\otimes\psi^{Sp,+}(u)\psi^{Sp,-}(v)\tau\right)=0.$

Clearly, $\psi^{Sp,+}(u)\psi^{Sp,-}(v)\tau$ is the solution of the SKP
hierarchy. A routine computation gives rise to
$\psi^{Sp,+}(u)\psi^{Sp,-}(v)=\frac{1}{1-uv}\frac{u}{u-v}\Gamma^{Sp}(u,v)$.
Therefore, $\Gamma^{Sp}(u,v)\tau$ is also a solution. ∎

######  Lemma 3.9.

It holds that

$\displaystyle[\widehat{\Omega},1\otimes\Gamma^{Sp}(p,q)+\Gamma^{Sp}(p,q)\otimes
1]=0,$ (3.33)

where $[A,B]=_{def}AB-BA$.

###### Proof.

Lemma can be calculated directly from the
$\Gamma^{Sp}(p,q)=(1-pq)\frac{p-q}{p}\psi^{Sp,+}(p)\psi^{Sp,-}(q)$. The
specific calculation process is not listed here. ∎

Let us consider the function

$\displaystyle\
\tau(x,y)=\tau(x,y;p,q,c)=\prod\limits_{i=1}^{n}e^{p_{i}c_{i}\Gamma^{Sp}(p_{i},q_{i})}\cdot
1,\quad p_{i},q_{i},c_{i}\in\mathbb{C},p_{i}\neq
q_{j},p_{i}\neq\frac{1}{q_{j}}for\ i\neq j,$ (3.34)

and set

$\displaystyle\ \eta_{i}=\sum\limits_{k\geq
1}(p_{i}^{k}-q_{i}^{k})\frac{p_{k}(x)}{k}.$ (3.35)

By (3.29), Eq.(3.34) can be rewritten as

$\displaystyle\tau(x,y;p,q,c)=\sum_{J\subset I}\left(\prod_{i\in
J}c_{i}(1-q_{i}^{2})\right)\left(\prod\limits_{{i,j\in
J}\atop{i<j}}A_{ij}\right)\exp\left(\sum\limits_{i\in J}\eta_{i}\right),$
(3.36)

where $I=\\{1,2,\ldots,n\\}$ .

######  Proposition 3.10.

The function $\tau(x,y;p,q,c)$ in (3.36) is a solution of the SKP hierarchy,
which we call the $n$-soliton solutions.

###### Proof.

Suppose that $\tau$ is a solution of the SKP hierarchy. We put
$\widehat{\tau}=\left(1+pc\Gamma^{Sp}(p,q)\right)\tau$. It follows from Lemma
3.8 and 3.9 that

$\displaystyle\widehat{\Omega}(\widehat{\tau}\otimes\widehat{\tau})$
$\displaystyle=$
$\displaystyle\widehat{\Omega}(\tau\otimes\tau)+pc\widehat{\Omega}(\tau\otimes\Gamma^{Sp}(p,q)\tau+\Gamma^{Sp}(p,q)\tau\otimes\tau)+p^{2}c^{2}\widehat{\Omega}(\Gamma^{Sp}(p,q)\tau\otimes\Gamma^{Sp}(p,q)\tau)$
(3.37) $\displaystyle=$ $\displaystyle
pc\widehat{\Omega}[(1\otimes\Gamma^{Sp}(p,q))(\tau\otimes\tau)+(\Gamma^{Sp}(p,q)\otimes
1)(\tau\otimes\tau)]$ $\displaystyle=$ $\displaystyle
pc\widehat{\Omega}(1\otimes\Gamma^{Sp}(p,q)+\Gamma^{Sp}(p,q)\otimes
1)(\tau\otimes\tau)$ $\displaystyle=$ $\displaystyle
pc(1\otimes\Gamma^{Sp}(p,q)+\Gamma^{Sp}(p,q)\otimes
1)\widehat{\Omega}(\tau\otimes\tau)$ $\displaystyle=$ $\displaystyle 0.$

Hence $\widehat{\tau}$ is a solution of the SKP hierarchy. Note that $\tau=1$
solves the SKP hierarchy, it is easy to see that the $n$-soliton solutions
defined in (3.34) is really a solution of the SKP hierarchy.

∎

##  4\. Polynomial tau-functions and $n$-soliton solutions of the OKP
hierarchy

In this section, we firstly construct quantum fields of orthogonal Schur
functions and deduce the relationship between these operators. Meanwhile, the
generating functions of the orthogonal Schur functions have been investigated.
Moreover, by applying the quantum field presentation of the OKP hierarchy, the
polynomial tau-functions and the soliton solutions have been presented.

###  4.1. Quantum fields presentation of orthogonal Schur functions and the
OKP hierarchy

Introduce the quantum fields defined by

$\displaystyle\psi^{O,+}(u)=u^{-1}(1-u^{2})R(u)H(u)E^{\bot}(-u)E^{\bot}(-\frac{1}{u})=\sum\limits_{k\in\mathbb{Z}+\frac{1}{2}}\psi^{O,+}_{k}u^{-k-\frac{1}{2}},$
(4.1)
$\displaystyle\psi^{O,-}(u)=R^{-1}(u)E(-u)H^{\bot}(u)H^{\bot}(\frac{1}{u})=\sum\limits_{k\in\mathbb{Z}+\frac{1}{2}}\psi^{O,-}_{k}u^{-k-\frac{1}{2}}.$

######  Proposition 4.1.

It can be checked that $\psi^{O,+}(u),\psi^{O,-}(u)$ satisfy the relations

$\displaystyle\psi^{O,\pm}(u)\psi^{O,\pm}(v)+\psi^{O,\pm}(v)\psi^{O,\pm}(u)=0,$
(4.2)
$\displaystyle\psi^{O,+}(u)\psi^{O,-}(v)+\psi^{O,-}(v)\psi^{O,+}(u)=\delta(u,v).$

Equivalently, Eq.(4.2) can be expressed as charged free fermions relation

$\displaystyle\psi^{O,\pm}_{k}\psi^{O,\pm}_{l}+\psi^{O,\pm}_{l}\psi^{O,\pm}_{k}=0,$
(4.3)
$\displaystyle\psi^{O,+}_{k}\psi^{O,-}_{l}+\psi^{O,-}_{l}\psi^{O,+}_{k}=\delta_{k,-l}.$

######  Remark 4.2.

From the formula (2) and (2.7), we easily get the bosonic form of the fields
$\psi^{O,\pm}(u)$:

$\displaystyle\psi^{O,+}(u)=u^{-1}(1-u^{2})R(u)\exp\left(\sum\limits_{n\geq
1}\frac{p_{n}}{n}u^{n}\right)\exp\left(-\sum\limits_{n\geq
1}\frac{\partial}{\partial
p_{n}}\frac{1}{u^{n}}\right)\exp\left(-\sum\limits_{n\geq
1}\frac{\partial}{\partial p_{n}}u^{n}\right),$ (4.4)
$\displaystyle\psi^{O,-}(u)=R^{-1}(u)\exp\left(-\sum\limits_{n\geq
1}\frac{p_{n}}{n}u^{n}\right)\exp\left(\sum\limits_{n\geq
1}\frac{\partial}{\partial
p_{n}}\frac{1}{u^{n}}\right)\exp\left(\sum\limits_{n\geq
1}\frac{\partial}{\partial p_{n}}u^{n}\right).$

Hence we obtain $\psi^{O,+}_{i}(z^{m})=0$ if $i>-m-\frac{1}{2}$ and
$\psi^{O,-}_{i}(z^{m})=0$ if $i>m-\frac{1}{2}$.

######  Definition 4.3.

For an unknown function $\tau=\tau(x)$, the bilinear equation

$\displaystyle\widetilde{\Omega}(\tau\otimes\tau)=0,$ (4.5)

is called the OKP hierarchy, where

$\displaystyle\widetilde{\Omega}=\sum\limits_{k\in\mathbb{Z}+\frac{1}{2}}\psi_{k}^{O,+}\otimes\psi_{-k}^{O,-}.$
(4.6)

######  Lemma 4.4.

Let $\widetilde{X}=\sum\limits_{i>N}C_{i}\psi_{i}^{O,+}$, where
$C_{i}\in\mathbb{C},N\in\mathbb{Z}$. Then $\widetilde{X}^{2}=0$.

######  Lemma 4.5.

Let $\widetilde{X}=\sum\limits_{i>N}C_{i}\psi_{i}^{O,+}$, where
$C_{i}\in\mathbb{C},N\in\mathbb{Z}$. Then
$\widetilde{\Omega}(\widetilde{X}\otimes\widetilde{X})=(\widetilde{X}\otimes\widetilde{X})\widetilde{\Omega}$.

######  Corollary 4.6.

Let $\tau\in\mathcal{B}^{m}$ be a tau-function of the OKP hierarchy, and let
$\widetilde{X}=\sum\limits_{i>N}C_{i}\psi_{i}^{O,+}$, where
$C_{i}\in\mathbb{C},N\in\mathbb{Z}$. Then
$\widetilde{\tau}=\widetilde{X}\tau\in\mathcal{B}^{m+1}$ is also a tau-
functions of the OKP hierarchy.

###### Proof.

The proof of the Lemma 4.4, 4.5 and Corollary 4.6 is quite similar to the
Lemma 3.4, 3.5 and Corollary 3.6, so is omitted. ∎

###  4.2. Generating functions and the polynomial tau-functions of the OKP
hierarchy

It is known that orthogonal Schur functions are tau-functions of the OKP
hierarchy [23]. Let $\widetilde{G}(u_{1},\ldots,u_{l})$ be a generating
function of the orthogonal Schur function in $\mathbf{u}=(u_{1},\ldots,u_{l})$
defined by

$\displaystyle\ \widetilde{G}(u_{1},\ldots,u_{l})=\prod\limits_{1\leq i<j\leq
l}\left(u_{i}-u_{j}\right)\prod\limits_{1\leq i\leq j\leq
l}\left(1-u_{i}u_{j}\right)\prod\limits_{i=1}^{l}H(u_{i}).$ (4.7)

By Proposition 2.1, we obatin

$\displaystyle\psi^{O,+}(u_{1})\cdots\psi^{O,+}(u_{l})(z^{k}f)=z^{k+l}u_{1}^{k}\cdots
u_{l}^{k}\widetilde{G}(u_{1},\ldots,u_{l}).$ (4.8)

Consider the set of formal Laurent series
$\widetilde{A}_{1}(u),\ldots,\widetilde{A}_{l}(u)$, define the formal Laurent
series
$\widetilde{T}_{i}(u)=\widetilde{A}_{i}(u)H(u)=\sum\limits_{p\in\mathbb{Z}}\widetilde{T}_{i,p}u^{p},\
i=1,\ldots,l$. Besides, let $\widetilde{T}(u_{1},\ldots,u_{l})$ be a formal
Laurent series in $(u_{1},\ldots,u_{l})$ defined by

$\displaystyle\ \widetilde{T}(u_{1},\ldots,u_{l})=\prod\limits_{1\leq i<j\leq
l}\left(u_{i}-u_{j}\right)\prod\limits_{1\leq i\leq j\leq
l}\left(1-u_{i}u_{j}\right)\prod\limits_{i=1}^{l}\widetilde{A}_{i}(u_{i})H(u_{i}).$
(4.9)

For any vector
$\boldsymbol{\zeta}=(\zeta_{1},\ldots,\zeta_{l})\in\mathbb{Z}^{l}$,
$\widetilde{T}_{\boldsymbol{\zeta}}$ is the coefficient of the following
expansion

$\displaystyle\
\widetilde{T}(u_{1},\ldots,u_{l})=\sum\limits_{\boldsymbol{\zeta}\in\mathbb{Z}^{l}}\widetilde{T}_{\boldsymbol{\zeta}}u_{1}^{\zeta_{1}}\cdots
u_{l}^{\zeta_{l}}.$ (4.10)

######  Theorem 4.7.

  * 1)

Formal Laurent series $\widetilde{T}(u_{1},\ldots,u_{l})$ can be written as

$\displaystyle\widetilde{T}(u_{1},\ldots,u_{l})=\det\left[\left(u_{i}^{l-j}-u_{i}^{l+j}\right)\widetilde{T}_{i}(u_{i})\right]_{1\leq
i,j\leq l}.$ (4.11)

  * 2)

For any vector
$\boldsymbol{\zeta}=(\zeta_{1},\ldots,\zeta_{l})\in\mathbb{Z}^{l}$, the
coefficient $\widetilde{T}_{\boldsymbol{\zeta}}$ of $u_{1}^{\zeta_{1}}\cdots
u_{l}^{\zeta_{l}}$ in (4.10) is given by

$\displaystyle\widetilde{T}_{\boldsymbol{\zeta}}=\det\left[\widetilde{T}_{i,\zeta_{i}-l-j}-\widetilde{T}_{i,\zeta_{i}-l+j}\right]_{1\leq
i,j\leq l}.$ (4.12)

  * 3)

$\widetilde{T}_{\boldsymbol{\zeta}}$ is a polynomial tau-function of the OKP
hierarchy.

###### Proof.

  * 1)

According to Vandermonde-like identity [38]

$\displaystyle\det\left[u_{i}^{k-j}-u_{i}^{k+j}\right]$ $\displaystyle=$
$\displaystyle\prod\limits_{1\leq i<j\leq
k}\left(u_{i}-u_{j}\right)\prod\limits_{1\leq i\leq j\leq
k}\left(1-u_{i}u_{j}\right)$ (4.13) $\displaystyle=$
$\displaystyle\sum\limits_{{\sigma\in S_{k}}\atop{\varepsilon_{i}=\pm
1}}sgn(\sigma)\varepsilon_{1}\cdots\varepsilon_{k}u_{1}^{k-\varepsilon_{1}\sigma(1)}\cdots
u_{k}^{k-\varepsilon_{k}\sigma(k)},$

we have

$\displaystyle\widetilde{T}(u_{1},\ldots,u_{l})$ $\displaystyle=$
$\displaystyle\prod\limits_{1\leq i<j\leq
l}\left(u_{i}-u_{j}\right)\prod\limits_{1\leq i\leq j\leq
l}\left(1-u_{i}u_{j}\right)\prod\limits_{i=1}^{l}\widetilde{A}_{i}(u_{i})H(u_{i})$
(4.14) $\displaystyle=$
$\displaystyle\det\left[u_{i}^{l-j}-u_{i}^{l+j}\right]\prod\limits_{i=1}^{l}\widetilde{T}_{i}(u_{i})=\det\left[\left(u_{i}^{l-j}-u_{i}^{l+j}\right)\widetilde{T}_{i}(u_{i})\right]_{1\leq
i,j\leq l}.$

  * 2)

Noticing that

$\displaystyle\widetilde{T}(u_{1},\ldots,u_{l})$
$\displaystyle=\det\left[\sum\limits_{p_{i}\in\mathbb{Z}}\widetilde{T}_{i,p_{i}}\left(u_{i}^{l+p_{i}-j}-u_{i}^{l+p_{i}+j}\right)\right]$
$\displaystyle=\sum\limits_{p_{i}\in\mathbb{Z}}\sum\limits_{{\sigma\in
S_{l}}\atop{\varepsilon_{i}=\pm
1}}sgn(\sigma)\varepsilon_{1}\cdots\varepsilon_{l}\widetilde{T}_{1,p_{1}}u_{1}^{l+p_{1}-\varepsilon_{1}\sigma(1)}\cdots\widetilde{T}_{l,p_{l}}u_{l}^{l+p_{l}-\varepsilon_{l}\sigma(l)}$
$\displaystyle=\sum\limits_{\zeta_{i}\in\mathbb{Z}}\sum\limits_{{\sigma\in
S_{l}}\atop{\varepsilon_{i}=\pm
1}}sgn(\sigma)\varepsilon_{1}\cdots\varepsilon_{l}\widetilde{T}_{1,\zeta_{1}-l+\varepsilon_{1}\sigma(1)}\cdots\widetilde{T}_{l,\zeta_{l}-l+\varepsilon_{l}\sigma(l)}u_{1}^{\zeta_{1}}\cdots
u_{l}^{\zeta_{l}}$
$\displaystyle=\sum\limits_{\zeta_{i}\in\mathbb{Z}}\det\left[\widetilde{T}_{i,\zeta_{i}-l-j}-\widetilde{T}_{i,\zeta_{i}-l+j}\right]_{1\leq
i,j\leq l}u_{1}^{\zeta_{1}}\cdots u_{l}^{\zeta_{l}}.$ (4.15)

Obviously, the coefficient $\widetilde{T}_{\boldsymbol{\zeta}}$ of
$u_{1}^{\zeta_{1}}\cdots u_{l}^{\zeta_{l}}$ is
$\det\left[\widetilde{T}_{i,\zeta_{i}-l-j}-\widetilde{T}_{i,\zeta_{i}-l+j}\right]_{1\leq
i,j\leq l}$.

  * 3)

From (4.7), it is straightforward to show that

$\displaystyle\widetilde{A}_{1}(u_{1})\cdots\widetilde{A}_{l}(u_{l})\psi^{O,+}(u_{1})\cdots\psi^{O,+}(u_{l})(z^{k}\cdot
1)=z^{l+k}u_{1}^{k}\cdots u_{l}^{k}\widetilde{T}(u_{1},\ldots,u_{l}).$ (4.16)

Let $\widetilde{A}_{j}(u)=\sum\limits_{M_{j}\leq r\leq
N_{j}}\widetilde{A}_{j,r-\frac{1}{2}}u^{r}(\widetilde{A}_{j,r-\frac{1}{2}}\in\mathbb{C},M_{j},N_{j},r\in\mathbb{Z},j=1,\ldots,l)$
be a power series expansion of the variable $u$. Therefore,
$\widetilde{T}_{\boldsymbol{\zeta}}$ can be written as

$\displaystyle\widetilde{T}_{\boldsymbol{\zeta}}=z^{-l-k}\widetilde{X}_{1}\cdots\widetilde{X}_{l}(z^{k}\cdot
1),$ (4.17)

where

$\displaystyle\widetilde{X}_{j}=\sum\limits_{M_{j}-\zeta_{j}-k-\frac{1}{2}\leq
i_{j}\leq
N_{j}-\zeta_{j}-k-\frac{1}{2}}\widetilde{A}_{j,\zeta_{j}+k+i_{j}}\psi^{O,+}_{i_{j}},\quad
j=1,\cdots,l.$ (4.18)

By Remark 4.2 and Corollary 4.6, it should be pointed out that the coefficient
$\widetilde{T}_{\boldsymbol{\zeta}}$ is a tau-function of the OKP hierarchy
with $k=0$. Since $\widetilde{T}_{\boldsymbol{\zeta}}$ is a finite linear
combination of $\psi_{i_{1}}^{O,+}\cdots\psi_{i_{l}}^{O,+}(1)$, it is a
polynomial tau-function.

∎

By changing $\widetilde{A}_{j}(u)\rightarrow
u^{\zeta_{j}}\widetilde{A}_{j}(u)$, we obtain

$\displaystyle\widetilde{A}_{i}(u)=u^{N_{i}}\widetilde{h}_{i}\sum\limits_{k=0}^{\infty}\widetilde{a}_{i,k}u^{k},\
\
N_{i}\in\mathbb{Z},\widetilde{h}_{i},\widetilde{a}_{i,k}\in\mathbb{C},\widetilde{a}_{i,0}=1,\widetilde{h}_{i}\neq
0,i=1,\ldots,l.$ (4.19)

From (2.3), $\sum\limits_{k=0}^{\infty}\widetilde{a}_{i,k}u^{k}$ can be
expressed as

$\displaystyle\sum\limits_{k=0}^{\infty}\widetilde{a}_{i,k}u^{k}=\exp\left(\sum\limits_{l=1}^{\infty}\widetilde{c}_{i,l}u^{l}\right),\
\ \text{and}\ \
\widetilde{a}_{i,k}=S_{k}(\widetilde{c}_{i,1},\widetilde{c}_{i,2},\ldots),$
(4.20)

where $\\{\widetilde{c}_{i,l}\\}$ are constants in $\mathbb{C}$. Then based on
(2), we get

$\displaystyle\ \widetilde{T}_{i}(u)$ $\displaystyle=$
$\displaystyle\widetilde{A}_{i}(u)H(u)=u^{N_{i}}\widetilde{h}_{i}\sum\limits_{l=0}^{\infty}S_{l}(t_{1}+\widetilde{c}_{i,1},t_{2}+\widetilde{c}_{i,2},\ldots)u^{l}.$
(4.21)

Hence
$\widetilde{T}_{i,p}=\widetilde{h}_{i}S_{p-N_{i}}(t_{1}+\widetilde{c}_{i,1},t_{2}+\widetilde{c}_{i,2},\ldots),\
i=1,\ldots,l$. From Theorem 4.7, polynomial tau-functions of the OKP hierarchy
have the form

$\displaystyle\widetilde{T}_{\boldsymbol{\zeta}}$ $\displaystyle=$
$\displaystyle\det\left[\widetilde{T}_{i,\zeta_{i}-l-j}-\widetilde{T}_{i,\zeta_{i}-l+j}\right]$
$\displaystyle=$ $\displaystyle\det\left[\widetilde{h}_{i}S_{\zeta_{i}-l-j-
N_{i}}(t_{1}+\widetilde{c}_{i,1},t_{2}+\widetilde{c}_{i,2},\ldots)-\widetilde{h}_{i}S_{\zeta_{i}-l+j-N_{i}}(t_{1}+\widetilde{c}_{i,1},t_{2}+\widetilde{c}_{i,2},\ldots)\right]$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{l}(\widetilde{h}_{i})\det\left[S_{\zeta_{i}-l-j-
N_{i}}(t_{1}+\widetilde{c}_{i,1},t_{2}+\widetilde{c}_{i,2},\ldots)-S_{\zeta_{i}-l+j-N_{i}}(t_{1}+\widetilde{c}_{i,1},t_{2}+\widetilde{c}_{i,2},\ldots)\right]_{i,j=1,\ldots,l}.$

Under the reduction $\widetilde{c}_{i,l}=0$, $\widetilde{h}_{i}=-1$ and
$l+N_{i}=i$, $\widetilde{T}_{\boldsymbol{\zeta}}$ lead to the orthogonal Schur
functions [23]. Thus the polynomial tau-functions (4.2) of the OKP hierarchy
can be reduced to the solution of the OKP hierarchy in [23], which are the
zero mode of an appropriate combinatorial generating functions.

###  4.3. $N$-soliton solutions of the OKP hierarchy

Let

$\displaystyle\Gamma^{O}(p,q)=p^{-1}(1-p^{2})R(p)R^{-1}(q)H(p)E(-q)E^{\bot}(-p)H^{\bot}(q)E^{\bot}(-\frac{1}{p})H^{\bot}(\frac{1}{q}).$
(4.23)

From Proposition 2.1, it is easy to check that

$\displaystyle\Gamma^{O}(p_{i},q_{i})\Gamma^{O}(p_{j},q_{j})=A_{ij}:\Gamma^{O}(p_{i},q_{i})\Gamma^{O}(p_{j},q_{j}):,$
(4.24)

where

$\displaystyle
A_{ij}=\frac{(1-p_{i}p_{j})(1-q_{i}q_{j})(p_{i}-p_{j})(q_{i}-q_{j})}{(1-q_{i}p_{j})(1-p_{i}q_{j})(q_{i}-p_{j})(p_{i}-q_{j})}.$
(4.25)

In particular $\Gamma^{O}(p,q)^{2}=0$; therefore
$e^{pc\Gamma^{O}(p,q)}=1+pc\Gamma^{O}(p,q)$.

######  Lemma 4.8.

If $\tau$ is a solution of the OKP hierarchy, then $\Gamma^{O}(u,v)\tau$ is
also a solution.

######  Lemma 4.9.

It holds that

$\displaystyle[\widetilde{\Omega},1\otimes\Gamma^{O}(p,q)+\Gamma^{O}(p,q)\otimes
1]=0.$ (4.26)

Considering the following function

$\displaystyle\
\tau(x,y)=\tau(x,y;p,q,c)=\prod\limits_{i=1}^{n}e^{p_{i}c_{i}\Gamma^{O}(p_{i},q_{i})}\cdot
1,\quad p_{i},q_{i},c_{i}\in\mathbb{C},p_{i}\neq
q_{j},p_{i}\neq\frac{1}{q_{j}}for\ i\neq j.$ (4.27)

Let us set

$\displaystyle\ \eta_{i}=\sum\limits_{k\geq
1}(p_{i}^{k}-q_{i}^{k})\frac{p_{k}(x)}{k}.$ (4.28)

From Eq.(4.24), Eq.(4.27) can be rewritten as

$\displaystyle\tau(x,y;p,q,c)=\sum_{J\subset I}\left(\prod_{i\in
J}c_{i}(1-p_{i}^{2})\right)\left(\prod\limits_{{i,j\in
J}\atop{i<j}}A_{ij}\right)\exp\left(\sum\limits_{i\in J}\eta_{i}\right),$
(4.29)

where $I=\\{1,2,\ldots,n\\}$ .

######  Proposition 4.10.

The function $\tau(x,y;p,q,c)$ in (4.29) is a solution of the OKP hierarchy,
which we call the $n$-soliton solutions.

###### Proof.

Lemma 4.8, 4.9 and Proposition 4.10 can be proved with the similar procedure
as in Lemma 3.8, 3.9 and Proposition 3.10. ∎

##  5\. Polynomial tau-functions of the BUC hierarchy

In this section, the quantum fields of the generalized $Q$-functions shall be
developed. By using neutral fermions, we construct an integrable BUC hierarchy
characterized by the generalized $Q$-functions. Based upon the generating
functions of the polynomial tau-functions of the BUC hierarchy, it is showed
that the polynomial tau-function of the BUC hierarchy is a zero mode of
certain generating functions.

###  5.1. Quantum fields presentation of the generalized $Q$-functions and the
BUC hierarchy

Introduce another class of symmetric functions $q_{k}(x_{1},x_{2},\ldots)$ by

$\displaystyle Q(u)=\sum\limits_{k\in\mathbb{Z}}q_{k}u^{k}=E(u)H(u),$ (5.1)

here $q_{k}=\sum\limits_{i=0}^{k}e_{i}h_{k-i}$ for $k>0$, $q_{0}=1$ and
$q_{k}=0$ for $k<0$.

Define

$\displaystyle Q(u)=S(u)^{2},\quad\text{where}\
S(u)=\exp\left(\sum\limits_{n\in N_{odd}}\frac{p_{n}}{n}u^{n}\right),$ (5.2)
$\displaystyle S^{\bot}(u)=\exp\left(\sum\limits_{n\in
N_{odd}}\frac{\partial}{\partial p_{n}}\frac{1}{u^{n}}\right),\quad
N_{odd}=\\{1,3,5,\ldots\\}.$

######  Proposition 5.1.

The following commutation relations about generating functions hold (cf.[33])

$\displaystyle H^{\bot}(u)Q(v)=\frac{u+v}{u-v}Q(v)H^{\bot}(u),$ (5.3)
$\displaystyle E^{\bot}(u)Q(v)=\frac{u+v}{u-v}Q(v)E^{\bot}(u),$ $\displaystyle
S^{\bot}(u)Q(v)=\frac{u+v}{u-v}Q(v)S^{\bot}(u).$

Define the formal distributions $\varphi(u)$ and $\overline{\varphi}(u)$ of
operators acting on the boson Fock space
$\mathcal{B}_{odd}=\mathbb{C}[p_{1},p_{3},p_{5},\ldots]$

$\displaystyle\varphi(u)=Q(u)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u)=\sum\limits_{j\in\mathbb{Z}}\varphi_{j}u^{-j},$
(5.4)
$\displaystyle\overline{\varphi}(u)=Q^{\prime}(u)S^{\bot}(-\frac{1}{u})S^{\prime\bot}(-u)=\sum\limits_{j\in\mathbb{Z}}\overline{\varphi}_{j}u^{-j},$

where $Q^{\prime}(u)$ means the generating functions for the
$q_{k}(\mathbf{y})$ and their adjoint operators hold for the variable
$\mathbf{y}$, the operators $\varphi_{j}$ and $\overline{\varphi}_{j}$ are the
neutral fermions.

Let

$\displaystyle f(u,v)=\frac{u-v}{u+v}=1+2\sum\limits_{k\geq
1}(-1)^{k}\frac{v^{k}}{u^{k}},\quad|u|>|v|,$ (5.5)

then

$\displaystyle
f(u,v)+f(v,u)=(v-u)\delta(v,-u)=2\sum\limits_{k\in\mathbb{Z}}\frac{v^{k}}{(-u)^{k}}=2v\delta(v,-u).$
(5.6)

######  Proposition 5.2.

$\varphi(u)$ and $\overline{\varphi}(u)$ satisfy the following relations

$\displaystyle\varphi(u)\varphi(v)+\varphi(v)\varphi(u)=2v\delta(v,-u),$ (5.7)
$\displaystyle\overline{\varphi}(u)\overline{\varphi}(v)+\overline{\varphi}(v)\overline{\varphi}(u)=2v\delta(v,-u),$
$\displaystyle\varphi(u)\overline{\varphi}(v)-\overline{\varphi}(v)\varphi(u)=0.$

Eq.(5.7) can alos be expressed as neutral fermions relation

$\displaystyle\varphi_{m}\varphi_{n}+\varphi_{n}\varphi_{m}=2(-1)^{m}\delta_{m+n,0},$
(5.8)
$\displaystyle\overline{\varphi}_{m}\overline{\varphi}_{n}+\overline{\varphi}_{n}\overline{\varphi}_{m}=2(-1)^{m}\delta_{m+n,0},$
$\displaystyle\varphi_{m}\overline{\varphi}_{n}-\overline{\varphi}_{n}\varphi_{m}=0.$

###### Proof.

We only prove the first formula of (5.7) and (5.8), other formulas cab be
proved similarly. In terms of Proposition 5.1 and (5.6), we have

$\displaystyle\ \varphi(u)\varphi(v)$ $\displaystyle=$ $\displaystyle
Q(u)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u)Q(v)S^{\prime\bot}(-\frac{1}{v})S^{\bot}(-v)$
(5.9) $\displaystyle=$
$\displaystyle\frac{-u+v}{-u-v}Q(u)Q(v)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u)S^{\prime\bot}(-\frac{1}{v})S^{\bot}(-v),$
$\displaystyle\varphi(v)\varphi(u)$ $\displaystyle=$ $\displaystyle
Q(v)S^{\prime\bot}(-\frac{1}{v})S^{\bot}(-v)Q(u)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u)$
(5.10) $\displaystyle=$
$\displaystyle\frac{-v+u}{-v-u}Q(v)Q(u)S^{\prime\bot}(-\frac{1}{v})S^{\bot}(-v)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u),$

therefore,

$\displaystyle\ \varphi(u)\varphi(v)+\varphi(v)\varphi(u)$ $\displaystyle=$
$\displaystyle\left(\frac{u-v}{u+v}+\frac{v-u}{v+u}\right)Q(u)Q(v)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u)S^{\prime\bot}(-\frac{1}{v})S^{\bot}(-v)$
(5.11) $\displaystyle=$
$\displaystyle(v-u)\delta(v,-u)Q(u)Q(v)S^{\prime\bot}(-\frac{1}{u})S^{\bot}(-u)S^{\prime\bot}(-\frac{1}{v})S^{\bot}(-v)$
$\displaystyle=$ $\displaystyle 2v\delta(v,-u).$

Expanding $\varphi(u)\varphi(v)+\varphi(v)\varphi(u)=2v\delta(v,-u)$ into

$\displaystyle\sum\limits_{m\in\mathbb{Z}}\varphi_{m}u^{-m}\sum\limits_{n\in\mathbb{Z}}\varphi_{n}v^{-n}+\sum\limits_{n\in\mathbb{Z}}\varphi_{n}v^{-n}\sum\limits_{m\in\mathbb{Z}}\varphi_{m}u^{-m}=2v\sum\limits_{k\in\mathbb{Z}}\frac{v^{k}}{(-u)^{k+1}},$
(5.12)

and taking the coefficient of $u^{-m}v^{-n}$ at both ends of the above
formula, we derive the first formula of (5.8). ∎

######  Remark 5.3.

From the formula (5.2), we derive the bosonic form of the quantum fields
$\varphi(u)$ and $\overline{\varphi}(u)$ as follows

$\displaystyle\varphi(u)$ $\displaystyle=$
$\displaystyle\exp\left(\sum\limits_{n\in\mathbb{N}_{odd}}\frac{2p_{n}}{n}u^{n}\right)\exp\left(-\sum\limits_{n\in\mathbb{N}_{odd}}\frac{\partial}{\partial
p_{n}^{\prime}}u^{n}\right)\exp\left(-\sum\limits_{n\in\mathbb{N}_{odd}}\frac{\partial}{\partial
p_{n}}\frac{1}{u^{n}}\right),$ $\displaystyle\overline{\varphi}(u)$
$\displaystyle=$
$\displaystyle\exp\left(\sum\limits_{n\in\mathbb{N}_{odd}}\frac{2p_{n}^{\prime}}{n}u^{n}\right)\exp\left(-\sum\limits_{n\in\mathbb{N}_{odd}}\frac{\partial}{\partial
p_{n}}u^{n}\right)\exp\left(-\sum\limits_{n\in\mathbb{N}_{odd}}\frac{\partial}{\partial
p_{n}^{\prime}}\frac{1}{u^{n}}\right).$ (5.13)

Hence one can check that $\varphi_{m}(1)=0(m>0)$, $\varphi_{0}(1)=1$ and
$\overline{\varphi}_{n}(1)=0(n>0)$, $\overline{\varphi}_{0}(1)=1$.

######  Definition 5.4.

The BUC hierarchy is the system of bilinear relations

$\displaystyle\Omega(\tau\otimes\tau)=\overline{\Omega}(\tau\otimes\tau)=(\tau\otimes\tau),$
(5.14)

where

$\displaystyle\Omega=\sum\limits_{n\in\mathbb{Z}}\varphi_{n}\otimes(-1)^{n}\varphi_{-n},\
\ \ \
\overline{\Omega}=\sum\limits_{n\in\mathbb{Z}}\overline{\varphi}_{n}\otimes(-1)^{n}\overline{\varphi}_{-n}.$
(5.15)

From Remark 5.3, it is easy to see that $\tau=1$ is a tau-function of the BUC
hierarchy. Similarly, if the solution of (5.14) is a polynomial function of
the variables $(p_{1},p_{3},\ldots)$, we say it is a polynomial tau-function.
Now we consider other forms of tau-functions of the BUC hierarchy.

######  Lemma 5.5.

Let $X=\sum\limits_{n\geq N}A_{n}\varphi_{n}$, $Y=\sum\limits_{m\geq
M}B_{m}\overline{\varphi}_{m}$, where $A_{n},B_{m}\in\mathbb{C}$ and
$N,M\in\mathbb{Z}$. Then

$\displaystyle X^{2}=\left\\{\begin{aligned} &\sum\limits_{N\leq
k\leq-N}(-1)^{k}A_{k}A_{-k},\quad N<0,\\\ &A_{0}^{2},\quad\quad\quad N=0,\\\
&0,\quad\quad\quad\ \ N>0.\end{aligned}\right.,\quad\quad
Y^{2}=\left\\{\begin{aligned} &\sum\limits_{M\leq
l\leq-M}(-1)^{l}B_{l}B_{-l},\quad M<0,\\\ &B_{0}^{2},\quad\quad\quad M=0,\\\
&0,\quad\quad\quad\ \ M>0.\end{aligned}\right.$ (5.16)

######  Lemma 5.6.

$\displaystyle\Omega(X\otimes X)=(X\otimes
X)\Omega,\quad\quad\quad\Omega(Y\otimes Y)=(Y\otimes Y)\Omega,$
$\displaystyle\overline{\Omega}(X\otimes X)=(X\otimes
X)\overline{\Omega},\quad\quad\quad\overline{\Omega}(Y\otimes Y)=(Y\otimes
Y)\overline{\Omega}.$ (5.17)

###### Proof.

Using the similar approach in [34], we can prove the Lemma. ∎

######  Corollary 5.7.

Let $\tau\in\mathcal{B}_{odd}$ be a tau-function of the BUC hierarchy, and let
$X=\sum\limits_{n\geq N}A_{n}\varphi_{n}$, $Y=\sum\limits_{m\geq
M}B_{m}\overline{\varphi}_{m}$, where $A_{n},B_{m}\in\mathbb{C}$ and
$N,M\in\mathbb{Z}$. Then $\tau^{\prime}=X\tau$ and $\tau^{\prime\prime}=Y\tau$
are also tau-functions of the BUC hierarchy.

###### Proof.

The proof method of this Corollary is similar to that of Corollary 3.6, so it
will not be described in detail here. ∎

###  5.2. Generating functions and polynomial tau-functions of the BUC
hierarchy

Let $Q(\mathbf{u},\mathbf{v})$ be a generating function of the BUC hierarchy
in $(\mathbf{u},\mathbf{v})=(u_{1},\ldots,u_{r},v_{1},\ldots,v_{s})$ defined
by

$\displaystyle Q(\mathbf{u},\mathbf{v})=\prod\limits_{1\leq i<j\leq
r}\frac{u_{i}-u_{j}}{u_{i}+u_{j}}\prod\limits_{1\leq i<j\leq
s}\frac{v_{i}-v_{j}}{v_{i}+v_{j}}\prod\limits_{{1\leq i\leq r}\atop{1\leq
j\leq
s}}\frac{1-u_{i}v_{j}}{1+u_{i}v_{j}}\prod\limits_{i=1}^{r}Q(u_{i})\prod\limits_{j=1}^{s}Q^{\prime}(v_{j}).$
(5.18)

From Proposition 5.1 and $S^{\bot}(u)(1)=1$, we have

$\displaystyle\varphi(u_{1})\cdots\varphi(u_{r})\overline{\varphi}(v_{1})\cdots\overline{\varphi}(v_{s})(1)=Q(\mathbf{u},\mathbf{v}).$
(5.19)

It is expanded into rational function form
$Q(\mathbf{u},\mathbf{v})=\sum\limits_{{\boldsymbol{\alpha}\in\mathbb{Z}^{r}}\atop{\boldsymbol{\beta}\in}\mathbb{Z}^{s}}Q_{\boldsymbol{\alpha},\boldsymbol{\beta}}u_{1}^{\alpha_{1}}\cdots
u_{r}^{\alpha_{r}}v_{1}^{\beta_{1}}\cdots v_{s}^{\beta_{s}}$, then

$\displaystyle
Q_{\boldsymbol{\alpha},\boldsymbol{\beta}}=\varphi_{-\alpha_{1}}\cdots\varphi_{-\alpha_{r}}\overline{\varphi}_{-\beta_{1}}\cdots\overline{\varphi}_{-\beta_{s}}(1).$
(5.20)

In the following, in order to express the family of generation functions of
the BUC hierarchy as a certain Pfaffian, we denote the variables as
$\check{\mathbf{u}}=_{def}(u_{1},u_{2},\ldots,u_{r}),\
\check{\mathbf{v}}=_{def}(u_{-1}^{-1},u_{-2}^{-1},\ldots,u_{-s}^{-1})$.
Consider the set of Laurent polynomial $A_{1}(u),\ldots,A_{l}(u)$ and
$B_{-s}(u^{-1}),\ldots,B_{-1}(u^{-1})$, define a formal distribution

$\displaystyle\ T(\check{\mathbf{u}},\check{\mathbf{v}})$ $\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{r}A_{j}(u_{j})\prod\limits_{i=-s}^{-1}B_{i}(u_{i}^{-1})Q(\check{\mathbf{u}},\check{\mathbf{v}})$
(5.21) $\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{r}A_{j}(u_{j})\prod\limits_{i=-s}^{-1}B_{i}(u_{i}^{-1})\prod\limits_{{-s\leq
i<j\leq r}\atop{i,j\neq
0}}f(u_{i},u_{j})\prod\limits_{i=-s}^{-1}Q^{\prime}(u_{i}^{-1})\prod\limits_{j=1}^{r}Q(u_{j}).$

For any
$\boldsymbol{\gamma}=(\gamma_{1},\ldots,\gamma_{r},\gamma_{-1},\ldots,\gamma_{-s})\in\mathbb{Z}^{r+s}$,
$T_{\boldsymbol{\gamma}}$ is the coefficient of the following expansion

$\displaystyle
T(\check{\mathbf{u}},\check{\mathbf{v}})=\sum\limits_{\boldsymbol{\gamma}\in\mathbb{Z}^{r+s}}T_{\boldsymbol{\gamma}}u_{1}^{\gamma_{1}}\cdots
u_{r}^{\gamma_{r}}u_{-1}^{\gamma_{-1}}\cdots u_{-s}^{\gamma_{-s}}.$ (5.22)

We recall that if $A=[a_{ij}]$ is a skew symmetric matrix of even size
$2n\times 2n$, its determinant is a perfect square: $\det[A]=Pf[A]^{2}$, where

$\displaystyle
Pf[A]=\sum\limits_{\omega}sgn(\omega)a_{\omega(1)\omega(2)}\cdots
a_{\omega(2n-1)\omega(2n)},$ (5.23)

summed over $\omega\in S_{2n}$ such that $\omega(2r-1)<\omega(2r)$ for $1\leq
r\leq n$, and $\omega(2r-1)<\omega(2r+1)$ for $1\leq r\leq n-1$. In addition,
it is well-known that

$\displaystyle Pf\left[\frac{u_{i}-u_{j}}{u_{i}+u_{j}}\right]_{1\leq i,j\leq
2n}=\prod\limits_{1\leq i<j\leq 2n}\frac{u_{i}-u_{j}}{u_{i}+u_{j}}.$ (5.24)

Introduce the skew symmetric matrix $F=[f_{i,j}]_{{-2s\leq i,j\leq
2r}\atop{i,j\neq 0}}$, where

$\displaystyle f_{i,j}=\left\\{\begin{aligned} &f(u_{i},u_{j}),\quad\quad\quad
i<j,\\\ &0,\quad\quad\quad\quad\quad\quad i=j,\\\ &-f(u_{j},u_{i}),\quad\ \
i>j.\end{aligned}\right.$ (5.25)

Obviously,

$\displaystyle Pf[F]=\prod\limits_{{-2s\leq i<j\leq 2r}\atop{i,j\neq
0}}f(u_{i},u_{j})=\prod\limits_{{-2s\leq i<j\leq 2r}\atop{i,j\neq
0}}\frac{u_{i}-u_{j}}{u_{i}+u_{j}}.$ (5.26)

Define the formal distributions

$\displaystyle
T^{(i)}(u_{i}^{-1})=B_{i}(u_{i}^{-1})Q^{\prime}(u_{i}^{-1}),\quad
i\in\\{-2s,\ldots,-1\\},$ (5.27) $\displaystyle
T^{(j)}(u_{j})=A_{j}(u_{j})Q(u_{j}),\quad\quad\quad\ \ j\in\\{1,\ldots,2r\\},$

and

$\displaystyle
T^{(i,j)}=f_{ij}T^{(i)}T^{(j)}=\sum\limits_{m,n\in\mathbb{Z}}T_{m,n}^{(i,j)}u_{i}^{m}u_{j}^{n},$
(5.28)

where $T^{(i)}$ denotes $T^{(i)}(u_{i}^{-1})$ for negative $i$ and
$T^{{i}}(u_{i})$ for positive $i$.

######  Theorem 5.8.

  * 1)

The formal distribution
$T(u_{1},\ldots,u_{2r},u_{-1}^{-1},\ldots,u_{-2s}^{-1})$ can be expressed as

$\displaystyle
T(u_{1},\ldots,u_{2r},u_{-1}^{-1},\ldots,u_{-2s}^{-1})=Pf[T^{(i,j)}]_{{-2s\leq
i,j\leq 2r}\atop{i,j\neq 0}}.$ (5.29)

  * 2)

The coefficient $T_{\check{\boldsymbol{\gamma}}}$ about the expansion in
Eq.(5.29) can be written as

$\displaystyle
T_{\breve{\boldsymbol{\gamma}}}=Pf[T^{(i,j)}_{\gamma_{i},\gamma_{j}}]_{{-2s\leq
i,j\leq 2r}\atop{i,j\neq 0}},$ (5.30)

where
$\check{\boldsymbol{\gamma}}=(\gamma_{1},\ldots,\gamma_{2r},\gamma_{-1},\ldots,\gamma_{-2s})\in\mathbb{Z}^{2r+2s}$.

  * 3)

For any
$\boldsymbol{\gamma}=(\gamma_{1},\ldots,\gamma_{r},\gamma_{-1},\ldots,\gamma_{-s})\in\mathbb{Z}^{r+s}$,
the coefficient $T_{\boldsymbol{\gamma}}$ about $u_{1}^{\gamma_{1}}\cdots
u_{r}^{\gamma_{r}}u_{-1}^{\gamma_{-1}}\cdots u_{-s}^{\gamma_{-s}}$ in
Eq.(5.22) is a polynomial tau-function of the BUC hierarchy.

  * 4)

There is a set of Laurent polynomials
$A_{1}(u),\ldots,A_{r}(u),B_{-s}(u^{-1}),\ldots,B_{-1}(u^{-1})$ such that
$\tau$ is the zero-mode of the Eq.(5.22) if $\tau$ is a polynomial tau-
function of the BUC hierarchy.

###### Proof.

  * 1)

A direct calculation gives rise to

$\displaystyle
T(u_{1},u_{2},\ldots,u_{2r},u_{-1}^{-1},u_{-2}^{-1},\ldots,u_{-2s}^{-1})$
(5.31) $\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{2r}A_{j}(u_{j})\prod\limits_{i=-2s}^{-1}B_{i}(u_{i}^{-1})\prod\limits_{{-2s\leq
i<j\leq 2r}\atop{i,j\neq
0}}f(u_{i},u_{j})\prod\limits_{i=-2s}^{-1}Q^{\prime}(u_{i}^{-1})\prod\limits_{j=1}^{2r}Q(u_{j})$
$\displaystyle=$ $\displaystyle
Pf[F]\prod\limits_{i=-2s}^{-1}T^{(i)}(u_{i}^{-1})\prod\limits_{j=1}^{2r}T^{(j)}(u_{j})$
$\displaystyle=$ $\displaystyle\sum\limits_{\sigma\in
S_{2s+2r}}sgn(\sigma)f_{\sigma(-2s)\sigma(-2s+1)}\cdots
f_{\sigma(-2)\sigma(-1)}f_{\sigma(1)\sigma(2)}\cdots
f_{\sigma(2r-1)\sigma(2r)}\cdot$
$\displaystyle\prod\limits_{i=-2s}^{-1}T^{(i)}(u_{i}^{-1})\prod\limits_{j=1}^{2r}T^{(j)}(u_{j})$
$\displaystyle=$ $\displaystyle\sum\limits_{\sigma\in
S_{2s+2r}}sgn(\sigma)f_{\sigma(-2s)\sigma(-2s+1)}T^{(\sigma(-2s))}T^{(\sigma(-2s+1))}\cdots
f_{\sigma(-2)\sigma(-1)}T^{(\sigma(-2))}T^{(\sigma(-1))}$ $\displaystyle
f_{\sigma(1)\sigma(2)}T^{(\sigma(1))}T^{(\sigma(2))}\cdots
f_{\sigma(2r-1)\sigma(2r)}T^{(\sigma(2r-1))}T^{(\sigma(2r))}$ $\displaystyle=$
$\displaystyle Pf[f_{ij}T^{(i)}T^{(j)}]_{{-2s\leq i,j\leq 2r}\atop{i,j\neq
0}}=Pf[T^{(i,j)}]_{{-2s\leq i,j\leq 2r}\atop{i,j\neq 0}}.$

  * 2)

By the definition of $T^{(i,j)}$, after a straightforward calculation, we
obtain

$\displaystyle Pf[T^{(i,j)}]_{{-2s\leq i,j\leq 2r}\atop{i,j\neq
0}}=Pf\left[\sum\limits_{\gamma_{i},\gamma_{j}}T^{(i,j)}_{\gamma_{i},\gamma_{j}}u_{i}^{\gamma_{i}}u_{j}^{\gamma_{j}}\right]_{{-2s\leq
i,j\leq 2r}\atop{i,j\neq 0}}$ (5.32) $\displaystyle=$
$\displaystyle\sum\limits_{\sigma\in
S_{2s+2r}}sgn(\sigma)\sum\limits_{\gamma_{i},\gamma_{j}}T^{(\sigma(-2s),\sigma(-2s+1))}_{\gamma_{\sigma(-2s)},\gamma_{\sigma(-2s+1)}}u_{\sigma(-2s)}^{\gamma_{\sigma(-2s)}}u_{\sigma(-2s+1)}^{\gamma_{\sigma(-2s+1)}}\cdots
T^{(\sigma(-2),\sigma(-1))}_{\gamma_{\sigma(-2)},\gamma_{\sigma(-1)}}u_{\sigma(-2)}^{\gamma_{\sigma(-2)}}u_{\sigma(-1)}^{\gamma_{\sigma(-1)}}$
$\displaystyle
T^{(\sigma(1),\sigma(2))}_{\gamma_{\sigma(1)},\gamma_{\sigma(2)}}u_{\sigma(1)}^{\gamma_{\sigma(1)}}u_{\sigma(2)}^{\gamma_{\sigma(2)}}\cdots
T^{(\sigma(2r-1),\sigma(2r))}_{\gamma_{\sigma(2r-1)},\gamma_{\sigma(2r)}}u_{\sigma(2r-1)}^{\gamma_{\sigma(2r-1)}}u_{\sigma(2r)}^{\gamma_{\sigma(2r)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\gamma}}Pf[T^{(i,j)}_{\gamma_{i},\gamma_{j}}]_{{-2s\leq
i,j\leq 2r}\atop{i,j\neq 0}}u_{1}^{\gamma_{1}}\cdots
u_{2r}^{\gamma_{2r}}u_{-1}^{\gamma_{-1}}\cdots u_{-2s}^{\gamma_{-2s}}.$

Clearly, the coefficient of $u_{1}^{\gamma_{1}}\cdots
u_{2r}^{\gamma_{2r}}u_{-1}^{\gamma_{-1}}\cdots u_{-2s}^{\gamma_{-2s}}$ in
Eq.(5.29) is $Pf[T^{(i,j)}_{\gamma_{i},\gamma_{j}}]_{{-2s\leq i,j\leq
2r}\atop{i,j\neq 0}}$.

  * 3)

Let $A_{j}(u)=\sum\limits_{M_{j}\leq k\leq N_{j}}A_{j,k}u^{j}$ and
$B_{i}(u^{-1})=\sum\limits_{U_{i}\leq m\leq V_{i}}B_{i,m}u^{-m}$ be power
series expansions about variable $u$, where $j=1,\ldots,l,i=-s,\ldots,-1$ and
$M_{j},N_{j},U_{i},V_{i}\in\mathbb{Z}$. From (5.19), we can get

$\displaystyle
T(\check{\mathbf{u}},\check{\mathbf{v}})=\prod\limits_{j=1}^{r}A_{j}(u_{j})\prod\limits_{i=-s}^{-1}B_{i}(u_{i}^{-1})\varphi(u_{1})\cdots\varphi(u_{r})\overline{\varphi}(u_{-1}^{-1})\cdots\overline{\varphi}(u_{-s}^{-1})(1)$
$\displaystyle=\sum\limits_{M_{1}\leq k_{1}\leq
N_{1}}\sum\limits_{l_{1}\in\mathbb{Z}}A_{1,k_{1}}\varphi_{l_{1}}u_{1}^{k_{1}-l_{1}}\cdots\sum\limits_{M_{r}\leq
k_{r}\leq
N_{r}}\sum\limits_{l_{r}\in\mathbb{Z}}A_{r,k_{r}}\varphi_{l_{r}}u_{r}^{k_{r}-l_{r}}\sum\limits_{U_{-1}\leq
m_{-1}\leq V_{-1}}\sum\limits_{n_{-1}\in\mathbb{Z}}B_{-1,m_{-1}}$
$\displaystyle\overline{\varphi}_{n_{-1}}u_{-1}^{n_{-1}-m_{-1}}\cdots\sum\limits_{U_{-s}\leq
m_{-s}\leq
V_{-s}}\sum\limits_{n_{-s}\in\mathbb{Z}}B_{-s,m_{-s}}\overline{\varphi}_{n_{-s}}u_{-s}^{n_{-s}-m_{-s}}(1)$
$\displaystyle=\sum\limits_{\gamma\in\mathbb{Z}^{r+s}}\sum\limits_{M_{1}-\gamma_{1}\leq
l_{1}\leq
N_{1}-\gamma_{1}}A_{1,\gamma_{1}+l_{1}}\varphi_{l_{1}}u_{1}^{\gamma_{1}}\cdots\sum\limits_{M_{r}-\gamma_{r}\leq
l_{r}\leq
N_{r}-\gamma_{r}}A_{r,\gamma_{r}+l_{r}}\varphi_{l_{r}}u_{r}^{\gamma_{r}}$
$\displaystyle\sum\limits_{U_{-1}+\gamma_{-1}\leq n_{-1}\leq
V_{-1}+\gamma_{-1}}B_{-1,n_{-1}-\gamma_{-1}}\overline{\varphi}_{n_{-1}}u_{-1}^{\gamma_{-1}}\cdots\sum\limits_{U_{-s}+\gamma_{-s}\leq
n_{-s}\leq
V_{-s}+\gamma_{-s}}B_{-s,n_{-s}-\gamma_{-s}}\overline{\varphi}_{n_{-s}}u_{-s}^{\gamma_{-s}}(1).$

Thus the coefficient $T_{\boldsymbol{\gamma}}$ of $u_{1}^{\gamma_{1}}\cdots
u_{r}^{\gamma_{r}}u_{-1}^{\gamma_{-1}}\cdots u_{-s}^{\gamma_{-s}}$ can be
written as $X_{1}\cdots X_{r}Y_{-1}\cdots Y_{-s}(1)$, where

$\displaystyle X_{i}=\sum\limits_{M_{i}-\gamma_{i}\leq l_{i}\leq
N_{i}-\gamma_{i}}A_{i,\gamma_{i}+l_{i}}\varphi_{l_{i}},\quad i=1,\ldots,r,$
(5.34) $\displaystyle Y_{i}=\sum\limits_{U_{i}+\gamma_{i}\leq n_{i}\leq
V_{i}+\gamma_{i}}B_{i,n_{i}-\gamma_{i}}\overline{\varphi}_{n_{i}},\quad
i=-s,\ldots,-1.$

By Remark 5.3 and Lemma 5.6, we conclude that the coefficient
$T_{\boldsymbol{\gamma}}$ is a tau-function of the BUC hierarchy.
$T_{\boldsymbol{\gamma}}$ is a polynomial tau-function because it is a finite
linear combination of
$\varphi_{l_{1}}\cdots\varphi_{l_{r}}\overline{\varphi}_{n_{-1}}\cdots\overline{\varphi}_{n_{-s}}(1)$.

  * 4)

Polynomial tau-function of the BUC hierarchy has the form

$\displaystyle\tau=X_{1}\cdots X_{r}Y_{-1}\cdots Y_{-s}(1),$ (5.35)

where
$[\lambda,\mu]=[(\lambda_{1},\lambda_{2},\ldots,\lambda_{r}),(-\lambda_{-1},-\lambda_{-2},\ldots,-\lambda_{-s})]$
is a pair of partitions, and

$\displaystyle X_{i}=\sum\limits_{-\lambda_{i}\leq m\leq
N_{i}}d_{m,i}\varphi_{m},\quad d_{m,i}\in\mathbb{C},b_{-\lambda_{i},i}\neq
0,N_{i}\in\mathbb{Z},i=1,\ldots,r,$ $\displaystyle
Y_{i}=\sum\limits_{-\lambda_{i}\leq h\leq
V_{i}}e_{h,i}\overline{\varphi}_{h},\quad
e_{h,i}\in\mathbb{C},e_{-\lambda_{i},i}\neq
0,V_{i}\in\mathbb{Z},i=-s,\ldots,-1.$ (5.36)

For a vector
$\boldsymbol{\gamma}=(\gamma_{1},\ldots,\gamma_{r},\gamma_{-1},\ldots,\gamma_{-s})\in\mathbb{Z}^{r+s}$,
we define $A_{i}(u)$ and $B_{i}(u^{-1})$ in the Eq.(5.21) to be the Laurent
polynomial with the following form

$\displaystyle A_{i}(u)=\sum\limits_{\gamma_{i}-\lambda_{i}\leq t\leq
N_{i}+\gamma_{i}}d_{t-\gamma_{i},i}u^{t},\quad\quad\quad\quad i=1,\ldots,r,$
$\displaystyle B_{i}(u^{-1})=\sum\limits_{-\gamma_{i}-\lambda_{i}\leq w\leq
V_{i}-\gamma_{i}}e_{w+\gamma_{i},i}u^{-w},\quad i=-s,\ldots,-1.$ (5.37)

By using $A_{1}(u),\ldots,A_{r}(u)$ and
$B_{-s}(u^{-1}),\ldots,B_{-1}(u^{-1})$, it is easy to verify that Eq. (5.34)
leads to (4)). The coefficient $T_{\boldsymbol{\gamma}}$ corresponds to the
polynomial tau-function (5.35). It is showed that $\tau$ is the zero-mode of
the series expansion of $T(\check{\mathbf{u}},\check{\mathbf{v}})$ with
$\gamma_{1}=\cdots=\gamma_{r}=\gamma_{-1}=\cdots=\gamma_{-s}=0$ .

∎

######  Corollary 5.9.

  * 1)

We have proved that the polynomial tau-functions of the BUC hierarchy are
zero-mode of certain generating functions
$T(\check{\mathbf{u}},\check{\mathbf{v}})$. By replacing $A_{j}(u)$ with
$u^{\gamma_{j}}A_{j}(u)\ (j=1,\ldots,r)$ and $B_{i}(u^{-1})$ with
$u^{-\gamma_{i}}B_{i}(u^{-1})\ (i=-s,\ldots,-1)$, we derive the any polynomial
tau-function as a coefficient of a given monomial $u_{1}^{\gamma_{1}}\cdots
u_{r}^{\gamma_{r}}u_{-1}^{\gamma_{-1}}\cdots u_{-s}^{\gamma_{-s}}$.

  * 2)

Introducing

$\displaystyle
A_{j}(u)=h_{j}\sum\limits_{i=0}^{M_{j}}a_{j,i}u^{i},\quad\quad\quad\
M_{j}\in\mathbb{Z},h_{j},a_{j,i}\in\mathbb{C},a_{j,0}=1,j=1,\ldots,r,$ (5.38)
$\displaystyle B_{i}(u^{-1})=g_{i}\sum\limits_{m=0}^{M_{i}}b_{i,m}u^{-m},\quad
M_{i}\in\mathbb{Z},g_{i},b_{i,m}\in\mathbb{C},b_{i,0}=1,i=-s,\ldots,-1,$

where $A_{1}(u),\ldots A_{r}(u),B_{-1}(u_{-1}),\ldots,B_{-s}(u_{-s})$ are non-
zero Laurent series defined in the $T(\check{\mathbf{u}},\check{\mathbf{v}})$.
By means of (2.3), we have

$\displaystyle\sum\limits_{i=0}^{M_{j}}a_{j,i}u^{i}=\exp\left(\sum\limits_{s=1}^{\infty}c_{j,s}u^{s}\right),\quad\quad\sum\limits_{m=0}^{M_{i}}b_{i,m}u^{-m}=\exp\left(\sum\limits_{l=1}^{\infty}c^{\prime}_{i,l}u^{-l}\right),$
(5.39)

and

$\displaystyle a_{j,i}=S_{i}(c_{j,1},c_{j,2},\ldots),\quad\quad
b_{i,m}=S_{m}(c^{\prime}_{i,1},c^{\prime}_{i,2},\ldots),$ (5.40)

where $c_{j,s}$ and $c^{\prime}_{i,l}$ are constants in $\mathbb{C}$.

Setting
$(\widetilde{x}_{1},\widetilde{x}_{2},\widetilde{x}_{3},\ldots)=(2p_{1},0,\frac{2}{3}p_{3},0,\ldots)$
and
$(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{2},\widetilde{x}^{\prime}_{3},\ldots)=(2p^{\prime}_{1},0,\frac{2}{3}p^{\prime}_{3},0,\ldots)$,
we get

$\displaystyle\ T^{(j)}(u_{j})$ $\displaystyle=$ $\displaystyle
A_{j}(u_{j})Q(u_{j})=h_{j}\exp\left(\sum\limits_{k\geq
1}c_{j,k}u_{j}^{k}\right)\exp\left(\sum\limits_{k\geq
1}\widetilde{x}_{k}u_{j}^{k}\right)$ $\displaystyle=$ $\displaystyle
h_{j}\sum\limits_{k=0}^{\infty}S_{k}(\widetilde{x}_{1}+c_{j,1},\widetilde{x}_{2}+c_{j,2},\ldots)u_{j}^{k},\quad
j=1,\ldots,r,$ $\displaystyle T^{(i)}(u_{i}^{-1})$ $\displaystyle=$
$\displaystyle
B_{i}(u_{i}^{-1})Q^{\prime}(u_{i}^{-1})=g_{i}\exp\left(\sum\limits_{l\geq
1}c^{\prime}_{i,l}u_{i}^{-l}\right)\exp\left(\sum\limits_{l\geq
1}\widetilde{x}^{\prime}_{l}u_{i}^{-l}\right)$ (5.41) $\displaystyle=$
$\displaystyle
g_{i}\sum\limits_{l=0}^{\infty}S_{l}(\widetilde{x}^{\prime}_{1}+c^{\prime}_{i,1},\widetilde{x}^{\prime}_{2}+c^{\prime}_{i,2},\ldots)u_{i}^{-l},\quad
i=-s,\ldots,-1.$

Hence, $T^{(i,j)}$ can be written as

$\displaystyle\ T^{(i,j)}=\left\\{\begin{aligned}
&h_{i}h_{j}\left(1+2\sum\limits_{k\geq
1}(-1)^{k}\frac{u_{j}^{k}}{u_{i}^{k}}\right)\sum\limits_{m,n\in\mathbb{Z}}S_{m}\left(\widetilde{x}+c_{i}\right)S_{n}\left(\widetilde{x}+c_{j}\right)u_{i}^{m}u_{j}^{n},\quad\quad\
0<i<j\leq 2r,\\\ &g_{i}g_{j}\left(1+2\sum\limits_{k\geq
1}(-1)^{k}\frac{u_{j}^{k}}{u_{i}^{k}}\right)\sum\limits_{m,n\in\mathbb{Z}}S_{m}\left(\widetilde{x}^{\prime}+c^{\prime}_{i}\right)S_{n}\left(\widetilde{x}^{\prime}+c^{\prime}_{j}\right)u_{i}^{-m}u_{j}^{-n},\quad-2s\leq
i<j<0,\\\ &h_{i}g_{j}\left(1+2\sum\limits_{k\geq
1}(-1)^{k}\frac{u_{j}^{k}}{u_{i}^{k}}\right)\sum\limits_{m,n\in\mathbb{Z}}S_{m}\left(\widetilde{x}^{\prime}+c^{\prime}_{i}\right)S_{n}\left(\widetilde{x}+c_{j}\right)u_{i}^{-m}u_{j}^{n},-2s\leq
i<0<j\leq 2r.\end{aligned}\right.$ (5.42)

Expanding $T^{(i,j)}$, we can obtain the expression of the $T^{(i,j)}_{m,n}$

$\displaystyle T^{(i,j)}_{m,n}=\left\\{\begin{aligned}
&2h_{i}h_{j}\mathcal{X}^{(1)}_{m,n}\left(\widetilde{x}+c_{i},\widetilde{x}+c_{j}\right),\quad\quad\quad
0<i<j\leq 2r,\\\
&2g_{i}g_{j}\mathcal{X}^{(2)}_{m,n}\left(\widetilde{x}^{\prime}+c^{\prime}_{i},\widetilde{x}^{\prime}+c^{\prime}_{j}\right),\quad\quad-2s\leq
i<j<0,\\\
&2h_{i}g_{j}\mathcal{X}^{(3)}_{m,n}\left(\widetilde{x}^{\prime}+c^{\prime}_{i},\widetilde{x}+c_{j}\right),\quad-2s\leq
i<0<j\leq 2r,\end{aligned}\right.$ (5.43)

where $T^{(i,j)}_{m,n}=-T^{(j,i)}_{m,n}$ for $i>j$, $T^{(i,i)}_{m,n}=0$, and

$\displaystyle\mathcal{X}^{(1)}_{m,n}\left(\widetilde{x}+c_{i},\widetilde{x}+c_{j}\right)=\frac{1}{2}S_{m}(\widetilde{x}+c_{i})S_{n}(\widetilde{x}+c_{j})+\sum\limits_{k\geq
1}(-1)^{k}S_{m+k}(\widetilde{x}+c_{i})S_{n-k}(\widetilde{x}+c_{j}),$
$\displaystyle\mathcal{X}^{(2)}_{m,n}\left(\widetilde{x}^{\prime}+c^{\prime}_{i},\widetilde{x}^{\prime}+c^{\prime}_{j}\right)=\frac{1}{2}S_{m}(\widetilde{x}^{\prime}+c^{\prime}_{i})S_{n}(\widetilde{x}^{\prime}+c^{\prime}_{j})+\sum\limits_{k\geq
1}(-1)^{k}S_{m-k}(\widetilde{x}^{\prime}+c^{\prime}_{i})S_{n+k}(\widetilde{x}^{\prime}+c^{\prime}_{j}),$
$\displaystyle\mathcal{X}^{(3)}_{m,n}\left(\widetilde{x}^{\prime}+c^{\prime}_{i},\widetilde{x}+c_{j}\right)=\frac{1}{2}S_{m}(\widetilde{x}^{\prime}+c^{\prime}_{i})S_{n}(\widetilde{x}+c_{j})++\sum\limits_{k\geq
1}(-1)^{k}S_{m-k}(\widetilde{x}^{\prime}+c^{\prime}_{i})S_{n-k}(\widetilde{x}+c_{j}).$
(5.44)

In order to facilitate expression, some new symbols will be introduced. Define
the skew-symmetric matrix $M=(m_{i,j})_{0<i,j\leq 2r}$ by putting each
$(i,j)$-th as
$m_{i,j}=2h_{i}h_{j}\mathcal{X}^{(1)}_{m,n}\left(\widetilde{x}+c_{i},\widetilde{x}+c_{j}\right)$
for $i<j$ and $m_{i,j}=-m_{j,i}$ for $i>j$, $m_{i,i}=0$. Similarly, define
another skew-symmetric matrix $\overline{M}=(\overline{m}_{i,j})_{-2s\leq
i,j<0}$, where
$\overline{m}_{i,j}=2g_{i}g_{j}\mathcal{X}^{(2)}_{m,n}\left(\widetilde{x}^{\prime}+c^{\prime}_{i},\widetilde{x}^{\prime}+c^{\prime}_{j}\right)$
for $i<j$. Introduce the third matrix $N=(n_{i,j})_{-2s\leq i<0,0<j\leq 2r}$,
where
$n_{i,j}=2h_{i}g_{j}\mathcal{X}^{(3)}_{m,n}\left(\widetilde{x}^{\prime}+c^{\prime}_{i},\widetilde{x}+c_{j}\right)$.
From Theorem 5.8, polynomial tau-functions of the BUC hierarchy have the form

$\displaystyle T_{\boldsymbol{\gamma}}=Pf\begin{bmatrix}\overline{M}&N\\\
-N^{T}&M\end{bmatrix}_{-2s\leq i,j\leq 2r.}$ (5.45)

######  Remark 5.10.

It is noted that the polynomial tau-functions of the BUC hierarchy reduce to
the solutions of the BKP hierarchy [34] with the reduction $\mathbf{y}=0$.

##  6\. Conclusions and discussions

In this paper, we have discussed exact solutions of the SKP, OKP and BUC
hierarchies including the polynomial-type and soliton-type solutions. It is
showed that the generating functions play a vital role in establishing the
polynomial tau-functions of the integrable systems. Furthermore, we expressed
the polynomial tau-functions of the SKP, OKP and BUC hierarchies as
determinant and Pfaffian forms, respectively. The results here are hoped to be
helpful for better understanding the essential properties of the SKP, OKP and
BUC hierarchies. It is known that symplectic universal character (SUC) and
orthogonal universal character (OUC) hierarchies are the extensions of the SKP
and OKP hierarchies. However, it should be pointed out that we have not
expressed the polynomial tau-functions of the SUC and OUC hierarchies as a
perfect determinant form due to the inappropriate quantum fields presentation
of SUC and OUC. We will concentrate on studying this interesting question in
the near future.

##  7\. Acknowledgements

This work is partially supported by National Natural Science Foundation of
China (Grant Nos. 11965014 and 12061051) and National Science Foundation of
Qinghai Province (Grant No. 2021-ZJ-708). The authors gratefully acknowledge
the support of Professor Ke Wu and Professor Weizhong Zhao (CNU, China).

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