Polynomial Tau-Functions of the n-th Sawada–Kotera Hierarchy

Abstract

We give a review of the B-type Kadomtsev–Petviashvili (BKP) hierarchy and find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada–Kotera hierarchy). The name comes from the fact that, for $n=3$, the simplest equation of the hierarchy is the famous Sawada–Kotera equation.

1. Introduction

The three most famous hierarchies of Lax equations on one function u are the Korteweg–de Vries (KdV) hierarchy, the Kaup–Kupershmidt hierarchy, and the Sawada–Kotera hierarchy. The Lax operators are, respectively,

$$\begin{array}{c}\hfill \mathcal{L}={\partial }^2+u,\end{array}$$

(1)

$$\begin{array}{c}\hfill \mathcal{L}={\partial }^3+u\partial +\frac{1}{2}{u}^{\prime },\end{array}$$

(2)

$$\begin{array}{c}\hfill \mathcal{L}={\partial }^3+u\partial .\end{array}$$

(3)

Let $t=(t_1,t_2,t_3,\dots )$ and $\tilde{t}=(t_1,t_3,t_5,\dots )$. Recall that the Kadomtsev–Petviashvili (KP) hierarchy is the following hierarchy of Lax equations on the pseudodifferential operator $L(t,\partial )=\partial +u_1\left(t\right){\partial }^{-1}+u_2\left(t\right){\partial }^{-2}+\cdots$ in $\partial =\frac{\partial }{\partial t_1}$ [1]:

$$\frac{\partial L(t,\partial )}{\partial t_k}=\left[{\left(L{(t,\partial )}^k\right)}_{\ge 0},L(t,\partial )\right],k=1,2,\dots .$$

(4)

The KdV hierarchy is the second reduced KP hierarchy, meaning that one imposes the following constraint on $L(t,\partial )$:

$$\mathcal{L}(t,\partial )=L{(t,\partial )}^2\mathrm{is} \mathrm{a} \mathrm{differential} \mathrm{operator}.$$

(5)

In this case, the operator $\mathcal{L}$ is defined by (1), with $u\left(t\right)=2u_1\left(t\right)$, and the KP hierarchy (4) reduces to the KdV hierarchy

$$\frac{\partial \mathcal{L}(t,\partial )}{\partial t_k}=\left[{\left(\mathcal{L}{(t,\partial )}^{\frac{k}{2}}\right)}_{\ge 0},\mathcal{L}(t,\partial )\right],k=1,3,5,\dots .$$

(6)

For even k, this equation is trivial; for $k=3$, Equation (6) is the KdV equation [2].

Recall that in order to construct solutions of the KP hierarchy and the reduced KP hierarchies, one introduces the tau-function $\tau \left(t\right)$, defined by [1,2]:

$$L(t,\partial )=P(t,\partial )\circ \partial \circ P{(t,\partial )}^{-1},$$

(7)

where $P(t,\partial )$ is a pseudodifferential operator, with the symbol

$$P(t,z)=\frac{1}{\tau \left(t\right)}exp\left(-\sum _{k=1}^{\infty }\frac{z^{-k}}{k}\frac{\partial }{\partial t_k}\right)\left(\tau \left(t\right)\right).$$

(8)

The tau-function has a geometric meaning as a point on an infinite-dimensional Grassmannian, and in [1], Sato showed that all Schur polynomials are tau-functions of the KP hierarchy. Recently, all polynomial tau-functions of the KP hierarchy and its n-reductions have been constructed in [3] (see also [4]).

The CKP hierarchy (KP hierarchy of type C) can be constructed by making use of the KP hierarchy, and assuming the additional constraint $L{(\tilde{t},\partial )}^{\ast }=-L(\tilde{t},\partial )$ (see, e.g., [5] for details). Its 3-reduction is defined by the constraint that $\mathcal{L}(\tilde{t},\partial )=L{(\tilde{t},\partial )}^3$ is a differential operator, and the corresponding hierarchy is

$$\frac{\partial \mathcal{L}(\tilde{t},\partial )}{\partial t_k}=\left[{\left(\mathcal{L}{(\tilde{t},\partial )}^{\frac{k}{3}}\right)}_{\ge 0},\mathcal{L}(\tilde{t},\partial )\right],k=1,3,5,\dots ,k\notin 3\mathbb{Z},$$

(9)

where $\mathcal{L}$ is given by (2). For $k=5$, we obtain the Kaup–Kupershmidt equation, the simplest non-trivial equation in this hierarchy. All polynomial tau-functions of (9) (and all n reductions of the CKP hierarchy) have been constructed in [5].

In the present paper, we construct all polynomial tau-functions of the n-reduced BKP hierarchies (KP hierarchy of type B). These are hierarchies of Lax equations on the differential operator

$$\mathcal{L}(\tilde{t},\partial )={\partial }^n+u_{n-2}\left(\tilde{t}\right){\partial }^{n-2}+\cdots +u_1\left(\tilde{t}\right)\partial ,$$

(10)

satisfying the constraint

$$\mathcal{L}{(\tilde{t},\partial )}^{\ast }={(-1)}^n{\partial }^{-1}\mathcal{L}(\tilde{t},\partial )\partial .$$

(11)

The n-th reduced BKP hierarchy is

$$\frac{\partial \mathcal{L}(\tilde{t},\partial )}{\partial t_k}=\left[{\left(\mathcal{L}{(\tilde{t},\partial )}^{\frac{k}{n}}\right)}_{\ge 0},\mathcal{L}(\tilde{t},\partial )\right],k=1,3,5,\dots ,k\notin n\mathbb{Z}.$$

(12)

We call it the n-th Sawada–Kotera hierarchy, since, for $n=3$, $\mathcal{L}$ is given by (3), and for $k=5$, Equation (12) is the Sawada–Kotera equation [6] (see Equation (33)).

In the present paper, using the description of polynomial tau-functions of the BKP hierarchy [4,7] (see Theorem 1), we find all polynomial tau-functions for the n-th Sawada–Kotera hierarchies (see Theorem 2), and, in particular, for the Sawada–Kotera hierarchy (see Corollary 1).

2. The BKP Hierarchy and Its Polynomial Tau-Functions

In this section, we recall the construction of the BKP hierarchy [8] and description of its polynomial tau-functions from [4,7].

Following Date, Jimbo, Kashiwara, and Miwa [8] (see also [7] for details), we introduce the BKP hierarchy in terms of the so-called twisted neutral fermions ${\varphi }_i$, $i\in \mathbb{Z}$, which are generators of a Clifford algebra over $\mathbb{C}$, satisfying the following anti-commutation relation:

$${\varphi }_i{\varphi }_j+{\varphi }_j{\varphi }_i={(-1)}^i{\delta }_{i,-j}.$$

(13)

Consider the right (resp., left) irreducible module $F=F_r$ (resp., $F_l$) over this algebra by the following action on the vacuum vector $|0⟩$ (resp., $⟨0|$):

$${\varphi }_0|0⟩=\frac{1}{\sqrt{2}}|0⟩,{\varphi }_j|0⟩=0\left(\mathrm{resp}.,⟨0|{\varphi }_0=\frac{1}{\sqrt{2}}⟨0|,⟨0|{\varphi }_{-j}=0\right),\mathrm{for}j>0.$$

(14)

The quadratic elements

$${\varphi }_j{\varphi }_k-{\varphi }_k{\varphi }_j\mathrm{for}j,k\in \mathbb{Z},j>k,$$

form a basis of the infinite-dimensional Lie algebra $s{o}_{\infty ,odd}$ over $\mathbb{C}$. Let $S{O}_{\infty ,odd}$ be the corresponding Lie group. We proved in ([9], Theorem 1.2a) that a non-zero element $\tau \in F$ lies in this Lie group orbit of the vacuum vector $|0⟩$ if and only if it satisfies the BKP hierarchy in the fermionic picture, i.e., the following equation in $F\otimes F$:

$$\sum _{j\in \mathbb{Z}}{(-1)}^j{\varphi }_j\tau \otimes {\varphi }_{-j}\tau =\frac{1}{2}\tau \otimes \tau .$$

(15)

Non-zero elements of F, satisfying (15), are called tau-functions of the BKP hierarchy in the fermionic picture.

The group $S{O}_{\infty ,odd}$ consists of elements G leaving the symmetric bilinear form

$$({\varphi }_j,{\varphi }_k)={(-1)}^j{\delta }_{j,-k}$$

(16)

on F invariant, i.e.,

$$(G{\varphi }_j,G{\varphi }_k)=({\varphi }_j,{\varphi }_k).$$

(17)

Stated differently,

$$G{\varphi }_k{G}^{-1}=\sum _{j_{\in }\mathbb{Z}}{a}_{jk}{\varphi }_k(\mathrm{finite} \mathrm{sum}) \mathrm{with}\sum _{j\in \mathbb{Z}}{(-1)}^j{a}_{jk}{a}_{-j\ell }={(-1)}^k{\delta }_{k,-\ell }.$$

(18)

The group orbit of the vacuum vector is the disjoint union of Schubert cells (see Section 3 of [7] for details). These cells are parametrized by the strict partitions $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k)$, with ${\lambda }_1>{\lambda }_2>.\cdots >{\lambda }_k>0$. Namely, the cell, attached to the partition $\lambda$ is

$$C_{\lambda }=\{v_1{v}_2\cdots v_{k-1}{v}_k|0⟩|v_j=\sum _{i\ge -{\lambda }_j}{a}_{ij}{\varphi }_i(\mathrm{finite} \mathrm{sum}) \mathrm{with}{a}_{-{\lambda }_j,j}\ne 0\}.$$

(19)

An element $\tau \in C_{\lambda }$ corresponds to the following point in the maximal isotropic Grassmannian (i.e., a maximal isotropic subspace of $V={⨁}_{j\in \mathbb{Z}}\mathbb{C}{\varphi }_j$):

$$\mathrm{Ann}\tau =\{v\in V|v{v}_1{v}_2\cdots v_{k-1}{v}_k|0⟩=0\}.$$

(20)

For instance, $\mathrm{Ann}|0⟩=\mathrm{span}\{{\varphi }_1,{\varphi }_2,\dots \}.$

Using the bosonization of Equation (15), one obtains a hierarchy of differential equations on $\tau$ ([4,7,8,10], Section 3). This bosonization is an isomorphism $\sigma$ between the spin module F and the polynomial algebra $B=\mathbb{C}\left[\tilde{t}\right]=\mathbb{C}[t_1,t_3,t_5,\dots ]$. Explicitly, we introduce the twisted neutral fermionic field

$$\varphi \left(z\right)=\sum _{j\in \mathbb{Z}}{\varphi }_j{z}^{-j},$$

and the bosonic field

$$\alpha \left(z\right)=\sum _{j\in \mathbb{Z}}{\alpha }_{2j+1}{z}^{-2j-1}=\frac{1}{2}:\varphi \left(z\right)\varphi (-z):,$$

where the normal ordering $::$ is defined by

$$:{\varphi }_j{\varphi }_k:={\varphi }_j{\varphi }_k-⟨0|{\varphi }_j{\varphi }_k|0⟩;$$

equivalently: $:{\varphi }_j{\varphi }_k:={\varphi }_j{\varphi }_k$ if $j\le k$ and $=-{\varphi }_k{\varphi }_j$ if $j>k$, except when $j=k=0$, then $:{\varphi }_0{\varphi }_0:=0$. The operators ${\alpha }_j$ satisfy the commutation relations of the Heisenberg Lie algebra

$$[{\alpha }_j,{\alpha }_k]=\frac{j}{2}{\delta }_{j-k},{\alpha }_i|0⟩=⟨0|{\alpha }_{-i}=0,\mathrm{for} i>0,$$

(21)

and its representation on F is irreducible ([9], Theorem 3.2). Using this, we obtain a vector space isomorphism $\sigma :F\to B$, uniquely defined by the following relations:

$$\sigma \left(\right|0⟩)=1,\sigma {\alpha }_j{\sigma }^{-1}=\frac{\partial }{\partial t_j},\sigma {\alpha }_{-j}{\sigma }^{-1}=\frac{j}{2}{t}_j,\mathrm{for} j>0\mathrm{odd}.$$

(22)

Explicitly ([9], Section 3.2):

$$\sigma \varphi \left(z\right){\sigma }^{-1}=\frac{1}{\sqrt{2}}exp\sum _{j=1}^{\infty }{t}_{2j-1}{z}^{2j-1}exp\sum _{j=1}^{\infty }-2\frac{\partial }{\partial t_{2j-1}}\frac{z^{-2j+1}}{2j-1}.$$

(23)

Since (15) can be rewritten as

$${\mathrm{Res}}_z\varphi \left(z\right)\tau \otimes \varphi (-z)\tau \frac{dz}{z}=\frac{1}{2}\tau \otimes \tau ,$$

under the isomorphism $\sigma$, Equation (15) turns into:

$${\mathrm{Res}}_z{e}^{{\sum }_{j=1}^{\infty }(t_{2j-1}-t_{2j-1}^{\prime })z^{2j-1}}{e}^{{\sum }_{j=1}^{\infty }2\left(\frac{\partial }{\partial t_{2j-1}^{\prime }}-\frac{\partial }{\partial t_{2j-1}}\right)\frac{z^{-2j+1}}{2j-1}}\tau \left(\tilde{t}\right)\tau \left({\tilde{t}}^{\prime }\right)\frac{dz}{z}=\tau \left(\tilde{t}\right)\tau \left({\tilde{t}}^{\prime }\right),$$

(24)

where $\tilde{t}=(t_1,t_3,t_5,\dots )$ and ${\tilde{t}}^{\prime }=(t_1^{\prime },t_3^{\prime },t_5^{\prime },\dots )$. Therefore, $\tau \left(\tilde{t}\right)$ is the vacuum expectation value

$$\tau \left(\tilde{t}\right)=\sigma \tau {\sigma }^{-1}=⟨0|e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}\tau .$$

(25)

Furthermore, by making a change of variables, as in [10] (p. 972), viz. $t_{2k-1}=x_{2k-1}-y_{2k-1}$ and $t_{2k-1}^{\prime }=x_{2k-1}^{\prime }-y_{2k-1}^{\prime }$, and using the elementary Schur polynomials $s_j\left(r\right)$, which are defined by

$$exp\sum _{k=1}^{\infty }{r}_k{z}^k=\sum _{j=0}^{\infty }{s}_j\left(r\right)z^j,$$

(26)

we can rewrite (24), where we assume $x_{2k}=y_{2k}=0$:

$$\sum _{j=1}^{\infty }{s}_j(-2\tilde{y})s_j\left(2{\tilde{\partial }}_y\right)\tau (\tilde{x}-\tilde{y})\tau (\tilde{x}+\tilde{y})=0,$$

(27)

where $\tilde{y}=(y_1,0,y_3,0,\dots )$ and ${\tilde{\partial }}_y=(\frac{\partial }{\partial y_1},0,\frac{1}{3}\frac{\partial }{\partial y_3},0,\frac{1}{5}\frac{\partial }{\partial y_5},\dots )$. Using Taylor’s formula, we thus obtain the BKP hierarchy of Hirota bilinear equations [10] (p. 972):

$$\sum _{j=1}^{\infty }{s}_j(-2\tilde{y})s_j\left(2{\tilde{\partial }}_u\right)exp\sum _{j=1}^{\infty }{y}_{2j-1}\frac{\partial }{\partial u_{2j-1}}\tau (\tilde{x}-\tilde{u})\tau (\tilde{x}+\tilde{u}){|}_{\tilde{u}=0}=0.$$

(28)

Using the notation $p\left(D\right)f·g=p(\frac{\partial }{\partial u_1},\frac{\partial }{\partial u_2},\dots )f(\tilde{x}+\tilde{u})g(\tilde{x}-\tilde{u}){|}_{\tilde{u}=0}$, this turns into

$$\sum _{j=1}^{\infty }{s}_j(-2\tilde{y})s_j\left(2\tilde{D}\right)e^{{\sum }_{j=1}^{\infty }{y}_{2j-1}{D}_{2j-1}}\tau ·\tau =0.$$

(29)

The simplest equation in this hierarchy is ([10], Appendix 3):

$$(D_1^6-5D_1^3{D}_3-5D_3^2+9D_1{D}_5)\tau ·\tau =0.$$

(30)

If we assume that our tau-function does not depend on $t_3$, then this gives

$$(D_1^6+9D_1{D}_5)\tau ·\tau =0.$$

(31)

Letting $x=t_1$, $t=\frac{1}{9}{t}_5$, and

$$u(x,t)=2\frac{{\partial }^{2}log\tau (x,t)}{\partial x^2},$$

(32)

and viewing the remaining $t_j$ as parameters, Equation (31) turns into the famous Sawada–Kotera equation [6]:

$$u_t+15(u{u}_{xxx}+u_x{u}_{xx}+3u^2{u}_x)+u_{xxxxx}=0.$$

(33)

Another approach is by using the wave function; see [8] (p. 345),

$$\begin{array}{cc}\hfill w(\tilde{t},z)& =\frac{1}{\tau \left(t\right)}exp\sum _{j=1}^{\infty }{t}_{2j-1}{z}^{2j-1}exp-\sum _{j=1}^{\infty }2\frac{\partial }{\partial t_{2j-1}}\frac{z^{-2j+1}}{2j-1}\tau \left(t\right)\hfill \\ \hfill & =P(\tilde{t},z)e^{{\sum }_{j=1}^{\infty }\left(t_{2j-1}\right)z^{2j-1}},\hfill \end{array}$$

(34)

where $P(\tilde{t},z)=1+{\sum }_{j=1}^{\infty }{p}_j\left(\tilde{t}\right)z^{-j}$, and, in particular,

$$p_1\left(\tilde{t}\right)=-2\frac{\partial log\tau \left(\tilde{t}\right)}{\partial t_1}.$$

(35)

Letting $P(\tilde{t},\partial )$ be the pseudodifferential operator in $\partial =\frac{\partial }{\partial t_1}$ with the symbol $P(\tilde{t},z)$, Equation (24) turns into

$${\mathrm{Res}}_{z}P(\tilde{t},\partial )P({\tilde{t}}^{\prime },{\partial }^{\prime })e^{{\sum }_{j=1}^{\infty }(t_{2j-1}-t_{2j-1}^{\prime })z^{2j-1}}\frac{dz}{z}=1.$$

(36)

Now, using the fundamental lemma, Lemma 1.1 of [2] or Lemma 4.1 of [11], we deduce from (36):

$$\begin{array}{cc}\hfill & P(\tilde{t},\partial ){\partial }^{-1}P{(\tilde{t},\partial )}^{\ast }\partial =1,\hfill \\ \hfill & \frac{\partial P(\tilde{t},\partial )}{\partial t_{2j-1}}=-{\left(P(\tilde{t},\partial ){\partial }^{2j-1}P{(\tilde{t},\partial )}^{-1}{\partial }^{-1}\right)}_{<0}\partial P(\tilde{t},\partial ),j=1,2,\dots .\hfill \end{array}$$

(37)

Next, introducing the Lax operator

$$L(\tilde{t},\partial )=P(\tilde{t},\partial )\partial P{(\tilde{t},\partial )}^{-1}=\partial +u_1\left(\tilde{t}\right){\partial }^{-1}+u_2\left(\tilde{t}\right){\partial }^{-2}+\cdots ,$$

we deduce from (37) that L satisfies [8]

$$\begin{array}{cc}\hfill L{(\tilde{t},\partial )}^{\ast }& =-{\partial }^{-1}L(\tilde{t},\partial )\partial ,\hfill \\ \hfill \frac{\partial L(\tilde{t},\partial )}{\partial t_{2j-1}}& =\left[{\left(L(\tilde{t},\partial ){)}^{2j-1}\right)}_{\ge 0},L(\tilde{t},\partial )\right],j=1,2,\dots .\hfill \end{array}$$

(38)

Note that, since $u_1\left(\tilde{t}\right)=-\frac{\partial p_1\left(\tilde{t}\right)}{\partial t_1}$ and the fact that $p_1\left(t\right)$ is given by (35), we find that

$$u_1\left(\tilde{t}\right)=2\frac{{\partial }^{2}log\tau \left(\tilde{t}\right)}{\partial t_1^2},$$

(39)

which explains the choice (32) of $u(x,t)$ to obtain the Sawada–Kotera equation from the Hirota bilinear Equation (31).

To obtain the second equation of (38), we use (37) and the first equation of (38), which is equivalent (see [8], (p. 356)) to the fact that $L{(\tilde{t},\partial )}^{2j-1}$, for $j=1,2,\dots$, has zero constant term. Let us prove that the first equation of (38) indeed implies this fact. We have

$$L^k{\partial }^{-1}={(-{\partial }^{-1}{L}^{\ast }\partial )}^k{\partial }^{-1}={(-1)}^k{\partial }^{-1}{L}^{\ast k}={(-1)}^{k+1}{\left(L^k{\partial }^{-1}\right)}^{\ast }.$$

Now, using the fact that the constant term of $L^k$ is equal to

$${\mathrm{Res}}_{\partial }{L}^k{\partial }^{-1}=-{\mathrm{Res}}_{\partial }{\left(L^k{\partial }^{-1}\right)}^{\ast }={\mathrm{Res}}_{\partial }{(-1)}^{k+1}{\left(L^k{\partial }^{-1}\right)}^{\ast }={(-1)}^k{\mathrm{Res}}_{\partial }{L}^k{\partial }^{-1},$$

we find that the constant term of $L^k$ is zero whenever k is odd.

Remark 1.

Note that this also means that we can replace the second equation of (38) by, cf. [12],

$$\frac{\partial L(\tilde{t},\partial )}{\partial t_{2j-1}}=\left[{\left(L(\tilde{t},\partial ){)}^{2j-1}\right)}_{\ge 1},L(\tilde{t},\partial )\right],j=1,2,\dots .$$

In the formulation of Kupershmidt [12], this means that L satisfies not only the KP equation for the odd times, but also his formulation of the modified KP hierachy (only for the odd times).

Next, we describe polynomial tau-functions $\tau (t_1,t_3,\dots )$ of the BKP hierarchy obtained in ([7], Theorem 6) (see also [4]). For that, given integers a and b, $a>b\ge 0$, let

$$\begin{array}{cc}\hfill {\chi }_{a,b}(t,t^{\prime })& =\frac{1}{2}{s}_a\left(t^{\prime }\right)s_b\left(t\right)+\sum _{j=1}^b{(-1)}^j{s}_{a+j}\left(t^{\prime }\right)s_{b-j}\left(t\right),\hfill \\ \hfill {\chi }_{b,a}(t,t^{\prime })& =-{\chi }_{a,b}(t,t^{\prime }),{\chi }_{a,a}(t,t^{\prime })=0,\hfill \end{array}$$

(40)

and let ${\chi }_{a,b}(t,t^{\prime })=0$ if $b<0$. Then

Theorem 1

([7], Theorem 6). All polynomial tau-functions of the BKP hierarchy (24), up to a scalar multiple, are equal to

$${\tau }_{\lambda }\left(\tilde{t}\right)=Pf{\left({\chi }_{{\lambda }_i,{\lambda }_j}(\tilde{t}+c_i,\tilde{t}+c_j)\right)}_{1\le i,j\le 2n},$$

(41)

where $Pf$ is the Pfaffiann of a skew-symmetric matrix, $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{2n})$ is an extended strict partition, i.e., ${\lambda }_1>{\lambda }_2>\cdots >{\lambda }_{2n}\ge 0$, $\tilde{t}=(t_1,0,t_3,0,\dots )$, $c_i=(c_{1i},c_{2i},c_{3i},\dots )$, $c_{ij}\in \mathbb{C}$.

Remark 2.

The connection between the set of strict partitions and the extended strict partitions is as follows. If $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k)$ is a strict partition and k is even, then this partition is equal to the extended strict partition λ. However, if k is odd, the Pfaffian of a $k\times k$ anti-symmetric matrix is equal to 0, hence, in that case, we extend λ by the element 0, i.e., the corresponding extended strict partition is then $({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k,0)$.

3. The $\mathit{n}$-th Sawada–Kotera Hierarchy and Its Polynomial Tau-Functions

As we have seen in Section 2, a necessary condition for a tau-function to give a solution of the Sawada–Kotera equation is that $\frac{\partial \tau \left(\tilde{t}\right)}{\partial t_3}=0$. This means that the tau-function lies in a smaller group orbit of the vacuum vector $|0⟩$. Instead of the $S{O}_{\infty ,odd}$ orbit of the vacuum vector $|0⟩$, we consider the twisted loop group $G_3^{\left(2\right)}$, corresponding to the affine Lie algebra $s{l}_3^{\left(2\right)}$, to obtain the 3-reduced BKP hierarchy [8]. More generally (see also [8]), when $n=2k+1>1$ is odd, the $2k+1$-reduced hierarchy is related to the twisted loop group $G_{2k+1}^{\left(2\right)}$ corresponding to the twisted affine Lie algebra $s{l}_{2k+1}^{\left(2\right)}$. When $n=2k>2$ is even, one has the twisted loop group $G_{2k}^{\left(2\right)}$ corresponding to the affine Lie algebra $s{o}_{2k}^{\left(2\right)}$ [9,13] (see [14] (Chapter 7) for the construction of these Lie algebras). Elements G in this twisted loop group not only satisfy (18), which implies ${\sum }_{j\in \mathbb{Z}}{(-1)}^j{a}_{kj}{a}_{\ell ,-j}={(-1)}^k{\delta }_{k,-\ell }$, but also the n-periodicity condition $a_{i+n,j+n}=a_{ij}$. This means that these group elements also commute with the operator

$$\sum _{i\in \mathbb{Z}}{(-1)}^{pn-i}{\varphi }_i\otimes {\varphi }_{pn-i},\mathrm{for} p=1,2,3,\dots ,$$

namely

$$\begin{array}{cc}\hfill (G\otimes G)\sum _{i\in \mathbb{Z}}{(-1)}^{pn-i}{\varphi }_i\otimes {\varphi }_{pn-i}& =\sum _{i\in \mathbb{Z}}{(-1)}^{pn-i}G{\varphi }_i{G}^{-1}\otimes G{\varphi }_{pn-i}{G}^{-1}G\hfill \\ & =\sum _{i,j,k\in \mathbb{Z}}{(-1)}^{pn-i}{a}_{ji}{a}_{k,pn-i}{\varphi }_{j}G\otimes {\varphi }_{k}G\hfill \\ & =\sum _{i,j,k\in \mathbb{Z}}{(-1)}^{pn-i}{a}_{ji}{a}_{k-pn,-i}{\varphi }_{j}G\otimes {\varphi }_{k}G\hfill \\ & =\sum _{j\in \mathbb{Z}}{(-1)}^{pn-j}{\varphi }_{j}G\otimes {\varphi }_{pn-j}G\hfill \\ & =\sum _{j\in \mathbb{Z}}{(-1)}^{pn-j}{\varphi }_j\otimes {\varphi }_{pn-j}(G\otimes G).\hfill \end{array}$$

Since ${\sum }_{i\in \mathbb{Z}}{(-1)}^{pn-i}{\varphi }_i|0⟩\otimes {\varphi }_{pn-i}|0⟩=0$, we find that the elements $\tau$ in the orbit of the vacuum vector of this twisted loop not only satisfy (15), but also satisfy the conditions

$$\sum _{j\in \mathbb{Z}}{(-1)}^{pn-j}{\varphi }_j\tau \otimes {\varphi }_{pn-j}\tau =0,p=1,2,\dots .$$

(42)

This means that $\tau \left(\tilde{t}\right)=\sigma \left(\tau \right)$ not only satisfies (24), but also the conditions

$${\mathrm{Res}}_z{z}^{pn-1}{e}^{{\sum }_{j=1}^{\infty }(t_{2j-1}-t_{2j-1}^{\prime })z^{2j-1}}{e}^{{\sum }_{j=1}^{\infty }2\left(\frac{\partial }{\partial t_{2j-1}^{\prime }}-\frac{\partial }{\partial t_{2j-1}}\right)\frac{z^{-2j+1}}{2j-1}}\tau \left(\tilde{t}\right)\tau \left({\tilde{t}}^{\prime }\right)dz=0,p=1,2,\dots .$$

(43)

From (43), one deduces, using the fundamental Lemma ([11], Lemma 4.1) and the first equation of (37), that

$${\left(P(\tilde{t},\partial ){\partial }^{pn-1}P{(\tilde{t},\partial )}^{\ast }\right)}_{<0}={\left(P(\tilde{t},\partial ){\partial }^{pn}P{(\tilde{t},\partial )}^{-1}{\partial }^{-1}\right)}_{<0}=0.$$

Thus, the Lax operator $L(\tilde{t},\partial )$ satisfies

$${\left(L{(\tilde{t},\partial )}^{pn}\right)}_{\le 0}=0,p=1,2,\dots .$$

(44)

Hence, $\mathcal{L}(\tilde{t},\partial )=L{(\tilde{t},\partial )}^n$ is a monic differential operator with zero constant term. Moreover, $\mathcal{L}(\tilde{t},\partial )$ is equal to (10), and, by the first formula of (37), we have the relation (11).

Now, if n is odd, one can use the the Sato–Wilson equation, i.e., the second equation of (37), to find that

$$\frac{\partial P(\tilde{t},\partial )}{\partial t_{(2j-1)n}}=0,j=1,2,\dots .$$

From this we find that the tau-function satisfies

$$\frac{\partial \tau \left(\tilde{t}\right)}{\partial t_{(2j-1)n}}={\lambda }_j\tau \left(\tilde{t}\right),{\lambda }_j\in \mathbb{C},\mathrm{for} j=1,2,\dots .$$

(45)

Since we consider only polynomial tau-functions, we find that for odd n:

$$\frac{\partial \tau \left(\tilde{t}\right)}{\partial t_{(2j-1)n}}=0,j=1,2,\dots .$$

(46)

If n is even, there is no such restriction, because the Sato–Wilson Equation (37) is only defined for odd flows. However, the additional Equation (43) still holds and gives additional constraints on the tau-function.

Proposition 1.

For n odd, Equation (46) for $j=1$ and the BKP hierarchy (24) on $\tau \left(\tilde{t}\right)$ are equivalent to (24) and (43).

Proof. 

We only have to show that (46) for $j=1$ and (24) imply (43). For this, differentiate (24) by $t_n$ and use (46); this gives Equation (43) for $p=1$. Next, differentiate (43) for $p=1$ again by $t_n$ and use again (46); this gives (43) for $p=2$, etc. □

Remark 3.

If n is odd, Proposition 1 gives that a polynomial BKP tau-function is n-th Sawada–Kotera tau-function if and only if τ satisfies $\frac{\partial \tau }{\partial t_n}=0.$

Since L satisfies the BKP hierarchy, $\mathcal{L}=L^n$ also satisfies the BKP hierarchy. For $n=3$, assuming the constraint that $\mathcal{L}$ is a differential operator, $\mathcal{L}$ is given by (3) and $\frac{\partial \mathcal{L}}{\partial t_3}=[{\left({\mathcal{L}}^{\frac{5}{3}}\right)}_{\ge 0},\mathcal{L}]$ is the Sawada–Kotera equation (33). This leads to the following definition.

Definition 1.

Let $\mathcal{L}={\partial }^n+u_{n-2}{\partial }^{n-2}+\cdots +u_1\partial$ be a differential operator, satisfying (11). The system of Lax equations

$$\frac{\partial \mathcal{L}(\tilde{t},\partial )}{\partial t_{2j-1}}=\left[{\left(\mathcal{L}{(\tilde{t},\partial )}^{\frac{2j-1}{n}}\right)}_{\ge 0},\mathcal{L}(\tilde{t},\partial )\right],j=1,2,\dots ,$$

(47)

is called the n-th reduced BKP hierarchy or the n-th Sawada–Kotera hierarchy. For $n=3$, it is called the Sawada–Kotera hierarchy.

The geometric meaning of Equation (42) is that the space $\mathrm{Ann}\tau$ is invariant under the shift ${\Lambda }_n$, where

$${\Lambda }_n\left({\varphi }_i\right)={\varphi }_{i+n}.$$

(48)

As in the $S{O}_{\infty ,odd}$ case, all polynomial tau-functions in this n-reduced case lie in some Schubert cell. Such a Schubert cell has a “lowest” element $w_{\lambda }$, for $\lambda$ a certain strict partition. This element can be obtained from the vacuum vector by the action of the Weyl group corresponding to $G_n^{\left(2\right)}$. The element

$$w_{\lambda }={\varphi }_{-{\lambda }_1}{\varphi }_{-{\lambda }_2}\cdots {\varphi }_{-{\lambda }_k}|0⟩$$

(49)

lies in the Weyl group orbit of $|0⟩$, corresponding to $S{O}_{\infty ,odd}$, (see [15]), however, not all such elements lie in the Weyl group orbit of $|0⟩$ for $G_n^{\left(2\right)}$. For this, consider

$$\mathrm{Ann}{w}_{\lambda }=\mathrm{span}\{{\varphi }_{-{\lambda }_1},{\varphi }_{-{\lambda }_2},\dots ,{\varphi }_{-{\lambda }_k}\}\oplus \mathrm{span}\left\{{\varphi }_i\right|i>0,i\ne {\lambda }_j,j=1,\dots ,k\}.$$

(50)

The element $w_{\lambda }$ lies in the $G_n^{\left(2\right)}$ Weyl group orbit if and only if $\mathrm{Ann}{w}_{\lambda }$ is invariant under the action of ${\Lambda }_n$, which means that the (${\lambda }_1+1$ shifted) set

$$V_{\lambda }=\{{\lambda }_1+{\lambda }_i+1|,i=1,\dots ,k\}\cup \{{\lambda }_1-j+1|0<j<{\lambda }_1,j\ne {\lambda }_i\mathrm{for} i=1,\dots ,k\},$$

(51)

must satisfy the $-n$ shift condition, i.e.,

$$\mathrm{if} {\mu }_j\in V_{\lambda },\mathrm{then} {\mu }_j-n\in V_{\lambda }\mathrm{or} {\mu }_j-n\le 0.$$

(52)

Only the elements $w_{\lambda }$, for which the corresponding $V_{\lambda }$ satisfies condition (52), lie in the $G_n^{\left(2\right)}$ group orbit.

Example 1.

(a) For $n=2$, the only strict partition λ that satisfies condition (52) is $\lambda =\varnothing$.

• (b) For $n=3$, the only strict partitions λ that satisfy condition (52) are

  $$(3m+1,3m-2,3m-5,\dots ,4,1)and(3m+2,3m-1,3m-4,\dots ,5,2),m\in {\mathbb{Z}}_{\ge 0}.$$

Remark 4.

Note that (52) means that λ is a strict partition that is the union of strict partitions $(nm+a_i,n(m-1)+a_i,,\dots ,n+a_i,a_i)$, with $1\le a_i<n$ and $1\le i<n$, such that $a_j-n\ne -a_{\ell }$. In other words, $a_j+a_{\ell }\ne n$. Hence there are at most $\left[\frac{n}{2}-1\right]$ such $a_i$.

To a strict partition $\lambda$ that satisfies condition (52), the corresponding Schubert cell is then obtained through the action on a $w_{\lambda }$ by an upper-triangular matrix in the group $G_n^{\left(2\right)}$. This produces, up to a constant factor, elements

$$v_{\lambda }=v_1{v}_2\cdots v_k|0⟩,\mathrm{where} v_j={\varphi }_{-{\lambda }_j}+\sum _{i\ge 1-{\lambda }_j}{a}_{ij}{\varphi }_i(\mathrm{finite} \mathrm{sum}),$$

(53)

and

$$(v_j,v_{\ell })=0,\mathrm{for} j,\ell =1,\dots ,k,\mathrm{and} \mathrm{if}{\lambda }_i={\lambda }_j-n,\mathrm{then} v_i={\Lambda }_n\left(v_j\right).$$

(54)

We first express the constants $a_{ij}$ in terms of other constants by letting

$$a_{ij}=s_{i+{\lambda }_j}\left(c_{{\overline{\lambda }}_j}\right),\mathrm{where} \mathrm{the} s_i\mathrm{are} \mathrm{elementary} \mathrm{Schur} \mathrm{polynomials}.$$

Here, we use that

$$1+\sum _{i=1-{\lambda }_j}^{\infty }{a}_{ij}{z}^{i+{\lambda }_j}=exp\left(\sum _{k=1}^{\infty }{c}_{k,{\overline{\lambda }}_j}{z}^k\right),$$

hence, for every ${\overline{\lambda }}_j$, one can recursively obtain the $c_{k,{\overline{\lambda }}_j}$. Since $a_{ij}=0$ for $i>>0$, one only has a finite number of $c_{k,{\overline{\lambda }}_j}$. Thus,

$$v_j={\varphi }_{-{\lambda }_j}+\sum _{i>-{\lambda }_j}{s}_{i+{\lambda }_j}\left(c_{{\overline{\lambda }}_j}\right){\varphi }_i,$$

(55)

where $c_{{\overline{\lambda }}_j}=(c_{1,{\overline{\lambda }}_j},c_{2,{\overline{\lambda }}_j},c_{3,{\overline{\lambda }}_j},\dots )$. Here, ${\overline{\lambda }}_j={\lambda }_{j}modn$, which means that there are at most $\left[\frac{n}{2}-1\right]$ of such infinite series of constants (see Remark 4) and the $v_j$ satisfy the condition

$$\mathrm{if} {\lambda }_i={\lambda }_j-n,\mathrm{then} v_i={\Lambda }_n\left(v_j\right).$$

(56)

We can now use the isomorphism $\sigma$ to calculate the bosonization of elements $v_{\lambda }$. For this, we use formula (25) and apply this to $v_{\lambda }$ (which is given by (53) with $v_j$ given by (55)). Now, using (22) and (23) and the fact that

$$e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}\frac{\partial }{\partial s_{2j-1}}}{e}^{{\sum }_{j=1}^{\infty }{s}_{2j-1}{z}^{2j-1}}=e^{{\sum }_{j=1}^{\infty }(t_{2j-1}+s_{j-1})z^{2j-1}}{e}^{{\sum }_{j=1}^{\infty }{t}_{2j-1}\frac{\partial }{\partial s_{2j-1}}}$$

we find that

$$e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}\varphi \left(z\right)e^{-{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}=e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{z}^{2j-1}}\varphi \left(z\right).$$

Thus, using (55), we find that

$$\begin{array}{cc}\hfill v_j\left(\tilde{t}\right):=& e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}{v}_j{e}^{-{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}\hfill \\ \hfill =& e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}({\varphi }_{-{\lambda }_j}+\sum _{i\ge 1-{\lambda }_j}{s}_{i+{\lambda }_j}\left(c_{{\overline{\lambda }}_j}\right){\varphi }_i)e^{-{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}\hfill \\ \hfill =& e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}\mathrm{Res}\sum _{\ell =0}^{\infty }{s}_{\ell }\left(c_{{\overline{\lambda }}_j}\right)z^{\ell -{\lambda }_j}\varphi \left(z\right)e^{-{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}\frac{dz}{z}\hfill \\ \hfill =& \mathrm{Res}\sum _{\ell =0}^{\infty }{s}_{\ell }\left(c_{{\overline{\lambda }}_j}\right)z^{\ell -{\lambda }_j}\varphi \left(z\right)e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{z}^{2j-1}}\frac{dz}{z}\hfill \\ \hfill =& \mathrm{Res}{z}^{-{\lambda }_j}{e}^{{\sum }_{i=1}^{\infty }{c}_{i,{\overline{\lambda }}_j}{z}^i+{\sum }_{j=1}^{\infty }{t}_{2j-1}{z}^{2j-1}}\varphi \left(z\right)\frac{dz}{z}\hfill \\ \hfill =& \mathrm{Res}{z}^{-{\lambda }_j}\sum _{k=0}^{\infty }{s}_k(\tilde{t}+c_{{\overline{\lambda }}_j})z^k\sum _{i\in \mathbb{Z}}{\varphi }_i{z}^{-i}\frac{dz}{z}\hfill \\ \hfill =& {\varphi }_{-{\lambda }_j}+\sum _{i\ge 1-{\lambda }_j}{s}_{i+{\lambda }_j}(\tilde{t}+c_{{\overline{\lambda }}_j}){\varphi }_i.\hfill \end{array}$$

Since $e^{{\sum }_{j=1}^{\infty }{t}_{2j-1}{\alpha }_{2j-1}}|0⟩=0$, we find that the corresponding tau-function is equal to the vacuum expectation value

$$\tau \left(\tilde{t}\right)=⟨0|v_1\left(\tilde{t}\right)v_2\left(\tilde{t}\right)\cdots v_k\left(\tilde{t}\right)|0⟩.$$

(57)

If $k=2m$, then this is the Pfaffian of a $2m\times 2m$ skew-symmetric matrix. If $k=2m-1$, we use the fact that

$$⟨0|v_1\left(\tilde{t}\right)v_2\left(\tilde{t}\right)\cdots v_k\left(\tilde{t}\right)|0⟩=2⟨0|v_1\left(\tilde{t}\right)v_2\left(\tilde{t}\right)\cdots v_k\left(\tilde{t}\right){\varphi }_0|0⟩$$

and again we find a Pfaffian. We thus arrive at the main theorem.

Theorem 2.

All polynomial tau-functions of the n-th Sawada–Kotera hierarchy are, up to a scalar factor, equal to the Pfaffian

$${\tau }_{\lambda }\left(\tilde{t}\right)=Pf{\left({\chi }_{{\lambda }_i,{\lambda }_j}(\tilde{t}+c_{{\overline{\lambda }}_i},\tilde{t}+c_{{\overline{\lambda }}_j})\right)}_{1\le i,j\le 2m},$$

(58)

where $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{2m})$, $m=0,1,\dots$, is an extended strict partition, which satisfies the $-n$ shift condition (52) for (51). The polynomials ${\chi }_{a,b}$ are given by (40). Here, as before, $\tilde{t}=(t_1,0,t_3,0,\dots )$, and $c_{{\overline{\lambda }}_i}=(c_{1,{\overline{\lambda }}_i},c_{2,{\overline{\lambda }}_i},c_{3,{\overline{\lambda }}_i},\dots )$ are arbitrary constants, where we replace, recursively, for all $j=1,2,\dots 2m$ (respectively, for all $j=1,2,\dots 2m-1$), when ${\lambda }_{2m}\ne 0$ (resp., ${\lambda }_{2m}=0$), the constants $c_{{\lambda }_j+{\lambda }_{\ell },{\overline{\lambda }}_j}$, for $j\le \ell \le 2m$ (resp., $j\le \ell <2m$) as follows:

(1)

If ${\overline{\lambda }}_j\ne {\overline{\lambda }}_{\ell }$, then

$$\begin{array}{cc}\hfill & c_{{\lambda }_j+{\lambda }_{\ell },{\overline{\lambda }}_j}=-{(-1)}^{{\lambda }_j+{\lambda }_{\ell }}\times \hfill \\ \hfill & s_{{\lambda }_j+{\lambda }_{\ell }}(c_{1,{\overline{\lambda }}_{\ell }}-c_{1,{\overline{\lambda }}_1},c_{2,{\overline{\lambda }}_{\ell }}+c_{2,{\overline{\lambda }}_1},\dots ,c_{{\lambda }_j+{\lambda }_{\ell }-1,{\overline{\lambda }}_{\ell }}+{(-1)}^{{\lambda }_j+{\lambda }_{\ell }-1}{c}_{{\lambda }_j+{\lambda }_{\ell }-1,{\overline{\lambda }}_j},c_{{\lambda }_j+{\lambda }_{\ell },{\overline{\lambda }}_{\ell }}).\hfill \end{array}$$

(59)

(2)

If ${\overline{\lambda }}_j={\overline{\lambda }}_{\ell }$ and ${\lambda }_j+{\lambda }_{\ell }$ is even, then

$$c_{{\lambda }_j+{\lambda }_{\ell },{\overline{\lambda }}_j}=-\frac{1}{2}{s}_{\frac{{\lambda }_j+{\lambda }_{\ell }}{2}}(2c_{2,{\overline{\lambda }}_j},2c_{4,{\overline{\lambda }}_j},\cdots ,2c_{{\lambda }_j+{\lambda }_{\ell }-2},0).$$

(60)

Proof. 

We still need to use the fact that all vectors $v_i$, for $i=1,\dots ,k$, form an isotropic subspace. So, assume that $1\le j\le \ell \le k$ and that $v_j$ and $v_{\ell }$ are given by (55). Then

$$\begin{array}{cc}\hfill 0=(v_j,v_{\ell })=& {(-1)}^{{\lambda }_j}\sum _{i=0}^{{\lambda }_j+{\lambda }_{\ell }}{(-1)}^i{s}_i\left(c_{{\overline{\lambda }}_j}\right)s_{{\lambda }_j+{\lambda }_{\ell }-i}\left(c_{{\overline{\lambda }}_{\ell }}\right)\hfill \\ \hfill =& {(-1)}^{{\lambda }_j}{s}_{{\lambda }_j+{\lambda }_{\ell }}(c_{i,{\overline{\lambda }}_{\ell }}+{(-1)}^i{c}_{i,{\overline{\lambda }}_j}).\hfill \end{array}$$

(61)

Here, we have used the fact that the coefficient of $z^m$ of

$$e^{{\sum }_{i=1}^{\infty }{x}_i{z}^i+y^i{(-z)}^i}=e^{{\sum }_{i=1}^{\infty }{x}_i{z}^i}{e}^{{\sum }_{i=1}^{\infty }{y}^i{(-z)}^i}$$

is equal to $s_m(x_i+{(-1)}^i{y}_i)={\sum }_{j=0}^m{(-1)}^{m-j}{s}_j\left(x\right)s_{m-j}\left(y\right)$.

So, we need to investigate condition (61). Here we have two possibilities, viz. ${\overline{\lambda }}_j\ne {\overline{\lambda }}_{\ell }$ and ${\overline{\lambda }}_j={\overline{\lambda }}_{\ell }$. If ${\overline{\lambda }}_j\ne {\overline{\lambda }}_{\ell }$, then using the fact that $s_i\left(x\right)=x_i+$ terms not containing $x_i$, we find (59). If ${\overline{\lambda }}_j={\overline{\lambda }}_{\ell }$, then notice that $s_{{\lambda }_j+{\lambda }_{\ell }}$ only depends on the $c_{i,{\overline{\lambda }}_j}$ with i even. Thus, all elementary Schur polynomials $s_{2i+1}$ in only the even variables are equal to zero. This means that if ${\lambda }_j+{\lambda }_{\ell }$ is odd, there is no restriction on the constants, but if ${\lambda }_j+{\lambda }_{\ell }$ is even, we find that

$$c_{{\lambda }_j+{\lambda }_{\ell },{\overline{\lambda }}_j}=-\frac{1}{2}{s}_{{\lambda }_j+{\lambda }_{\ell }}(0,2c_{2,{\overline{\lambda }}_j},0,2c_{4,{\overline{\lambda }}_j},0,2c_{6,{\overline{\lambda }}_j},\dots ,0,2c_{{\lambda }_j+{\lambda }_{\ell }-2,{\overline{\lambda }}_j},0,0).$$

Note that this restriction coincides with (60). □

Remark 5.

Since ${\chi }_{{\lambda }_j,{\lambda }_{\ell }}$ is given by (40), the constant $c_{2{\lambda }_1,{\overline{\lambda }}_1}$ does not appear in (58) and the substitution (60) for $c_{2{\lambda }_1,{\overline{\lambda }}_1}$ is void.

For $n=3$, see Example 1(b), we only have one infinite series of constants $c_{i,{\overline{\lambda }}_1}$, which means that we only have the substitutions (60). We therefore find:

Corollary 1.

All polynomial tau-functions of the Sawada–Kotera hierarchy are, up to a non-zero constant factor,

$${\tau }_{\lambda }\left(\tilde{t}\right)=Pf{\left({\chi }_{{\lambda }_i,{\lambda }_j}(\tilde{t}+c,\tilde{t}+c)\right)}_{1\le i,j\le 2m},$$

(62)

where λ is one of the following extended strict partitions:

1. 

$(6m+1,6m-2,6m-5,\dots ,4,1)$;

2. 

$(6m-2,6m-5,6m-8,\dots ,4,1,0)$;

3. 

$(6m+2,6m-1,6m-4,\dots ,5,2)$;

4. 

$(6m-1,6m-4,6m-7,\dots ,5,2,0)$,

where $m=0,1,\dots$, and $c=(c_1,c_2,c_3,\dots )$ are arbitrary constants in which we substitute recursively, $c_2=0,$ and $c_8,c_{14},c_{20},\dots ,c_{12m-4}$ in case 1; $c_2=0$, and $c_8,c_{14},c_{20},\dots ,c_{12m-10}$ in case 2; $c_4,c_{10},c_{16},\dots ,c_{12m-2}$ in case 3; and $c_4,c_{10},c_{16},\dots ,c_{12m-8}$ in case 4, respectively, by the following formula

$$c_{2k}=-\frac{1}{2}{s}_k(2c_2,2c_4,\dots 2c_{2k-2},0)for k>1.$$

(63)

Remark 6.

If we choose $c=\tilde{c}=(c_1,0,c_3,0,c_5,\dots )$, then the corresponding Sawada–Kotera tau-function ${\tau }_{\lambda }\left(\tilde{t}\right)$ is equal, up to a non-zero constant factor, to a Schur Q-function $Q_{\lambda }(\tilde{t}+\tilde{c})$ (cf. [15]).

Example 2.

All the Sawada–Kotera tau-functions related to the partition $\lambda =(5,2)$ are given, up to a multiplicative constant, by

$$\begin{array}{cc}& c_1^7+1120c_7+7c_1^6{t}_1-280c_2^3{t}_1-280c_5{t}_1^2+140c_2^2{t}_1^3+t_1^7++35c_1^3{(2c_2+t_1^2)}^2+\hfill \\ & +7c_1{t}_1^5(2c_2+3t_1^2)+35c_1^4(2c_2{t}_1+t_1^3)-280t_1^2{t}_5-7c_1^2(40c_5-60c_2^2{t}_1-20c_2{t}_1^3-3t_1^5+40t_5)+\hfill \\ & +14c_2(120c_5+t_1^5+120t_5)-7c_1(40c_2^3-60c_2^2{t}_1^2-10c_2{t}_1^4+t_1(80c_5-t_1^5+80t_5))+1120t_7,\hfill \end{array}$$

with $c_i\in \mathbb{C}$, arbitrary. (We have eliminated $c_4$ by the substitution $c_4=-c_2^2$; all the other constants that do not appear, disappear automatically).

4. Conclusions

Three classes of solutions of soliton equations have been extensively studied in the literature: rational solutions, soliton solutions, and theta function solutions. Soliton solutions are constructed by making use of vertex operators [2,6,10,11,13,14,16,17]. Theta function solutions are obtained by Krichever’s method [18,19]. Rational solutions are obtained by constructing polynomial tau-functions using the group transformations method, introduced by Sato [1]. In the present paper, we describe all polynomial tau-functions for the n-th Sawada–Kotera hierarchy, and, in particular, for the Sawada–Kotera equation.