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,
Let
and
. Recall that the Kadomtsev–Petviashvili (KP) hierarchy is the following hierarchy of Lax equations on the pseudodifferential operator
in
[
1]:
The KdV hierarchy is the second reduced KP hierarchy, meaning that one imposes the following constraint on
:
In this case, the operator
is defined by (
1), with
, and the KP hierarchy (
4) reduces to the KdV hierarchy
For even
k, this equation is trivial; for
, 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
, defined by [
1,
2]:
where
is a pseudodifferential operator, with the symbol
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
(see, e.g., [
5] for details). Its 3-reduction is defined by the constraint that
is a differential operator, and the corresponding hierarchy is
where
is given by (2). For
, 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
satisfying the constraint
The
n-th reduced BKP hierarchy is
We call it the
n-th Sawada–Kotera hierarchy, since, for
,
is given by (3), and for
, 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
,
, which are generators of a Clifford algebra over
, satisfying the following anti-commutation relation:
Consider the right (resp., left) irreducible module
(resp.,
) over this algebra by the following action on the vacuum vector
(resp.,
):
The quadratic elements
form a basis of the infinite-dimensional Lie algebra
over
. Let
be the corresponding Lie group. We proved in ([
9], Theorem 1.2a) that a non-zero element
lies in this Lie group orbit of the vacuum vector
if and only if it satisfies the BKP hierarchy in the fermionic picture, i.e., the following equation in
:
Non-zero elements of
F, satisfying (
15), are called tau-functions of the BKP hierarchy in the fermionic picture.
The group
consists of elements
G leaving the symmetric bilinear form
on
F invariant, i.e.,
Stated differently,
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
, with
. Namely, the cell, attached to the partition
is
An element
corresponds to the following point in the maximal isotropic Grassmannian (i.e., a maximal isotropic subspace of
):
For instance,
Using the bosonization of Equation (
15), one obtains a hierarchy of differential equations on
([
4,
7,
8,
10], Section 3). This bosonization is an isomorphism
between the spin module
F and the polynomial algebra
. Explicitly, we introduce the twisted neutral fermionic field
and the bosonic field
where the normal ordering
is defined by
equivalently:
if
and
if
, except when
, then
. The operators
satisfy the commutation relations of the Heisenberg Lie algebra
and its representation on
F is irreducible ([
9], Theorem 3.2). Using this, we obtain a vector space isomorphism
, uniquely defined by the following relations:
Explicitly ([
9], Section 3.2):
Since (
15) can be rewritten as
under the isomorphism
, Equation (
15) turns into:
where
and
. Therefore,
is the vacuum expectation value
Furthermore, by making a change of variables, as in [
10] (p. 972), viz.
and
, and using the elementary Schur polynomials
, which are defined by
we can rewrite (
24), where we assume
:
where
and
. Using Taylor’s formula, we thus obtain the BKP hierarchy of Hirota bilinear equations [
10] (p. 972):
Using the notation
, this turns into
The simplest equation in this hierarchy is ([
10], Appendix 3):
If we assume that our tau-function does not depend on
, then this gives
Letting
,
, and
and viewing the remaining
as parameters, Equation (
31) turns into the famous Sawada–Kotera equation [
6]:
Another approach is by using the wave function; see [
8] (p. 345),
where
, and, in particular,
Letting
be the pseudodifferential operator in
with the symbol
, Equation (
24) turns into
Now, using the fundamental lemma, Lemma 1.1 of [
2] or Lemma 4.1 of [
11], we deduce from (
36):
Next, introducing the Lax operator
we deduce from (
37) that
L satisfies [
8]
Note that, since
and the fact that
is given by (
35), we find that
which explains the choice (
32) of
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
, for
, has zero constant term. Let us prove that the first equation of (
38) indeed implies this fact. We have
Now, using the fact that the constant term of
is equal to
we find that the constant term of
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],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
of the BKP hierarchy obtained in ([
7], Theorem 6) (see also [
4]). For that, given integers
a and
b,
, let
and let
if
. Then
Theorem 1 ([
7], Theorem 6)
. All polynomial tau-functions of the BKP hierarchy (24), up to a scalar multiple, are equal towhere is the Pfaffiann of a skew-symmetric matrix, is an extended strict partition, i.e., , , , . Remark 2. The connection between the set of strict partitions and the extended strict partitions is as follows. If 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 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 .
3. The -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
. This means that the tau-function lies in a smaller group orbit of the vacuum vector
. Instead of the
orbit of the vacuum vector
, we consider the twisted loop group
, corresponding to the affine Lie algebra
, to obtain the 3-reduced BKP hierarchy [
8]. More generally (see also [
8]), when
is odd, the
-reduced hierarchy is related to the twisted loop group
corresponding to the twisted affine Lie algebra
. When
is even, one has the twisted loop group
corresponding to the affine Lie algebra
[
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
, but also the
n-periodicity condition
. This means that these group elements also commute with the operator
namely
Since
, we find that the elements
in the orbit of the vacuum vector of this twisted loop not only satisfy (
15), but also satisfy the conditions
This means that
not only satisfies (
24), but also the conditions
From (
43), one deduces, using the fundamental Lemma ([
11], Lemma 4.1) and the first equation of (
37), that
Thus, the Lax operator
satisfies
Hence,
is a monic differential operator with zero constant term. Moreover,
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
From this we find that the tau-function satisfies
Since we consider only polynomial tau-functions, we find that for odd
n:
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 and the BKP hierarchy (24) on are equivalent to (24) and (43). Proof. We only have to show that (
46) for
and (
24) imply (
43). For this, differentiate (
24) by
and use (
46); this gives Equation (
43) for
. Next, differentiate (
43) for
again by
and use again (
46); this gives (
43) for
, 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
Since
L satisfies the BKP hierarchy,
also satisfies the BKP hierarchy. For
, assuming the constraint that
is a differential operator,
is given by (3) and
is the Sawada–Kotera equation (
33). This leads to the following definition.
Definition 1. Let be a differential operator, satisfying (11). The system of Lax equationsis called the n-th reduced BKP hierarchy or the n-th Sawada–Kotera hierarchy. For , it is called the Sawada–Kotera hierarchy. The geometric meaning of Equation (
42) is that the space
is invariant under the shift
, where
As in the
case, all polynomial tau-functions in this
n-reduced case lie in some Schubert cell. Such a Schubert cell has a “lowest” element
, for
a certain strict partition. This element can be obtained from the vacuum vector by the action of the Weyl group corresponding to
. The element
lies in the Weyl group orbit of
, corresponding to
, (see [
15]), however, not all such elements lie in the Weyl group orbit of
for
. For this, consider
The element
lies in the
Weyl group orbit if and only if
is invariant under the action of
, which means that the (
shifted) set
must satisfy the
shift condition, i.e.,
Only the elements
, for which the corresponding
satisfies condition (
52), lie in the
group orbit.
Example 1. (a) For , the only strict partition λ that satisfies condition (52) is . (b) For , the only strict partitions λ that satisfy condition (52) are
Remark 4. Note that (52) means that λ is a strict partition that is the union of strict partitions , with and , such that . In other words, . Hence there are at most such . To a strict partition
that satisfies condition (
52), the corresponding Schubert cell is then obtained through the action on a
by an upper-triangular matrix in the group
. This produces, up to a constant factor, elements
and
We first express the constants
in terms of other constants by letting
Here, we use that
hence, for every
, one can recursively obtain the
. Since
for
, one only has a finite number of
. Thus,
where
. Here,
, which means that there are at most
of such infinite series of constants (see Remark 4) and the
satisfy the condition
We can now use the isomorphism
to calculate the bosonization of elements
. For this, we use formula (
25) and apply this to
(which is given by (
53) with
given by (
55)). Now, using (
22) and (
23) and the fact that
we find that
Thus, using (
55), we find that
Since
, we find that the corresponding tau-function is equal to the vacuum expectation value
If
, then this is the Pfaffian of a
skew-symmetric matrix. If
, we use the fact that
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 Pfaffianwhere , , is an extended strict partition, which satisfies the shift condition (52) for (51). The polynomials are given by (40). Here, as before, , and are arbitrary constants, where we replace, recursively, for all (respectively, for all ), when (resp., ), the constants , for (resp., ) as follows: - (1)
- (2)
If and is even, then
Proof. We still need to use the fact that all vectors
, for
, form an isotropic subspace. So, assume that
and that
and
are given by (
55). Then
Here, we have used the fact that the coefficient of
of
is equal to
.
So, we need to investigate condition (
61). Here we have two possibilities, viz.
and
. If
, then using the fact that
terms not containing
, we find (
59). If
, then notice that
only depends on the
with
i even. Thus, all elementary Schur polynomials
in only the even variables are equal to zero. This means that if
is odd, there is no restriction on the constants, but if
is even, we find that
Note that this restriction coincides with (
60). □
Remark 5. Since is given by (40), the constant does not appear in (58) and the substitution (60) for is void. For
, see Example 1(b), we only have one infinite series of constants
, 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,where λ is one of the following extended strict partitions: - 1.
;
- 2.
;
- 3.
;
- 4.
,
where , and are arbitrary constants in which we substitute recursively, and in case 1; , and in case 2; in case 3; and in case 4, respectively, by the following formula Remark 6. If we choose , then the corresponding Sawada–Kotera tau-function is equal, up to a non-zero constant factor, to a Schur Q-function (cf. [15]). Example 2. All the Sawada–Kotera tau-functions related to the partition are given, up to a multiplicative constant, bywith , arbitrary. (We have eliminated by the substitution ; all the other constants that do not appear, disappear automatically).