On Addition Formulae of KP, mKP and BKP Hierarchies
In this paper we study the addition formulae of the KP, the mKP and the BKP hierarchies. We prove that the total hierarchies are equivalent to the simplest equations of their addition formulae. In the case of the KP and the mKP hierarchies those results had previously been proved by Noumi, Takasaki...
Збережено в:
| Дата: | 2013 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2013
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149208 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Addition Formulae of KP, mKP and BKP Hierarchies / Y. Shigyo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149208 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1492082019-02-20T01:26:00Z On Addition Formulae of KP, mKP and BKP Hierarchies Shigyo, Y. In this paper we study the addition formulae of the KP, the mKP and the BKP hierarchies. We prove that the total hierarchies are equivalent to the simplest equations of their addition formulae. In the case of the KP and the mKP hierarchies those results had previously been proved by Noumi, Takasaki and Takebe by way of wave functions. Here we give alternative and direct proofs for the case of the KP and mKP hierarchies. Our method can equally be applied to the BKP hierarchy. 2013 Article On Addition Formulae of KP, mKP and BKP Hierarchies / Y. Shigyo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H70; 37K10; 37K20 DOI: http://dx.doi.org/10.3842/SIGMA.2013.035 http://dspace.nbuv.gov.ua/handle/123456789/149208 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we study the addition formulae of the KP, the mKP and the BKP hierarchies. We prove that the total hierarchies are equivalent to the simplest equations of their addition formulae. In the case of the KP and the mKP hierarchies those results had previously been proved by Noumi, Takasaki and Takebe by way of wave functions. Here we give alternative and direct proofs for the case of the KP and mKP hierarchies. Our method can equally be applied to the BKP hierarchy. |
| format |
Article |
| author |
Shigyo, Y. |
| spellingShingle |
Shigyo, Y. On Addition Formulae of KP, mKP and BKP Hierarchies Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Shigyo, Y. |
| author_sort |
Shigyo, Y. |
| title |
On Addition Formulae of KP, mKP and BKP Hierarchies |
| title_short |
On Addition Formulae of KP, mKP and BKP Hierarchies |
| title_full |
On Addition Formulae of KP, mKP and BKP Hierarchies |
| title_fullStr |
On Addition Formulae of KP, mKP and BKP Hierarchies |
| title_full_unstemmed |
On Addition Formulae of KP, mKP and BKP Hierarchies |
| title_sort |
on addition formulae of kp, mkp and bkp hierarchies |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149208 |
| citation_txt |
On Addition Formulae of KP, mKP and BKP Hierarchies / Y. Shigyo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT shigyoy onadditionformulaeofkpmkpandbkphierarchies |
| first_indexed |
2025-07-12T21:38:58Z |
| last_indexed |
2025-07-12T21:38:58Z |
| _version_ |
1837478806439329792 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 035, 16 pages
On Addition Formulae of KP, mKP
and BKP Hierarchies
Yoko SHIGYO
Department of Mathematics, Tsuda College, Kodaira, Tokyo, 187-8577, Japan
E-mail: yoko.shigyo@gmail.com
Received December 12, 2012, in final form April 04, 2013; Published online April 23, 2013
http://dx.doi.org/10.3842/SIGMA.2013.035
Abstract. In this paper we study the addition formulae of the KP, the mKP and the BKP
hierarchies. We prove that the total hierarchies are equivalent to the simplest equations of
their addition formulae. In the case of the KP and the mKP hierarchies those results had
previously been proved by Noumi, Takasaki and Takebe by way of wave functions. Here we
give alternative and direct proofs for the case of the KP and mKP hierarchies. Our method
can equally be applied to the BKP hierarchy.
Key words: KP hierarchy; modified KP hierarchy; BKP hierarchy
2010 Mathematics Subject Classification: 14H70; 37K10; 37K20
1 Introduction
Our purpose is to prove that some integrable hierarchies are equivalent to the simplest equations
of their addition formulae. In this paper we study the KP, the modified KP (mKP) and the
BKP hierarchies.
The (bilinear) KP hierarchy [12] is an infinite system of bilinear equations for τ(x), x =
(x1, x2, . . . ), given, in the generating form, by∮
e−2ξ(y,λ)τ
(
x− y −
[
λ−1
])
τ
(
x+ y +
[
λ−1
]) dλ
2πi
= 0.
Namely, if we expand the left hand side in y = (y1, y2, . . . ), then we get Hirota’s bilinear
equations, which contain, as the simplest equation, the Kadomtsev–Petviashvili (KP) equation
in the bilinear form(
D4
1 + 3D2
2 − 4D1D3
)
τ · τ = 0,
where Di is the Hirota’s bilinear operator defined by
(
Di1
1 D
i2
2 · · ·
)
τ · τ =
((
∂
∂y1
)i1 ( ∂
∂y2
)i2
· · ·
)
(τ(x+ y)τ(x− y))
∣∣∣
y=0
, y = (y1, y2, . . . ).
If we put y =
(
m−1∑
i=1
[βi]−
m+1∑
i=1
[αi]
)
/2, [α] = (α, α2/2, α3/3, . . . ), instead of expanding in y, and
compute the integral by taking residues, then we get addition formulae [20]:
m+1∑
i=1
(−1)i−1Cαβτ
x+
m−1∑
j=1
[βj ] + [αi]
τ
x+
m+1∑
j=1,j 6=i
[αj ]
= 0, m ≥ 2, (1)
mailto:yoko.shigyo@gmail.com
http://dx.doi.org/10.3842/SIGMA.2013.035
2 Y. Shigyo
where Cαβ depend only on {αi}1≤i≤m+1 and {βi}1≤i≤m−1. The simplest case of addition formulae
is the case of m=2:
α12α34τ(x+ [α1] + [α2])τ(x+ [α3] + [α4])− α13α24τ(x+ [α1] + [α3])τ(x+ [α2] + [α4])
+ α14α23τ(x+ [α1] + [α4])τ(x+ [α2] + [α3]) = 0, (2)
where αij = αi−αj . It is surprising that the KP hierarchy itself is equivalent to (2). This fact has
been proved by Takasaki and Takebe [23] by way of the wave functions of the KP hierarchy. Here
we give an alternative and direct proof. First we show that the totality of addition formulae (1)
is equivalent to the KP hierarchy by using the properties of symmetric functions. If we call the
function of the form τ(x+[α1]+ · · ·+[αn]) the n-point function, then (2) is a relation among two
point functions. By shifting x appropriately we can consider (2) as an expression of a four point
function in terms of two point functions (see Proposition 2). Repeating this process, we can
derive the formulae which express the 2m-point function as a determinant of 2-point functions.
These formulae are called Fay’s determinant formulae in the case of theta function [8, 19]. In [8]
it is indicated without proofs that the determinant formulae can be obtained from the trisecant
formulae corresponding to (2). In this sense the determinant formulae (11) and their derivation
from (2) cannot be considered a new result1. Next we show that the Plücker’s relations for the
determinants appearing in these formulae are nothing but the addition formulae (1) for m-point
functions. In this way, we can prove that (2) is equivalent to the KP hierarchy. For the mKP
and the BKP hierarchies, similar results hold although there are some differences.
The mKP hierarchy [12] is an infinite system of differential equations for an infinite number
of functions τl(x), l ∈ Z. In this case there are an infinite number of the simplest addition
formulae:
α23τl(x+ [α1])τl+1(x+ [α2] + [α3])− α13τl(x+ [α2])τl+1(x+ [α1] + [α3])
+ α12τl(x+ [α3])τl+1(x+ [α1] + [α2]) = 0, l ∈ Z. (3)
It had been proved that (3) is equivalent to the mKP hierarchy in [17]. Here we prove the
equivalence in a similar strategy to the case of KP. A new feature of the present case is that
there exist addition formulae involving τl and τl+k for k ≥ 2. We prove that these addition
formulae are consequences of (3).
The BKP hierarchy is an infinite system of bilinear equations for τ(x), x = (x1, x3, . . . ). The
following is the simplest addition formula which has four terms:
α̃12α̃13α23τ(x+ 2[α1]o)τ(x+ 2[α2]o + 2[α3]o)
− α̃12α̃23α13τ(x+ 2[α2]o)τ(x+ 2[α1]o + 2[α3]o)
+ α̃13α̃23α12τ(x+ 2[α3]o)τ(x+ 2[α1]o + 2[α2]o)
− α12α13α23τ(x)τ(x+ 2[α1]o + 2[α2]o + 2[α3]o) = 0, (4)
where α̃ij = αi + αj and [α]o =
(
α, α3/3, α5/5, . . .
)
. We prove that (4) is equivalent to the
BKP hierarchy in a similar way to the KP hierarchy. In this case we use Pfaffians instead of
determinants to express n-point functions in terms of one and two point functions. To this end
we need the analogue of Sylvester’s theorem and the Plücker’s relations for Pfaffians [9, 18].
We have shown that the KP, the mKP and the BKP hierarchies are equivalent to the simplest
addition formulae. It is interesting to study whether, for other integrable hierarchies [3, 4, 5,
12, 21, 22], similar structure exists. There exists a result for the Toda hierarchy [21, 24]. But
the problem arises to specify what are the fundamental equations in general. To consider these
problems, it will be effective to use free fermion descriptions of integrable hierarchies [3, 4, 5, 12].
1Takasaki K., Private communications.
On Addition Formulae of KP, mKP and BKP Hierarchies 3
It is also interesting to apply the results to the study of discrete differential geometries [1, 11]
and addition formulae for sigma functions [2, 7, 16].
This paper consists of three sections and three appendices. In Section 2, we consider the KP
hierarchy. The key point is to prove the equivalence between the KP hierarchy and its infinite
sequence of addition formulae. Since this case is fundamental, the details are given. Then we
consider the mKP hierarchy in Section 3. The arguments which are similar to the KP hierarchy
are omitted. In Section 4, we study the BKP hierarchy. The Pfaffians are necessary in this
case. Necessary properties of Pfaffians including the definition are reviewed in Appendices A, B
and C.
2 The addition formula for the τ -function of the KP hierarchy
Let
[α] =
(
α,
α2
2
,
α3
3
, . . .
)
, ξ(t, λ) =
∞∑
n=1
tnλ
n, t = (t1, t2, t3, . . . ).
The KP hierarchy is a system of equations for a function τ(t) [6, 12, 13] given by∮
eξ(t
′−t,λ)τ
(
t′ −
[
λ−1
])
τ
(
t+
[
λ−1
]) dλ
2πi
= 0. (5)
Here
∮
means a formal algebraic operator extracting the coefficient of z−1 of Laurent series:∮
dz
2πi
∞∑
n=−∞
anz
n = a−1.
Set t = x+ y, t′ = x− y. Then (5) becomes∮
e−2ξ(y,λ)τ
(
x− y −
[
λ−1
])
τ
(
x+ y +
[
λ−1
]) dλ
2πi
= 0. (6)
For an integer m ≥ 2, set
y =
1
2
(
m−1∑
i=1
[βi]−
m+1∑
i=1
[αi]
)
(7)
in (6). Then it becomes∮
exp
(
−ξ
(
m−1∑
i=1
[βi]−
m+1∑
i=1
[αi], λ
))
τ
(
x− 1
2
(
m−1∑
i=1
[βi]−
m+1∑
i=1
[αi]
)
−
[
λ−1
])
× τ
(
x+
1
2
(
m−1∑
i=1
[βi]−
m+1∑
i=1
[αi]
)
+
[
λ−1
]) dλ
2πi
= 0. (8)
By virtue of the identity
∞∑
n=1
xn
n
= − log(1− x),
the exponential factor in (8) reduces to a rational function of λ, αi, βi as
exp
(
−ξ
(
m−1∑
i=1
[βi]−
m+1∑
i=1
[αi], λ
))
=
m−1∏
i=1
(1− βiλ)
m+1∏
i=1
(1− αiλ)
.
4 Y. Shigyo
Computing the integral by taking residues at λ = α−1i , 1 ≤ i ≤ m + 1 [15] and shifting the
variable x as
x→ x+
1
2
(
m−1∑
i=1
[βi] +
m+1∑
i=1
[αi]
)
,
we get the following addition formulae of the τ -function [20]
m+1∑
i=1
(−1)i−1ζ(x;β1, . . . , βm−1, αi)ζ(x;α1, . . . , α̂i, . . . , αm+1) = 0, m ≥ 2, (9)
where
ζ(x;α1, . . . , αn) = ∆(α1, . . . , αn)τ(x+ [α1] + · · ·+ [αn]),
∆(α1, . . . , αn) =
∏
i<j
(αi − αj),
and α̂i means that αi should be removed.
Example 1. In the case of m = 2, we have
α12α34τ(x+ [α1] + [α2])τ(x+ [α3] + [α4])− α13α24τ(x+ [α1] + [α3])τ(x+ [α2] + [α4])
+ α14α23τ(x+ [α1] + [α4])τ(x+ [α2] + [α3]) = 0, (10)
where αij = αi − αj .
We call (10) ‘the three term equation’. We have derived (10) from (5). The fact that the
converse is true is proved by Takasaki and Takebe [23].
Theorem 1 ([23]). The three term equation (10) is equivalent to the KP hierarchy (5).
In [23] the theorem is proved by constructing the wave function of the KP-hierarchy. To do it
the differential Fay identity, which is a certain limit of (10), is used. Here we give an alternative
and direct proof of the theorem.
Proposition 1. The KP hierarchy (5) is equivalent to (9).
Proof. It is sufficient to prove that if (8) holds for any m ≥ 2 and arbitrary αi 6= 0, βi, then (6)
holds. Set the left hand side of (6) to be F (y). Expand F (y) in y as
F (y) =
∑
γ
Fγy
γ , yγ = yγ11 y
γ2
2 · · · , γ = (γ1, γ2, . . . ).
We consider the case βi = 0, 1 ≤ i ≤ m− 1 in (7) and set
y′ =
m+1∑
i=1
[αi].
We prove Fγ = 0 for any γ if F
(
−y′
2
)
= 0 for any m ≥ 2. We consider m fixed. Let us define
the weight of yi to be i and wt yγ =
∞∑
i=1
iγi. Decompose F according to weights as
F = F (0) + F (1) + F (2) + · · · , F (i) =
∑
wt yγ=i
Fγy
γ .
On Addition Formulae of KP, mKP and BKP Hierarchies 5
We substitute −y′
2 to y and get the homogeneous polynomial of degree i of α1, . . . , αm+1:
F (i)
(
−y
′
2
)
=
∑
γ1+···+γm+1=i
b(i)γ1...γm+1
αγ11 · · ·α
γm+1
m+1 .
Then F
(
−y′
2
)
= 0 is equivalent to F (i)
(
−y′
2
)
= 0 for any i. Notice that iy′i = αi1 + · · ·+ αim+1 is
a power sum symmetric function. Therefore y′1, . . . , y
′
m+1 are algebraically independent (see [14,
(2.12)]). If i ≤ m+ 1, then F (i)(y) is a polynomial at most of y1, . . . , ym+1. Thus F (i)
(
−y′
2
)
= 0
implies Fγ = 0 for any γ satisfying wt γ = i. Since m is arbitrary we have Fγ = 0 for any γ. �
Remark 1. In the course of the proof we actually prove the equivalence between (5) and (9)
with βi = 0 for any i (of course, in that case we have firstly to divide (9) by ∆(β1, . . . , βm−1)).
Proposition 2. The following formula follows from (10):
τ
(
x+
m∑
i=1
[βi]−
m∑
i=1
[αi]
)
τ(x)
=
m∏
i,j=1
(βi − αj)∏
i<j
αijβji
det
(
τ(x+ [βi]− [αj ])
(βi − αj)τ(x)
)
1≤i,j≤m
, m ≥ 2. (11)
Proof. We change the names of α variables in (10) as (α3, α4) → (β1, β2) and shift x to
x→ x− [α1]− [α2]. After that we solve it in τ(x+ [β1] + [β2]− [α1]− [α2]) we get m = 2 case
of (11).
Suppose that (11) holds in case of m = k:
τ
(
x+
k∑
i=1
[βi]−
k∑
i=1
[αi]
)
= τ(x)−k+1Ck det
(
τ(x+ [βi]− [αj ])
βi − αj
)
1≤i,j≤k
, (12)
where
Ck =
k∏
i,j=1
(βi − αj)∏
i<j
αijβji
.
In (12) we shift the variable x as
x→ x+ [βk+1]− [αk+1].
Then
τ
(
x+
k+1∑
i=1
[βi]−
k+1∑
i=1
[αi]
)
= τ(x+ [βk+1]− [αk+1])
−k+1Ck
× det
(
τ(x+ [βi] + [βk+1]− [αj ]− [αk+1])
βi − αj
)
1≤i,j≤k
. (13)
By (11) with m = 2,
τ(x+ [βi] + [βk+1]− [αj ]− [αk+1]) = τ(x)−1Aij ·Xij , (14)
6 Y. Shigyo
where
Aij =
(βi − αj)(βi − αk+1)(βk+1 − αj)(βk+1 − αk+1)
(αj − αk+1)(βk+1 − βi)
,
Xij = det
τ(x+ [βi]− [αj ])
βi − αj
τ(x+ [βi]− [αk+1])
βi − αk+1
τ(x+ [βk+1]− [αj ])
βk+1 − αj
τ(x+ [βk+1]− [αk+1])
βk+1 − αk+1
.
By substituting (14) to the determinant in the right hand side of (13), we have
det
(
τ(x+ [βi] + [βk+1]− [αj ]− [αk+1])
βi − αj
)
1≤i,j≤k
=
(
βk+1 − αk+1
τ(x)
)k k∏
i=1
(βk+1 − αi)(βi − αk+1)
(αi − αk+1)(βk+1 − βi)
det(Xij)1≤i,j≤k. (15)
Using Sylvester’s theorem (Appendix B),
det(Xij)1≤i,j≤k =
(
τ(x+ [βk+1]− [αk+1])
βk+1 − αk+1
)k−1
det
(
τ(x+ [βi]− [αj ])
βi − αj
)
1≤i,j≤k+1
. (16)
Substituting (16) and (15) into (13), we get the case of m = k + 1 of (11). �
Let us consider an N × m matrix A = (aij)1≤i≤N,1≤j≤m with N ≥ m and set, for 1 ≤
l1, . . . , lm ≤ N ,
A(l1, . . . , lm) = det(ali,j)1≤i,j≤m.
For any 1 ≤ k1, . . . , km−1, l1, . . . , lm+1 ≤ N these determinants satisfy Plücker’s relations:
m+1∑
i=1
(−1)i−1A(k1, . . . , km−1, li)A(l1, . . . , l̂i, . . . , lm+1) = 0. (17)
Proposition 3. The Plücker’s relations for the determinant of the right hand side of (11) give
the addition formulae (9).
Proof. Let m be fixed. Consider the ∞×m matrix A = (aij) with
aij =
τ(x+ [βi]− [αj ])
(βi − αj)τ(x)
.
Then A(k1, . . . , km−1, li) and A(l1, . . . , l̂i, . . . , lm+1) can be expressed by 2m point functions
by (11). We substitute them to (17) and shift the variable x as
x→ x+
m∑
s=1
[αs].
Then we get the additional formulae (9) by renaming the variables as (βk1 , . . . , βkm−1) →
(β1, . . . , βm−1), (βl1 , . . . , βlm+1)→ (α1, . . . , αm+1). �
Proof of Theorem 1. By Propositions 1, 2 and 3, we have (5) from (10). Thus Theorem 1 is
proved. �
On Addition Formulae of KP, mKP and BKP Hierarchies 7
3 mKP hierarchy
Let τl(t) (l ∈ Z) be τ -functions of the modified KP (mKP) hierarchy. We use the same notation
as that for KP hierarchy ([α], ξ(t, λ), etc.).
The mKP hierarchy is given by the bilinear equation [6, 12] of the form∮
eξ(t
′−t,λ)λl−l
′
τl
(
t′ −
[
λ−1
])
τl′
(
t+
[
λ−1
]) dλ
2πi
= 0, l ≥ l′. (18)
Set t = x+ y, t′ = x− y. Then (18) becomes∮
e−2ξ(y,λ)λl−l
′
τl
(
x− y −
[
λ−1
])
τl′
(
x+ y +
[
λ−1
]) dλ
2πi
= 0, l ≥ l′. (19)
Let l − l′ = k ≥ 0. Set
y =
1
2
(
m−2∑
i=1
[βi]−
m+k∑
i=1
[αi]
)
.
Then the exponential factor in (19) reduces to a rational function of λ, αi, βi as in the KP case:
exp
(
−ξ
(
m−2∑
i=1
[βi]−
m+k∑
i=1
[αi], λ
))
=
m−2∏
i=1
(1− βiλ)
m+k∏
i=1
(1− αiλ)
.
Computing the integral by taking residues at λ = α−1i , 1 ≤ i ≤ m+k and shifting the variable x
as
x→ x+
1
2
(
m−2∑
i=1
[βi] +
m+k∑
i=1
[αi]
)
,
we have the following addition formulae of the mKP hierarchy:
m+k∑
i=1
(−1)i−1ζl(x;β1, . . . , βm−2, αi)ζl+k(x;α1, . . . , α̂i, . . . , αm+k) = 0,
l ∈ Z, k ≥ 0, m ≥ 2, (20)
where
ζl(x;α1, . . . , αn) = ∆(α1, . . . , αn)τl
(
x+
n∑
i=1
[αi]
)
.
Example 2. The case l − l′ = 1 and m = 2 of (20) is
α23τl(x+ [α1])τl+1(x+ [α2] + [α3])− α13τl(x+ [α2])τl+1(x+ [α1] + [α3])
+ α12τl(x+ [α3])τl+1(x+ [α1] + [α2]) = 0. (21)
We call (21) ‘the three term equation of the mKP hierarchy’.
The following theorem is proved in [17].
Theorem 2 ([17]). The three term equation (21) is equivalent to the mKP hierarchy (18).
8 Y. Shigyo
We give an another proof which is similar to that of the KP hierarchy. The following propo-
sition can be proved as in the KP-case.
Proposition 4. The mKP hierarchy (18) is equivalent to (20).
Proposition 5. The following formula follows from (21) for n ≥ 2:
τl+1
(
x+
n∑
i=1
[αi]−
n−1∑
i=1
[βi]
)
τl(x)
= C det
τl(x+ [α1]− [β1])
(α1 − β1)τl(x)
· · · τl(x+ [α1]− [βn−1])
(α1 − βn−1)τl(x)
τl+1(x+ [α1])
τl(x)
τl(x+ [α2]− [β1])
(α2 − β1)τl(x)
· · · τl(x+ [α2]− [βn−1])
(α2 − βn−1)τl(x)
τl+1(x+ [α2])
τl(x)
...
. . .
...
...
τl(x+ [αn]− [β1])
(αn − β1)τl(x)
· · · τl(x+ [αn]− [βn−1])
(αn − βn−1)τl(x)
τl+1(x+ [αn])
τl(x)
, (22)
where
C = C(l, {αi}, {βi}) =
n∏
i=1
n−1∏
j=1
(αi − βj)(
n−1∏
i<j
βij
)(
n∏
i>j
αij
) .
Proof. The proof is similar to that of Proposition 2. Therefore we leave details to readers. �
Proposition 6. The Plücker’s relations for the determinant of the right hand side of (22)
give (20) with k = 1.
Proof. For m ≥ 1 consider the m× 2m matrix A = (aij)1≤i≤m, 1≤j≤2m given by
aij =
τl(x+ [αi]− [βj ])
αi − βj
, 1 ≤ j ≤ 2m− 1, ai,2m = τl+1(x+ [αi]).
For 1 ≤ r1, . . . , rm ≤ 2m we set
A(r1, . . . , rm) = det(ai,rj )1≤i,j≤m.
Then the Plücker’s relation gives k = 1 case of (20) by Propositions 2 and 5. �
By Propositions 5 and 6, equation (20) with k = 1 and arbitrary m ≥ 2 follows from (21).
The next lemma shows that (20) with k ≥ 2 and m ≥ 2 also follows from (21). The fact that (20)
with k = 0 follows from (21) is proved in [17]. We generalize the arguments in [17] for k ≥ 2.
Lemma 1. Equation (20) follows from (21).
Proof. We prove the lemma by induction on k. Suppose that equation (20) is valid for k and
any m ≥ 2. Shift the variable x as
x→ x−
m+k−1∑
j=1
[αj ],
On Addition Formulae of KP, mKP and BKP Hierarchies 9
and multiply the resulting equation by τl+k+1(x+ [αm+k+1]). Then we get
m+k−1∑
i=1
Aiτl
x− m+k−1∑
j 6=i
[αj ] +
m−2∑
j=1
[βj ]
τl+k (x+ [αm+k]− [αi]) τl+k+1(x+ [αm+k+1])
+Am+kτl
x+ [αm+k] +
m−2∑
j=1
[βj ]−
m+k−1∑
j=1
[αj ]
τl+k(x)τl+k+1(x+ [αm+k]) = 0, (23)
where
Ai = (−1)i−1∆(β1, . . . , βm−2, αi)∆(α1, . . . , α̂i, . . . , αm+k).
In (21) with l being replaced by l + k, make a shift x → x − [α3] and change the label of α as
(α1, α2, α3)→ (αm+k, αm+k+1, αi), 1 ≤ i ≤ m+ k − 1. Then we get
τl+k(x+ [αm+k]− [αi])τl+k+1(x+ [αm+k+1])
=
αm+k,i
αm+k+1,i
τl+k(x+ [αm+k+1]− [αi])τl+k+1(x+ [αm+k])
−
αm+k,m+k+1
αm+k+1,i
τl+k(x)τl+k+1(x+ [αm+k] + [αm+k+1]− [αi]). (24)
Substituting (24) to the summands of (23) and shifting x as x→ x+
m+k−1∑
j=1
[αj ], then we get
τl+k+1
x+
m+k∑
j=1
[αj ]
m+k−1∑
i=1
Ai
αm+k,i
αm+k+1,i
τl
x+ [αi] +
m−2∑
j=1
[βj ]
× τl+k
x+
m+k−1∑
j 6=i
[αj ] + [αm+k+1]
+ τl+k
x+
m+k−1∑
j=1
[αj ]
×
m+k−1∑
i=1
Ai
αm+k,m+k+1
αi,m+k+1
τl
x+ [αi] +
m−2∑
j=1
[βj ]
τl+k+1
x+
m+k+1∑
j 6=i
[αj ]
+Am+kτl
x+ [αm+k] +
m−2∑
j=1
[βj ]
τl+k+1
x+
m+k+1∑
j 6=m+k
[αj ]
= 0. (25)
We write (20) in the form
m+k−1∑
i=1
Aiτl
x+ [αi] +
m−2∑
j=1
[βj ]
τl+k
x+
m+k∑
j 6=i
[αj ]
= −Am+kτl
x+ [αm+k] +
m−2∑
j=1
[βj ]
τl+k
x+
m+k−1∑
j=1
[αj ]
. (26)
Change αm+k to αm+k+1 in (26). Notice that Ai, i < m+ k, changes to
(−1)i−1∆(β1, . . . , βm−2, αi)∆(α1, . . . , α̂i, . . . , αm+k−1, αm+k+1)
=
m+k−1∏
j=1
αj,m+k+1
αj,m+k
·Ai ·
αi,m+k
αi,m+k+1
10 Y. Shigyo
and Am+k changes to
(−1)m+k∆(β1, . . . , βm−2, αm+k+1)∆(α1, . . . , αm+k−1).
Then we can rewrite the first term of (25) as
τl+k+1
x+
m+k∑
j=1
[αj ]
m+k−1∑
i=1
Ai
αi,m+k
αi,m+k+1
τl
x+ [αi] +
m−2∑
j=1
[βj ]
× τl+k
x+
m+k−1∑
j 6=i
[αj ] + [αm+k+1]
= τl+k+1
x+
m+k∑
j=1
[αj ]
Bm+k+1
× τl
x+ [αm+k+1] +
m−2∑
j=1
[βj ]
τl+k
x+
m+k−1∑
j=1
[αj ]
, (27)
where
Bm+k+1 = (−1)m+k+1∆(αm+k+1, β1, . . . , βm−2)∆(α1, . . . , αm+k)
m+k−1∏
j=1
αj,m+k+1
.
Substitute (27) to (25) and divide the resulting equation by τl+k
(
x+
m+k−1∑
j=1
[αj ]
)
. We, then,
multiply it by
m+k−1∏
j=1
αj,m+k+1 and get the case of k + 1 of (20). �
4 BKP hierarchy
Let τ(t) be the τ -function of the BKP hierarchy. In this case, the time variable is t =
(t1, t3, t5, . . . ). We set
[α]o =
(
α,
α3
3
,
α5
5
, . . .
)
, ξ̃(t, λ) =
∞∑
n=1
t2n−1λ
2n−1.
The BKP hierarchy [6, 12] is defined by∮
eξ̃(t−t
′,λ)τ
(
t− 2
[
λ−1
]
o
)
τ
(
t′ + 2
[
λ−1
]
o
) dλ
2πiλ
= τ(t)τ(t′). (28)
Set t = x− y, t′ = x+ y. We get∮
e−2ξ̃(y,λ)τ
(
x− y − 2
[
λ−1
]
o
)
τ
(
x+ y + 2
[
λ−1
]
o
) dλ
2πiλ
= τ(x− y)τ(x+ y). (29)
Set
y =
n∑
i=1
[αi]o
in (29). Then we have∮
e
−2ξ̃
( n∑
i=1
[αi]o,λ
)
τ
(
x−
n∑
i=1
[αi]o − 2
[
λ−1
]
o
)
τ
(
x+
n∑
i=1
[αi]o + 2
[
λ−1
]
o
)
dλ
2πiλ
On Addition Formulae of KP, mKP and BKP Hierarchies 11
= τ
(
x−
n∑
i=1
[αi]o
)
τ
(
x+
n∑
i=1
[αi]o
)
.
By decomposing −2
∞∑
n=1
t2n−1λ
2n−1 as
−2
∞∑
n=1
t2n−1λ
2n−1 = −
∞∑
n=1
tnλ
n +
∞∑
n=1
tn(−λ)n,
we get
exp
(
−2ξ̃
(
n∑
i=1
[αi]o, λ
))
=
n∏
i=1
1− αiλ
1 + αiλ
.
Computing the integral by taking residues as before, shifting x as x +
n∑
i=1
[αi]o and dividing
by τ(x)2 we have
n∑
i=1
(−1)i−1
τ(x+ 2[αi]o)
τ(x)
A−1
1...̂i...n
τ
(
x+ 2
n∑
l=1, l 6=i
[αl]o
)
τ(x)
−A−11...n
τ
(
x+ 2
n∑
l=1
[αl]o
)
τ(x)
= 0, n odd, (30)
n−1∑
i=1
(−1)i−1
αi,n
α̃i,n
τ(x+ 2[αi]o + 2[αn]o)
τ(x)
A−1
1...̂i...n−1
τ
(
x+ 2
n−1∑
l=1, l 6=i
[αl]o
)
τ(x)
−A−11...n
τ
(
x+ 2
n∑
l=1
[αl]o
)
τ(x)
= 0, n even. (31)
Here A1...n is defined by
A1...n =
n∏
i<j
α̃ij
αij
, α̃ij = αi + αj , αij = αi − αj .
Example 3. The case n = 3 of (30) becomes
τ
(
x+ 2
3∑
i=1
[αi]o
)
τ(x)
= A123
{
τ(x+ 2[α1]o)
τ(x)
α23
α̃23
τ(x+ 2[α2]o + 2[α3]o)
τ(x)
−τ(x+ 2[α2]o)
τ(x)
α13
α̃13
τ(x+ 2[α1]o + 2[α3]o)
τ(x)
+
τ(x+ 2[α3]o)
τ(x)
α12
α̃12
τ(x+ 2[α1]o + 2[α2]o)
τ(x)
}
. (32)
We call equation (32) ‘the four term equation of the BKP hierarchy’.
12 Y. Shigyo
Example 4. The case of n = 4 of (31) is
τ
(
x+ 2
4∑
i=1
[αi]o
)
τ(x)
= A1234
{
α14
α̃14
τ(x+ 2[α1]o + 2[α4]o)
τ(x)
α23
α̃23
τ(x+ 2[α2]o + 2[α3]o)
τ(x)
− α24
α̃24
τ(x+ 2[α2]o + 2[α4]o)
τ(x)
α13
α̃13
τ(x+ 2[α1]o + 2[α3]o)
τ(x)
+
α34
α̃34
τ(x+ 2[α3]o + 2[α4]o)
τ(x)
α12
α̃12
τ(x+ 2[α1]o + 2[α2]o)
τ(x)
}
. (33)
As is proved in Proposition 8, equation (33) can be derived from equation (32).
Theorem 3. The four term equation (32) is equivalent to the BKP hierarchy (28).
We prove this theorem in a similar way to the case of the KP hierarchy.
In order to prove the theorem, we use Pfaffians. The definition and notation of Pfaffians
are reviewed in Appendix A.
Let us define the components of Pfaffians by
(0, j) =
τ(x+ 2[αj ]o)
τ(x)
, (i, j) =
αij
α̃ij
τ(x+ 2[αi]o + 2[αj ]o)
τ(x)
.
Then it is possible to rewrite (32) and (33) as
τ
(
x+ 2
3∑
i=1
[αi]o
)
τ(x)
= A123(0, 1, 2, 3), (34)
τ
(
x+ 2
4∑
i=1
[αi]o
)
τ(x)
= A1234(1, 2, 3, 4), (35)
respectively.
The following proposition can be proved in a similar manner to Proposition 1.
Proposition 7. The BKP hierarchy (28) is equivalent to (30) and (31).
Proposition 8. The following equations are implied by (32):
τ
(
x+ 2
n∑
i=1
[αi]o
)
τ(x)
= A1...n(0, 1, 2, . . . , n), n ≥ 3, odd, (36)
τ
(
x+ 2
n∑
i=1
[αi]o
)
τ(x)
= A1...n(1, 2, . . . , n), n ≥ 4, even. (37)
Proof. First we prove that (32) implies (33). Shift x in (32) as x→ x+ 2[α4]o:
τ
(
x+ 2
4∑
i=1
[αi]o
)
τ(x+ 2[α4]o)
= A123
τ(x)2
τ(x+ 2[α4]o)2
{
τ(x+ 2[α1]o + 2[α4]o)
τ(x)
α23
α̃23
τ(x+ 2[α2]o + 2[α3]o + 2[α4]o)
τ(x)
On Addition Formulae of KP, mKP and BKP Hierarchies 13
− τ(x+ 2[α2]o + 2[α4]o)
τ(x)
α13
α̃13
τ(x+ 2[α1]o + 2[α3]o + 2[α4]o)
τ(x)
+
τ(x+ 2[α3]o + 2[α4]o)
τ(x)
α12
α̃12
τ(x+ 2[α1]o + 2[α2]o + 2[α4]o)
τ(x)
}
. (38)
Use (34) to rewrite τ(x + 2[αi1 ]o + 2[αi2 ]o + 2[αi3 ]o) in (38). Then we get (35). We prove (36)
by induction on n. The case n = 3 is obvious. Suppose that (36) holds in case of n:
τ
(
x+ 2
n∑
i=1
[αi]o
)
τ(x)
= A1...n(0, 1, 2, . . . , n) = A1...nPf A, (39)
where A = (aij)0≤i,j≤n is a skew-symmetric matrix,
aij =
τ(x+ 2[αj ]o)
τ(x)
, i = 0,
αij
α̃ij
τ(x+ 2[αi]o + 2[αj ]o)
τ(x)
, i 6= 0, i < j.
In (39) we shift x as
x→ x+ 2[αn+1]o + 2[αn+2]o.
Then we have, using (34) and (35),
τ
(
x+ 2
n+2∑
i=1
[αi]o
)
τ(x)
= A1...n+2(n+ 1, n+ 2)−
n−1
2 Pf B, (40)
where B = (bij)0≤i<j≤n is a skew-symmetric matrix given by
bij = (n+ 1, n+ 2, i, j), i < j.
By the analogue of the Sylvester’ theorem for Pfaffians (Appendix B), we have
Pf((1, 2, . . . , 2r, i, j))2r+1≤i<j≤2m = (1, 2, . . . , 2r)m−r−1(1, 2, . . . , 2m).
Consider the case r = 1 and 2m = n+ 3 of this formula:
Pf((n+ 1, n+ 2, i, j))0≤i<j≤n = (n+ 1, n+ 2)
n−1
2 (n+ 1, n+ 2, 0, . . . , n). (41)
Substituting (41) to (40), we get
τ
(
x+ 2
n+2∑
i=1
[αi]o
)
τ(x)
= A1...n+2(0, 1, . . . , n+ 2).
The case of even n is similarly proved. �
Proposition 9. The Plücker’s relations for Pfaffians of the right hand side of (36) and (37)
give the addition formulae (30) and (31) respectively.
Proof. Using the Plücker’s relations (43) and (44) for Pfaffians given in Appendix C, the
proposition can easily be checked by direct calculations. �
14 Y. Shigyo
A Pfaffians
Let A = (aij)1≤i,j≤2m be a skew-symmetric matrix. Then the Pfaffian Pf A is defined by
detA = (Pf A)2, Pf A = a12a34 · · · a2m−1,2m + · · · .
Following [10] we denote Pf A by (1, 2, 3, . . . , 2m):
Pf A = (1, 2, 3, . . . , 2m).
It is directly defined by
(1, 2, 3, . . . , 2m) =
∑
sgn(i1, . . . , i2m) · (i1, i2)(i3, i4) · · · (i2m−1, i2m), (i, j) = aij ,
where the sum is over all permutations of (1, . . . , 2m) such that
i1 < i3 < · · · < i2m−1, i1 < i2, . . . , i2m−1 < i2m,
and sgn(i1, . . . , i2m) is the signature of the permutation (i1, . . . , i2m). The Pfaffian can be
expanded as
(1, 2, 3, . . . , 2m) =
2m∑
j=2
(−1)j(1, j)(2, 3, . . . , ĵ, . . . , 2m).
For example, in the case of m = 2,
(1, 2, 3, 4) = (1, 2)(3, 4)− (1, 3)(2, 4) + (1, 4)(2, 3).
B Sylvester’s theorem for determinants and Pfaffians
The following theorem is known as Sylvester’s theorem.
Theorem 4. Let r ≤ m, A = (aij)1≤i,j≤m and Ar = (aij)1≤i,j≤r. Set
B = (bij)r+1≤i,j≤m, bij = det
a11 . . . a1r a1j
...
. . .
...
...
ar1 . . . arr arj
ai1 . . . air aij
.
Then we get
detB = (detAr)m−r−1 detA.
Let A = (aij)1≤i,j≤2m be a skew-symmetric matrix and set (i, j) = aij . For r ≤ m let
P = (pij)2r+1≤i,j≤2m, pij = (1, 2, . . . , 2r, i, j) and Ir = {1, 2, . . . , 2r}. In general, for a subset
I ⊂ {1, 2, . . . , 2m} we set A(I) = (aij)i,j∈I and for i < j, k < l we denote by Aijkl be the square
matrix of degree 2(m − 1) which is obtained from A by removing i-th and j-th rows, k-th and
l-th columns.
Theorem 5 ([9]). For r ≤ m the following identity holds:
Pf P = (Pf A(Ir))
m−r−1Pf A.
On Addition Formulae of KP, mKP and BKP Hierarchies 15
C The Plücker relation for Pfaffians
There exist analogues of the Plücker’s relations for Pfaffians [18]. They are given by
L∑
l=1
(−1)l(i1, . . . , iK , jl)(j1, . . . , ĵl, . . . , jL)
+
K∑
k=1
(−1)k(i1, . . . , îk, . . . , iK)(j1, . . . , jL, ik) = 0, (42)
where K and L are odd. We understand that (∅) = 1.
For n odd, taking K = 1, L = n, i1 = 0 and j1, . . . , jn 6= 0 in (42), we get
n∑
l=1
(−1)l−1(0, jl)(j1, . . . , ĵl, . . . , jn)− (0, j1, . . . , jn) = 0. (43)
For n even, setting K = 1, L = n− 1 and i1 6= 0 in (42), we have
n−1∑
l=1
(−1)l−1(i1, jl)(j1, . . . , ĵl, . . . , jn−1)− (i1, j1, . . . , jn−1) = 0. (44)
Acknowledgements
I would like to thank Masatoshi Noumi and Takashi Takebe for permitting me to see the
manuscript [17] prior to its publication. I also thank Kanehisa Takasaki and Takashi Takebe
for insightful comments and their interests in the present work. I also thank Yasuhiro Ohta and
Soichi Okada for important comments about Pfaffians and informing me about reference [9].
Finally I am deeply grateful to Atsushi Nakayashiki for much advice. This research is supported
by JSPS Grant-in-Aid for Scientific Research (B) No. 23340037.
References
[1] Bobenko A.I., Suris Yu.B., Discrete differential geometry. Integrable structure, Graduate Studies in Mathe-
matics, Vol. 98, Amer. Math. Soc., Providence, RI, 2008.
[2] Buchstaber V.M., Enolski V.Z., Leykin D.V., Kleinian functions, hyperelliptic Jacobians and applications,
Rev. Math and Math. Phys. 10 (1997), no. 2, 1–125, solv-int/9603005.
[3] Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. I, J. Phys. Soc. Japan 51
(1982), 4116–4124.
[4] Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. II, J. Phys. Soc. Japan 51
(1982), 4125–4131.
[5] Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. V, J. Phys. Soc. Japan 52
(1982), 766–771.
[6] Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear In-
tegrable Systems – Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore,
1983, 39–119.
[7] Eilbeck J.C., Enolski V.Z., Gibbons J., Sigma, tau and Abelian functions of algebraic curves, J. Phys. A:
Math. Theor. 43 (2010), 455216, 20 pages, arXiv:1006.5219.
[8] Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag,
Berlin, 1973.
[9] Hirota R., Generalizations of determinant identities by Pfaffian, in Mathematical Theories and Applications
of Nonlinear Waves and Nonlinear Dynamics, Research Institute for Applied Mechanics, Kyushu University,
2004, 148–156.
http://dx.doi.org/10.1007/978-3-7643-8621-4
http://dx.doi.org/10.1007/978-3-7643-8621-4
http://arxiv.org/abs/solv-int/9603005
http://dx.doi.org/10.1143/JPSJ.51.4116
http://dx.doi.org/10.1143/JPSJ.51.4125
http://dx.doi.org/10.1143/JPSJ.52.766
http://dx.doi.org/10.1088/1751-8113/43/45/455216
http://dx.doi.org/10.1088/1751-8113/43/45/455216
http://arxiv.org/abs/1006.5219
16 Y. Shigyo
[10] Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge
University Press, Cambridge, 2004.
[11] Inoguchi J., Kajiwara K., Matsuura N., Ohta Y., Explicit solutions to the semi-discrete modified KdV
equation and motion of discrete plane curves, arXiv:1108.1328.
[12] Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983),
943–1001.
[13] Miwa T., Jimbo M., Date E., Solitons. Differential equations, symmetries and infinite-dimensional algebras,
Cambridge Tracts in Mathematics, Vol. 135, Cambridge University Press, Cambridge, 2000.
[14] Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The
Clarendon Press, Oxford University Press, New York, 1995.
[15] Miwa T., On Hirota’s difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 9–12.
[16] Nakayashiki A., Sigma function as a tau function, Int. Math. Res. Not. 2010 (2010), no. 3, 373–394,
arXiv:0904.0846.
[17] Noumi M., Takebe T., Algebraic analysis of integrable hierarchies, in preparation.
[18] Ohta Y., Bilinear theory of solitons with Pfaffian labels, Sūrikaisekikenkyūsho Kōkyūroku (1993), no. 822,
197–205.
[19] Raina A.K., Fay’s trisecant identity and conformal field theory, Comm. Math. Phys. 122 (1989), 625–641.
[20] Sato M., Sato Y., Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, in
Nonlinear PDE in Applied Science, North-Holland Math. Stud., Vol. 81, Editors H. Fujita, P. Lax, G. Strang,
Tokyo, 1982, 259–271.
[21] Takasaki K., Differential Fay identities and auxiliary linear problem of integrable hierarchies, in Exploring
New Structures and Natural Constructions in Mathematical Physics, Adv. Stud. Pure Math., Vol. 61, Math.
Soc. Japan, Tokyo, 2011, 387–441, arXiv:0710.5356.
[22] Takasaki K., Dispersionless Hirota equations of two-component BKP hierarchy, SIGMA 2 (2006), 057,
22 pages, nlin.SI/0604003.
[23] Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743–808,
hep-th/9405096.
[24] Teo L.P., Fay-like identities of the Toda lattice hierarchy and its dispersionless limit, Rev. Math. Phys. 18
(2006), 1055–1073, nlin.SI/0606059.
http://dx.doi.org/10.1017/CBO9780511543043
http://arxiv.org/abs/1108.1328
http://dx.doi.org/10.2977/prims/1195182017
http://dx.doi.org/10.3792/pjaa.58.9
http://dx.doi.org/10.1093/imrn/rnp135
http://arxiv.org/abs/0904.0846
http://dx.doi.org/10.1007/BF01256498
http://arxiv.org/abs/0710.5356
http://dx.doi.org/10.3842/SIGMA.2006.057
http://arxiv.org/abs/nlin.SI/0604003
http://dx.doi.org/10.1142/S0129055X9500030X
http://arxiv.org/abs/hep-th/9405096
http://dx.doi.org/10.1142/S0129055X06002838
http://arxiv.org/abs/nlin.SI/0606059
1 Introduction
2 The addition formula for the -function of the KP hierarchy
3 mKP hierarchy
4 BKP hierarchy
A Pfaffians
B Sylvester's theorem for determinants and Pfaffians
C The Plücker relation for Pfaffians
References
|