Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation
A multi-Poisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form ∂Xλ/∂t=[Xλ,Aλ] in the sense of the isospectral deformation, where Xλ,Aλ∈g depend rationally on the indeterminate λ called the spectral parameter. I...
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| description | A multi-Poisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form ∂Xλ/∂t=[Xλ,Aλ] in the sense of the isospectral deformation, where Xλ,Aλ∈g depend rationally on the indeterminate λ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation ∂Xλ/∂t=[Xλ,Aλ]+∂Aλ/∂λ in the sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed. This method gives a few new Painlevé systems of dimension four.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 025, 27 pages
Multi-Poisson Approach to the Painlevé Equations:
from the Isospectral Deformation
to the Isomonodromic Deformation
Hayato CHIBA
Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan
E-mail: chiba@imi.kyushu-u.ac.jp
Received October 11, 2016, in final form April 11, 2017; Published online April 15, 2017
https://doi.org/10.3842/SIGMA.2017.025
Abstract. A multi-Poisson structure on a Lie algebra g provides a systematic way to
construct completely integrable Hamiltonian systems on g expressed in Lax form ∂Xλ/∂t =
[Xλ, Aλ] in the sense of the isospectral deformation, where Xλ, Aλ ∈ g depend rationally on
the indeterminate λ called the spectral parameter. In this paper, a method for modifying the
isospectral deformation equation to the Lax equation ∂Xλ/∂t = [Xλ, Aλ] + ∂Aλ/∂λ in the
sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed.
This method gives a few new Painlevé systems of dimension four.
Key words: Painlevé equations; Lax equations; multi-Poisson structure
2010 Mathematics Subject Classification: 34M35; 34M45; 34M55
1 Introduction
A differential equation defined on a complex region is said to have the Painlevé property if any
movable singularity of any solution is a pole. Painlevé and his group classified second order
ODEs having the Painlevé property and found new six differential equations called the Painlevé
equations. Nowadays, it is known that they are written in Hamiltonian forms
(PJ) :
dq
dt
=
∂HJ
∂p
,
dp
dt
= −∂HJ
∂q
, J = I, . . . ,VI.
Among six Painlevé equations, the Hamiltonian functions of the first, second and fourth Painlevé
equations are polynomials in both of the independent variable t and the dependent variab-
les (q, p). They are given by
HI =
1
2
p2 − 2q3 − tq, (1.1)
HII =
1
2
p2 − 1
2
q4 − 1
2
tq2 − αq, (1.2)
HIV = −pq2 + p2q − 2pqt− αp+ βq, (1.3)
respectively, where α, β ∈ C are arbitrary parameters. Another important property of the
Painlevé equations is that they are expressed as Lax equations. Let Lλ and Aλ be square
matrices which depend rationally on the indeterminate λ called the spectral parameter. The
Painlevé equations are written in Lax form as
∂Lλ
∂t
= [Lλ, Aλ] +
∂Aλ
∂λ
, (1.4)
This paper is a contribution to the Special Issue on Symmetries and Integrability of Difference Equations.
The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html
mailto:chiba@imi.kyushu-u.ac.jp
https://doi.org/10.3842/SIGMA.2017.025
http://www.emis.de/journals/SIGMA/SIDE12.html
2 H. Chiba
for some choice of Lλ and Aλ. This equation arises from the compatibility condition of the two
differential systems
∂Ψ
∂λ
= LλΨ,
∂Ψ
∂t
= AλΨ.
Since the monodromy of the former system ∂Ψ/∂λ = LλΨ is independent of t if the equation (1.4)
is satisfied, (1.4) is called the isomonodromic deformation equation.
Another type of the Lax equation is of the form
∂Xλ
∂t
= [Xλ, Aλ], (1.5)
which is called the isospectral deformation equation because the eigenvalues of the matrix Xλ
is independent of t. There are several systematic ways to construct isospectral deformation
equations [1]. In particular, a Lie algebraic method have been often employed. Let g be a Lie
algebra. On the dual space g∗, there exists a canonical Poisson structure called the Lie–Poisson
structure. If g is equipped with a nondegenerate bilinear symmetric form, the Lie–Poisson
structure is also defined on g. Let P : T ∗g→ Tg be the Poisson tensor and F : g→ C a smooth
function. Then, the vector field PdF on g can be expressed as the Lax equation (1.5) with some
Xλ, Aλ ∈ g [1].
It is notable that the isospectral deformation equation (1.5) is completely integrable for
most examples, although the isomonodromic deformation equation (1.4) is not in general; it is
believed that solutions of an isomonodromic deformation equation define new functions called
the Painlevé transcendents.
In Nakamura [17], a way to obtain the isospectral deformation equation (1.5) from the isomon-
odromic deformation equation (1.4) by a certain scaling of the time t is proposed, which is called
the autonomous limit. She proved that the autonomous limits of 6-types of two dimensional
Painlevé equations and 40-types of four dimensional Painlevé equations are completely inte-
grable. Such relations of Painlevé systems with autonomous integrable systems are known
between Gaudin model and Schlesinger system, and also found in [13] (Painlevé–Calogero cor-
respondence).
The purpose in the present paper is opposite; a way to construct the isomonodromic defor-
mation equation (1.4) from the isospectral deformation equation (1.5) will be proposed. Let g
be a simple Lie algebra over C. Consider the set of g-valued polynomials of degree n
gn :=
{
Xλ := X0λ
n +X1λ
n−1 + · · ·+Xn |Xi ∈ g
}
,
with the indeterminate λ. This set gn is equipped with a structure of a Lie algebra by a certain
Lie bracket. At first, the isospectral deformation equation (1.5) on gn is constructed with the
aid of the bi-Poisson theory of Magri et al. [9, 14, 15, 16]. Isospectral deformation equations
obtained in this method are shown to be completely integrable (Theorem 2.4). Next, we restrict
the equations onto a symplectic leaf. Let ϕ1, . . . , ϕN be Casimir functions of an underlying
Poisson structure on gn. A symplectic leaf S is defined by the level surface of them as
S := {ϕi = αi (const) | i = 1, . . . , N}.
Restricted on the leaf S, the isospectral deformation equation (1.5) becomes an integrable Hamil-
tonian system. Since the matrix Xλ ∈ gn depends on the parameters α := (α1, . . . , αN ), it is
denoted as Xλ = Xλ(t, α).
Now suppose that there exists a parameter, say αj , such that the following condition holds
∂Xλ
∂αj
(t, α) =
∂Aλ
∂λ
. (1.6)
Multi-Poisson Approach to the Painlevé Equations 3
Equation (1.5) is put together with equation (1.6) to yield
∂Xλ
∂t
(t, α) +
∂Xλ
∂αj
(t, α) = [Xλ, Aλ] +
∂Aλ
∂λ
.
Define the Lax matrix Lλ by
Lλ := Xλ(t, α)|αj=t,
where the parameter αj satisfying the condition (1.6) is replaced by t. Then, the above equation
is rewritten as the isomonodromic deformation equation (1.4).
Remark that the isomonodromic deformation equation (1.4) is equivalent to the zero curvature
condition of the connection 1 form Lλdλ+Aλdt on a vector bundle over the (t, λ)-space, while
the condition (1.6) is the exactness condition of the connection 1 form Xλdλ+Aλdαj .
This method is demonstrated for the following three cases (I) g = sl(2,C), n = 2, (II) g =
sl(2,C), n = 3 and (III) g = so(5,C), n = 1. For the case (I), the first, second and fourth
Painlevé equations (1.1), (1.2), (1.3) will be obtained in Section 3.
More generally, for g = sl(2,C) with general n, it seems that one can obtain several Painlevé
hierarchies of dimension 2n − 2, including the first Painlevé hierarchy (PI)m [11, 12, 18],
the second-first Painlevé hierarchy (PII-1)m [5, 6, 11, 12], the second-second Painlevé hier-
archy (PII-2)m and the fourth Painlevé hierarchy (PIV)m [10, 11]. They are 2m-dimensional
Hamiltonian PDEs of the form (m = n− 1)
∂qj
∂ti
=
∂Hi
∂pj
,
∂pj
∂ti
= −∂Hi
∂qj
, j = 1, . . . ,m, i = 1, . . . ,m,
Hi = Hi(q1, . . . , qm, p1, . . . , pm, t1, . . . , tm)
consisting of m Hamiltonians H1, . . . ,Hm with m independent variables t1, . . . , tm. When m = 1
(the case (I)), (PI)1 and (PIV)1 are reduced to the first and fourth Painlevé equations, respec-
tively. Both of (PII-1)1 and (PII-2)1 coincide with the second Painlevé equation, while they are
different systems for m ≥ 2. When m = 2 (the case (II)), Hamiltonians of (PI)2, (PII-1)2, (PII-2)2
and (PIV)2 are given by
(PI)2
H1 = 2p2p1 + 3p22q1 + q41 − q21q2 − q22 − t1q1 + t2
(
q21 − q2
)
,
H2 = p21 + 2p2p1q1 − q51 + p22q2 + 3q31q2 − 2q1q
2
2
+ t1
(
q21 − q2
)
+ t2
(
t2q1 + q1q2 − p22
)
,
(1.7)
(PII-1)2
H1 = 2p1p2 − p32 − p1q21 + q22 − t1p2 + t2p1 + 2αq1,
H2 = −p21 + p1p
2
2 + p1p2q
2
1 + 2p1q1q2
+ t1p1 + t2
(
t2p1 − p1q21 + p1p2
)
− α(2p2q1 + 2q2 + 2t2q1),
(1.8)
(PII-2)2
H1 = p1p2 − p1q21 − 2p1q2 + p2q1q2 + q1q
2
2 + q2t1 + t2(q1q2 − p1) + αq1,
H2 = p21 − p1p2q1 + p22q2 − 2p1q1q2 − p2q22 + q21q
2
2
+ t1(q1q2 − p1)− t2
(
p1q1 + q22 + q2t2
)
+ αp2,
(1.9)
(PIV)2
H1 = p21 + p1p2 − p1q21 + p2q1q2 − p2q22 − t1p1 + t2p2q2 + αq2 + βq1,
H2 = p1p2q1 − 2p1p2q2 − p22q2 + p2q1q
2
2
+ p2q2t1 + t2
(
p1p2 − p2q22 + p2q2t2
)
+ (p1 − q1q2 + q2t2)α− βp2,
(1.10)
respectively, with arbitrary parameters α, β ∈ C. These systems will be obtained from the
case (II) g = sl(2,C), n = 3 in Section 4. In our method, such Hamiltonian PDEs are obtained
if there are several Hamiltonian systems written in Lax form (1.5), and if there are several
parameters satisfying (1.6); such parameters will be replaced by distinct times t1, t2, . . . .
4 H. Chiba
We will find other 4-dimensional Painlevé systems with Hamiltonian functions
H(1,1,2,0) = −p21q1 − 2p1q
2
1 + 2p1q2 − 2p1p2q2 − 2p2q1q2
+ (2p1q1 + 2p2q2)t+ (2α2 + 2β2)q1 + 2β2p1 + 2β3p2, (1.11)
H(−1,4,1,2) = p1 − p22 − 2p1q1q2 − p2q22 + 2β3q2 + 2β5q1 + p2t, (1.12)
HCosgrove = −4p1p2 − 2p22q1 −
73
128
q41 +
11
8
q21q2 −
1
2
q22 − q1t−
α2
48
(
q1 +
α2
6
)
q21, (1.13)
where αi, βi ∈ C are arbitrary parameters (the subscripts for parameters are related to the
weighted degrees so that the Hamiltonian functions become quasihomogeneous, see below). The
first two systems will be also obtained from the case (II). As far as the author knows, these
systems have not appeared in the literature. The last one HCosgrove will be obtained from the
case (III) g = so(5,C), n = 1 in Section 5. If we rewrite the system as a fourth order single
equation of q1 = y, we obtain
y′′′′ = 18yy′′ + 9(y′)2 − 24y3 + 16t+ αy
(
y +
1
9
α
)
. (1.14)
This equation was given in [7], denoted by F-VI, without a proof that it has the Painlevé property.
Since this system is obtained as the isomonodromic deformation equation in this paper, this
equation actually enjoys the Painlevé property. In [7] it is conjectured that this equation defines
a new Painlevé transcendent (i.e., it is not reduced to known equations). Another expression of
the Hamiltonian function of the same system is
H̃Cosgrove = 2p1p2 −
18
13
p22q1 −
2
169
q41 −
180
13
q21q2 + 6q22 − 8q1t+
8
9
α2q
3
1 +
8
27
α2
2q
2
1. (1.15)
The corresponding Hamiltonian system is also reduced to (1.14).
Note that all of the Hamiltonian functions above are polynomials in both of the independent
variables and the dependent variables. Furthermore, they are semi-quasihomogeneous functions.
In general, a polynomial H(x1, . . . , xn) is called a quasihomogeneous polynomial if there are
integers a1, . . . , an and h such that
H
(
λa1x1, . . . , λ
anxn
)
= λhH(x1, . . . , xn) (1.16)
for any λ ∈ C. A polynomial H is called a semi-quasihomogeneous if H is decomposed into two
polynomials as H = HP +HN , where HP satisfies (1.16) and HN satisfies
HN
(
λa1x1, . . . , λ
anxn
)
∼ o
(
λh
)
, |λ| → ∞.
The integer wdeg(H) := h is called the weighted degree of H with respect to the weight
wdeg(x1, . . . , xn) := (a1, . . . , an). For example, if we define degrees of variables by wdeg(q, p, t) =
(2, 3, 4) for HI, wdeg(q, p, t) = (1, 2, 2) for HII and wdeg(q, p, t) = (1, 1, 1) for HIV, then Hamil-
tonian functions have the weighted degrees 6, 4 and 3, respectively (Table 1). The weights for
four dimensional systems above are shown in Table 2. In this paper, these weights are naturally
obtained from a suitable definition of weights of entries of a matrix Xλ ∈ gn and the spectral
parameter λ. In particular, the weights of the Hamiltonian functions are closely related to the
exponents of simple Lie algebras because the Hamiltonian functions are essentially Ad-invariant
polynomials of simple Lie algebras. See [2, 3, 4] for the detailed study of the weights of the
Painlevé equations.
Multi-Poisson Approach to the Painlevé Equations 5
Table 1. Weights for two dimensional Painlevé equations.
wdeg(q, p, t) wdeg(H)
PI (2, 3, 4) 6
PII (1, 2, 2) 4
PIV (1, 1, 1) 3
Table 2. Weights for four dimensional Painlevé equations.
wdeg(q1, p1, q2, p2) wdeg(t1, t2) wdeg(H1, H2)
(PI)2 (2, 5, 4, 3) 6, 4 8, 10
(PII-1)2 (1, 4, 3, 2) 4, 2 6, 8
(PII-2)2 (1, 3, 2, 2) 3, 2 5, 6
(PIV)2 (1, 2, 1, 2) 2, 1 4, 5
H(1,1,2,0) (1, 1, 2, 0) 1 3
H(−1,4,1,2) (−1, 4, 1, 2) 2 4
HCosgrove (2, 5, 4, 3) 6 8
2 Settings
2.1 Lie–Poisson structure on gn
We define a multi-Poisson structure on a certain Lie algebra following Magri et al. [9, 14, 15, 16].
Let (g, [ · , · ]) be a simple Lie algebra over C. Consider the set of g-valued polynomials of degree n
gn :=
{
Xλ := X0λ
n +X1λ
n−1 + · · ·+Xn |Xi ∈ g
}
,
with the indeterminate λ. The bracket defined by
[Xλ, Yλ]n := [Xn, Yn] + λ([Xn, Yn−1] + [Xn−1, Yn]) + · · ·
+ λn ([X0, Yn] + [X1, Yn−1] + · · ·+ [Xn, Y0])
introduces the structure of a Lie algebra on gn. Note that [Xλ, Yλ]n coincides with [Xλ, Yλ]
expanded in λ and truncated at degree n.
It is known that the dual space g∗ of any Lie algebra g is equipped with a canonical Pois-
son structure called the Lie–Poisson structure. If a nondegenerate symmetric bilinear form
η : g× g→ C is defined on g, it induces a Lie–Poisson structure on g. For functions F,G : g→ C,
the Poisson bracket on g is defined by {F,G}(X) = η(X, [∇F (X),∇G(X)]), where ∇F (X) ∈ g
is defined through (dF )X(Y ) = η(∇F (X), Y ). To give the Lie–Poisson structure on gn, we
define a nondegenerate symmetric bilinear form η on gn by
η(Xλ, Yλ) :=
n∑
i=0
Tr(XiYn−i),
by which gn is identified with its dual. For a smooth function F : gn → C, define the gradient
∇F ∈ gn through (dF )(Yλ) = η(∇F, Yλ), and define ∇iF ∈ g by
∇F = (∇nF )λn + (∇n−1F )λn−1 + · · ·+∇0F.
Using them, the Lie–Poisson bracket on gn is given by
{F,G}0 := η(Xλ, [∇F,∇G]n)
6 H. Chiba
= Tr(X0 · [∇0F,∇0G]) + Tr (X1 · ([∇0F,∇1G] + [∇1F,∇0G])) + · · ·
+ Tr(Xn · ([∇0F,∇nG] + · · ·+ [∇nF,∇0G]))
= −Tr(∇0F · ([X0,∇0G] + [X1,∇1G] + · · ·+ [Xn,∇nG]))− · · ·
− Tr(∇n−1F · ([Xn−1,∇0G] + [Xn,∇1G]))− Tr(∇nF · [Xn,∇0G]).
The Poisson tensor (bivector) P0 : T ∗gn → Tgn is defined so that
{F,G}0 = dF (P0dG) = η(∇F, P0dG) =
n∑
i=0
Tr(∇iF · (P0dG)i).
This implies
−(P0dG)0 = [X0,∇0G] + [X1,∇1G] + · · ·+ [Xn,∇nG],
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
−(P0dG)n−1 = [Xn−1,∇0G] + [Xn,∇1G],
−(P0dG)n = [Xn,∇0G].
The following expression is useful
P0 : dG 7→ −
[X0, · ] [X1, · ] . . . [Xn, · ]
... . .
.
[Xn−1, · ] [Xn, · ]
[Xn, · ]
∇0G
...
∇n−1G
∇nG
=
[∇0G,X0] + [∇1G,X1] + · · ·+ [∇nG,Xn]
...
[∇0G,Xn−1] + [∇1G,Xn]
[∇0G,Xn]
.
It is also represented as a matrix as follows. Let A = A(X) be a representation matrix of the
mapping
T ∗g(' g)→ g, dG 7→ [X,∇G], G : g→ C, X ∈ g
with respect to some coordinates on g (here ∇G is the gradient on g). By the definition, −A is
a Poisson tensor of the Lie–Poisson structure on g. Since A(X) is linear in X, A(Xλ) is expanded
as A(Xλ) = λnA(X0) + λn−1A(X1) + · · · + A(Xn). Putting A(Xj) = Aj , P0 is represented as
an (n+ 1) dim(g)× (n+ 1) dim(g) matrix
P0 = −
A0 . . . An−1 An
A1 . . . An
... . .
.
An
.
In what follows, suppose dim(g) = d, rank(g) = h and let m1, . . . ,mh be exponents of g.
Let (y1, . . . , yd) be coordinates on g. It is known that the Casimir functions of the Lie–Poisson
structure on g (i.e., a function ϕ satisfying {F,ϕ} = 0 for any F : g → C) are the Ad-invariant
polynomials denoted by ϕi(y1, . . . , yd), i = 1, . . . , h, and they satisfy deg(ϕi) = mi + 1.
Let xj := (xj,1, . . . , xj,d) be coordinates on the j-th copy of g (coordinate expression for Xj)
and (x0, . . . , xn) coordinates on gn. We define the weighted degrees of variables to be
wdeg(xj) = wdeg(xj,α) = j, wdeg(λ) = 1.
Multi-Poisson Approach to the Painlevé Equations 7
Then, Xλ is quasihomogeneous (homogeneous in the weighted sense) of wdeg(Xλ) = n. Substi-
tuting yα = x0,αλ
n + x1,αλ
n−1 + · · ·+ xn,α into ϕi(y1, . . . , yd) and expanding it in λ provide
ϕi(y1, . . . , yd) = ϕi,0(x0, . . . , xn)λ(mi+1)n + ϕi,1(x0, . . . , xn)λ(mi+1)n−1 + · · ·
+ ϕi,(mi+1)n(x0, . . . , xn), i = 1, . . . , h,
which defines polynomials ϕi,j on gn satisfying
deg(ϕi,j) = mi + 1, wdeg(ϕi,j) = j.
Proposition 2.1.
(i) ϕi,j depends only on (x0, . . . , xj) for 0 ≤ j ≤ n− 1.
(ii) ϕi,j(x0, x1, . . . , xn) = ϕi,(mi+1)n−j(xn, . . . , x1, x0).
(iii) For each i, j, α, the derivative ∂ϕi,j+k/∂xk,α is independent of k = 0, . . . , n.
(iv) For each i, j, the gradient ∇kϕi,j+k is independent of k = 0, . . . , n.
(v) For each i, j, k, the equality
n∑
l=0
Al
∂ϕi,j+k−l
∂xk
=
n∑
l=0
Al
∂ϕi,j−l
∂x0
= 0
holds.
(vi) The Casimir functions of the Lie–Poisson structure P0 on gn are
ϕi,(mi+1)n−j , i = 1, . . . , h, j = 0, . . . , n.
Proof. (i) and (ii) follow from the definition of ϕi,j .
(iii) For yα =
n∑
k=0
λn−kxk,α, we have
∂ϕi
∂yα
=
∂xk,α
∂yα
∂
∂xk,α
(mi+1)n∑
j=0
λ(mi+1)n−jϕi,j
=
(mi+1)n∑
j=0
λmin−j+k ∂ϕi,j
∂xk,α
=
min+k∑
j=k
λmin−j+k ∂ϕi,j
∂xk,α
.
For the last equality, we used part (i) combined with part (ii). Thus we obtain
∂ϕi
∂yα
=
min∑
j=0
λmin−j ∂ϕi,j+k
∂xk,α
.
Since the left hand side is independent of k, so is each coefficient of λmin−j in the right hand
side. Part (iv) immediately follows from (iii).
(v) The first equality is a consequence of part (iii). Since ϕi(y) is a Casimir function of
the Lie–Poisson structure on g, Adϕi = 0, where A is a matrix defined before. Substituting
y =
n∑
k=0
λn−kxk yields
0 = A
∂ϕi
∂y
=
(
λnA0 + λn−1A1 + · · ·+An
)min∑
j=0
λmin−j ∂ϕi,j+k
∂xk
=
∑
j,l
λmin−j+n−lAl
∂ϕi,j+k
∂xk
=
min+l∑
j=l
λmin+n−j
n∑
l=0
Al
∂ϕi,j+k−l
∂xk
.
This proves the second equality of (v).
8 H. Chiba
To prove (vi), it is sufficient to show
A0 . . . An−1 An
A1 . . . An
... . .
.
An
∂ϕi,(mi+1)n−j/∂x0
∂ϕi,(mi+1)n−j/∂x1
...
∂ϕi,(mi+1)n−j/∂xn
= 0
for j = 0, . . . , n. This is verified with the aid of part (v). �
Example 2.2. For g = sl(2,C), we have d = 3, h = 1 and mi = m1 = 1. Denote a general
element Xλ ∈ gn as
Xλ = λnX0 + λn−1X1 + · · ·+Xn
= λn
(
u0 v0
w0 −u0
)
+ λn−1
(
u1 v1
w1 −u1
)
+ · · ·+
(
un vn
wn −un
)
.
Let (uj , vj , wj) be coordinates on the j-th copy of g and (u0, v0, w0, . . . , un, vn, wn) coordinates
on gn. Then,
∇jF =
1
2
∂F
∂uj
∂F
∂wj
∂F
∂vj
−1
2
∂F
∂uj
, Aj =
0 vj −wj
−vj 0 2uj
wj −2uj 0
.
The Casimir function on g is given by ϕi = ϕ = u2 + vw. Then, the functions ϕi,j = ϕj are
defined by expanding(
λnu0 + · · ·+ un
)2
+
(
λnv0 + · · ·+ vn
)(
λnw0 + · · ·+ wn
)
in λ. This gives
ϕj =
∑
k+l=j
(ukul + vkwl) , j = 0, . . . , 2n.
Note that they are coefficients of −detXλ. The Casimir functions of gn are given by ϕj for
j = n, . . . , 2n.
2.2 Multi-Poisson structure on g0
n
In general, a manifold M is called a bi-Poisson manifold if
(i) there are two Poisson brackets { , }0 and { , }1, and
(ii) the linear combination { , }0 + t{ , }1 is also a Poisson bracket for any t ∈ C.
See [9, 14, 15, 16] for applications of bi-Poisson manifolds to integrable systems. Here, we
introduce a bi-Poisson structure on gn following [14]. The shift operator Xλ 7→ Xλ+t defines an
automorphism of gn with a parameter t ∈ C. It induces a deformation, denoted by { , }t, of the
Lie–Poisson bracket { , }0. Let
{ , }t = { , }0 + t{ , }1 + · · ·+ tn+1{ , }n+1 + · · ·
Multi-Poisson Approach to the Painlevé Equations 9
be its expansion. Magnano and Magri [14] proved that each { , }i, i = 0, . . . , n + 1, and their
any linear combination satisfy the axiom of a Poisson bracket. Hence, gn has n+ 2 compatible
Poisson brackets and it becomes a multi-Poisson manifold. Their Poisson tensors are
P1 =
0 0 0 . . . 0
0 −A1 −A2 . . . −An
...
...
... . .
.
0 −An−1 −An
0 −An
,
Pk+1 =
0 0 . . . 0 0 . . . 0
0 A0
... . .
. ...
0 A0 . . . Ak−1
0 −Ak+1 . . . −An
...
... . .
.
0 −An
, k = 1, . . . , n− 1,
Pn+1 =
0 0 . . . 0 0
0 A0
0 A0 A1
... . .
. ...
...
0 A0 . . . An−2 An−1
.
(P0 is the same as before). Let g0n be a submanifold of gn defined by x0 = const;
g0n :=
{
Xλ = X0λ
n +X1λ
n−1 + · · ·+Xn |X0 = const
}
⊂ gn.
Since the first row and column of P1, . . . , Pn+1 are zero (i.e., x0 = (x0,1, . . . , x0,d) are Casimir
functions of them), the restrictions of them on g0n define a multi-Poisson structure on g0n, whose
brackets and tensors are again denoted by ({ , }i, Pi). The tensors are given by
P1 =
−A1 −A2 . . . −An
...
... . .
.
−An−1 −An
−An
,
Pk+1 =
A0
. .
. ...
A0 . . . Ak−1
−Ak+1 . . . −An
... . .
.
−An
, k = 1, . . . , n− 1,
Pn+1 =
A0
A0 A1
. .
. ...
...
A0 . . . An−2 An−1
.
For i = 1, . . . , h and j = 1, . . . , (mi + 1)n, define functions ψi,j on g0n by
ψi,j(x1, . . . , xn) := ϕi,j |g0n = ϕi,j |x0=const
(we do not define ψi,0 because ϕi,0 is constant on g0n).
10 H. Chiba
Proposition 2.3.
(i) Casimir functions of Pk+1 are ψi,j, i = 1, . . ., h, for j = 1, 2, . . ., k and for j = min+ k + 1,
min+ k + 2, . . . , (mi + 1)n.
(ii) Casimir functions of the combination λPk+1−Pk are ψi,j, i = 1, . . . , h, for j = 1, 2, . . . , k−1
and for j = min+ k + 1,min+ k + 2, . . . , (mi + 1)n, and
λminψi,k + λmin−1ψi,k+1 + · · ·+ ψi,min+k, i = 1, . . . , h.
(iii) Let F : g0n → C be a smooth function. The differential equation for the vector field (λPk+1−
Pk)dF is expressed in Lax form as
d
dt
Xλ = [Xλ,∇kF ], Xλ = λnX0 + λn−1X1 + · · ·+Xn.
(iv) Define the function Gi,k,j to be
Gi,k,j = −
(
λj−1ψi,k + λj−2ψi,k+1 + · · ·+ ψi,k+j−1
)
.
Then, the equality
Pk+1dψi,k+j = Pkdψi,k+j−1 = (λPk+1 − Pk)dGi,k,j
holds for i = 1, . . . , h, j = 1, . . . ,min and k = 1, . . . , n. In particular, the vector field
Pk+1dψi,k+j is independent of k and the equation for it is expressed in Lax form as
d
dt
Xλ = [Xλ,∇kGi,k,j ].
(v) The vector fields Pk+1dψi,k+j for i = 1, . . . , h and j = 1, . . . ,min commute with each other
(note that it is zero when j /∈ {1, . . . ,min}).
Proof. (i) and (ii) can be verified by a straightforward calculation with the aid of Proposi-
tion 2.1(v). To prove (iii), note that the vector field Pk+1dF is written as
Pk+1dF =
[X0, · ]
. .
. ...
[X0, · ] . . . [Xk−1, · ]
−[Xk+1, · ] . . . −[Xn, · ]
... . .
.
−[Xn, · ]
∇1F
...
∇kF
∇k+1F
...
∇nF
,
and similarly for PkdF . Using them, write down the equation of Xj for the vector field (λPk+1−
Pk)dF . For example, the equation for X1 is dX1/dt = λ[X0,∇kF ]− [X0,∇k−1F ]. Summing up
the equations of λn−jXj proves the desired result.
(iv) Since λminψi,k + · · ·+ ψi,min+k is the Casimir of λPk+1 − Pk, we have
(λPk+1 − Pk)d(λminψi,k + λmin−1ψi,k+1 + · · ·+ ψi,min+k) = 0.
Expanding this yields the first equality. The second equality is confirmed by a straightforward
calculation.
(v) Due to Part (iv), we can assume that k = n. Because of the property [Pn+1dF, Pn+1dG] =
Pn+1d{G,F} of a Poisson bracket (the left hand side is the Lie bracket for vector fields), it is
Multi-Poisson Approach to the Painlevé Equations 11
sufficient to show the equality {ψi′,j′ , ψi,j}n+1 = 0 for i, i′ = 1, . . . , h and j, j′ = 1, . . . , (mi+1)n.
When j = 1, . . . , n, it is trivial because ψi,j is the Casimir of Pn+1. Next, we have
{λminψi′,k + λmin−1ψi′,k+1 + · · ·+ ψi′,min+k, ψi,n+j}n+1
= 〈d(λminψi′,k + λmin−1ψi′,k+1 + · · ·+ ψi′,min+k), Pn+1dψi,n+j〉
= 〈d(λminψi′,k + λmin−1ψi′,k+1 + · · ·+ ψi′,min+k), Pk+1dψi,k+j〉
= 〈d(λminψi′,k + λmin−1ψi′,k+1 + · · ·+ ψi′,min+k), (λPk+1 − Pk)dGi,k,j〉
= −〈dGi,k,j , (λPk+1 − Pk)d(λminψi′,k + λmin−1ψi′,k+1 + · · ·+ ψi′,min+k)〉 = 0.
This provides
{ψi′,k, ψi,n+j}n+1 = · · · = {ψi′,min+k, ψi,n+j}n+1 = 0,
for any k = 1, . . . , n and any j = 1, . . . ,min, which completes the proof. �
Theorem 2.4. Suppose that the constant x0 for the definition of g0n is chosen so that the
functions {ψi,j}i,j are functionally independent. Then, the vector field Pk+1dψi,k+j, which is
independent of k, is completely integrable in the Liouville sense for any i and j.
Proof. Recall dim(g) = d and rank(g) = h. Thus, dim(g0n) = nd. Since Pk+1 has nh Casimir
functions, the dimension of a symplectic leaf S of Pk+1 is n(d−h). On the leaf S, the vector fields
{Pk+1dψi,k+j}i,j define n(d−h)-dimensional Hamiltonian systems, among which nonzero vector
fields are for i = 1, . . . , h and j = 1, . . . ,min. Further, these nonzero vector fields commute
with each other and they are linearly independent due to the assumption. The number of the
nonzero vector fields is
h∑
i=1
min =
1
2
(dim(g)− rank(g))n =
1
2
n(d− h) =
1
2
dim(S).
Hence, the Liouville theorem shows that the vector fields are integrable. �
In what follows, we suppose the above assumption; the constant x0 for the definition of g0n
is chosen so that the functions {ψi,j}i,j are functionally independent. That is, the differentials
{dψi,j}i,j are linearly independent except for finite points.
2.3 Symplectic reduction
The next purpose is to perform a symplectic reduction [9, 14, 15, 16].
Theorem 2.5. The h-dimensional distribution D defined by
D = span{Pkdψi,k | i = 1, . . . , h}
is integrable in the Frobenius sense. The vector fields Pkdψi,k are linear for i = 1, . . . , h.
Proof. The first statement follows from Proposition 2.3(v). Since Pkdψi,k is independent of k,
we obtain Pkdψi,k = P1dψi,1. Since wdeg(ψi,1) = 1, dψ1,i is a constant, while P1 is linear in
(x1, . . . , xn). �
The differential equation for Pkdψi,k = P1dψi,1 is given by
d
dt
Xλ = [Xλ,∇1Gi,1,1] = [∇1ψi,1, Xλ].
12 H. Chiba
Since ∇1ψi,1 is independent of λ, this is decomposed as
d
dt
Xk = [∇1ψi,1, Xk], k = 1, . . . , n.
In coordinates, it is expressed as
dxk
dt
= −Ak
∂ψi,1
∂x1
(x1), k = 1, . . . , n. (2.1)
Let us consider the orbit space π : g0n → g0n/D, which is a smooth manifold if points on g0n at
which dim(D) < h is removed if necessary. The Marsden–Ratiu reduction theorem [9] states
that the orbit space g0n/D is again a multi-Poisson manifold with compatible Poisson tensors
denoted by P̃1, . . . , P̃n+1. They are defined by P̃k = π∗Pkπ
∗. Let { , }k and { , }′k be Poisson
brackets associated with Pk and P̃k, respectively. For a function F on g0n which is constant along
each integral manifold of D, a function F̃ on g0n/D is well-defined through F̃ ◦ π = π∗F̃ = F .
Conversely, for a function F̃ on g0n/D, we can find a function F on g0n, which is constant along D,
such that F̃ ◦ π = F . Then, { , }′k is given by {F̃ , G̃}′k ◦ π = {F,G}k.
Because of Proposition 2.3(v), ψi,j is constant along each integral manifold of D and the
projection ψ̃i,j is well-defined. The projected vector field is given by P̃kdψ̃i,j = π∗(Pkdψi,j).
It is convenient to realize g0n/D as a submanifold of g0n. Let σ : g0n/D → g0n be a smooth
section. The image σ(g0n/D) is a submanifold of g0n which is diffeomorphic to g0n/D. In Propo-
sition 2.6(iii) below, g0n/D is identified with a submanifold in this manner.
Proposition 2.6.
(i) Casimir functions of P̃k+1 are ψ̃i,j, i = 1, . . . , h, for j = 1, 2, . . . , k + 1 and for j =
min+ k + 1,min+ k + 2, . . . , (mi + 1)n.
(ii) Casimir functions of the combination λP̃k+1 − P̃k are ψ̃i,j, i = 1, . . . , h, for j = 1, 2, . . . , k
and for j = min+ k + 1,min+ k + 2, . . . , (mi + 1)n, and
λmin−1ψ̃i,k+1 + λmin−2ψ̃i,k+2 + · · ·+ ψ̃i,min+k, i = 1, . . . , h.
(iii) For a smooth function F̃ : g0n/D → C, there exist scalar-valued functions β1, . . . , βh : g0n/D
→ C such that the equation for the vector field (λP̃k+1 − P̃k)dF̃ is expressed in Lax form
as
d
dt
X̃λ =
[
X̃λ,∇kF̃
]
−
h∑
i=1
βi
[
X̃λ,∇1ψ̃i,1
]
=
[
X̃λ,∇kF̃ −
h∑
i=1
βi∇1ψ̃i,1
]
,
where X̃λ = Xλ|g0n/D and ∇kF̃ = (∇kF )|g0n/D.
(iv) Define the function G̃i,k,j to be
G̃i,k,j = −
(
λj−1ψ̃i,k + λj−2ψ̃i,k+1 + · · ·+ ψ̃i,k+j−1
)
.
Then, the equality
P̃k+1dψ̃i,k+j = P̃kdψ̃i,k+j−1 =
(
λP̃k+1 − P̃k
)
dG̃i,k,j
holds for i = 1, . . . , h, j = 2, . . . ,min and k = 1, . . . , n.
(v) The vector fields P̃k+1dψ̃i,k+j for i = 1, . . . , h and j = 2, . . . ,min commute with each other
(note that it is zero when j /∈ {2, . . . ,min}).
Multi-Poisson Approach to the Painlevé Equations 13
Proof. (i) ψ̃i,j for j = 1, . . . , k and j = min+ k+ 1, . . . , (mi + 1)n are Casimir of P̃k+1 because
they are Casimir of Pk+1. For ψ̃i,k+1, we have{
F̃ , ψ̃i,k+1
}′
k+1
= {F,ψi,k+1}k+1 = 〈dF, Pk+1dψi,k+1〉 = (Pkdψi,k)(F ).
The right hand side becomes zero because F is constant along D.
(ii) The first statement (on ψ̃i,j) is trivial because they are common Casimir of P̃k+1 and P̃k.
The last function λmin−1ψ̃i,k+1 + · · · is a projection of the function given in Proposition 2.3(ii).
The results of (iv) and (v) are projections of those of Proposition 2.3(iv) and (v).
To prove (iii), g0n/D is identified with a submanifold of g0n as above. Put F = π∗F̃ . We have
to calculate the projection of the vector field [Xλ,∇kF ] onto g0n/D (see Proposition 2.3(iii)). At
first, we restrict the domain to g0n/D as
[Xλ,∇kF ]|g0n/D =
[
Xλ|g0n/D, (∇kF )|g0n/D
]
=
[
X̃λ,∇kF̃
]
.
Since this is not tangent to T (g0n/D), we calculate the projection of it according to the decompo-
sition Tg0n = T (g0n/D)⊕D. Then, (iii) follows from the fact that the distribution D is spanned
by the vector fields of the form [Xλ,∇1ψi,1]. �
2.4 Isospectral deformation to isomonodromic deformation
Now we have (min− 1)h distinct vector fields on g0n/D
j = 2: P̃1dψ̃i,2 = · · · = P̃k+1dψ̃i,k+2 = · · · = P̃n+1dψ̃i,n+2,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
j = j : P̃1dψ̃i,j = · · · = P̃k+1dψ̃i,k+j = · · · = P̃n+1dψ̃i,n+j ,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
j = min : P̃1dψ̃i,min = · · · = P̃k+1dψ̃i,k+min = · · · = P̃n+1dψ̃i,(mi+1)n.
They are (nd− h)-dimensional integrable systems. For fixed k, a symplectic leaf of the Poisson
structure P̃k is given by a level surface of Casimir functions as
Sk =
{
ψ̃i,j = αi,j (const) | i = 1, . . . , h, j = 1, . . . , k, j = min+ k, . . . , (mi + 1)n
}
.
Restricted on the symplectic leaf, the vector fields become (nd−nh−2h)-dimensional completely
integrable Hamiltonian systems of the form
P̃k+1dψ̃i,k+j :
d
dt
X̃λ =
[
X̃λ, Aλ
]
on Sk, (2.2)
Aλ := ∇kG̃i,k,j −
h∑
i=1
βi∇1ψ̃i,1.
Both of X̃λ and Aλ depend on parameters {αi,j}i,j which define the symplectic leaf. Thus, we
write X̃λ as X̃λ(t, α), where α denotes the collection of parameters αi,j .
Now suppose that there exists a parameter αi′,j′ such that the following condition holds
∂X̃λ
∂αi′,j′
(t, α) = λl
∂Aλ
∂λ
(2.3)
for some integer l. Equation (2.2) is put together with equation (2.3) to yield
∂X̃λ
∂t
(t, α) +
∂X̃λ
∂αi′,j′
(t, α) =
[
X̃λ, Aλ
]
+ λl
∂Aλ
∂λ
.
14 H. Chiba
Define the Lax matrix Lλ by
Lλ :=
1
λl
X̃λ(t, α)|αi′,j′=t,
where the parameter αi′,j′ satisfying the condition (2.3) is replaced by t. Then, the above
equation is rewritten as
dLλ
dt
= [Lλ, Aλ] +
∂Aλ
∂λ
, (2.4)
which is known as the isomonodromic deformation equation. It is known that a system written
as the isomonodromic deformation equation enjoys the Painlevé property. The function ψ̃i,k+j
restricted on Sk will be a Hamiltonian function of the Painlevé equation after replacing αi′,j′ 7→ t
and changing to Darboux’s coordinates if necessary.
For ABCD-type simple Lie algebras, the dimensions of Painlevé systems obtained in this
manner are summarized in Table 3. In particular, the dimension is 2 when g = sl2, n = 2, and
it is 4 when g = sl2, n = 3 or g = so5, n = 1. From the next sections, we will demonstrate
our method for these cases. In particular, the Hamiltonian functions shown in Section 1 will be
obtained.
Table 3. The dimensions of Painlevé systems.
dimension h = 1 h = 2 h = 3
Ah(h ≥ 1) slh+1 nh2 + nh− 2h 2n− 2 6n− 4 12n− 6
Bh(h ≥ 2) so2h+1 2nh2 − 2h − 8n− 4 18n− 6
Ch(h ≥ 3) sp2h 2nh2 − 2h − − 18n− 6
Dh(h ≥ 4) so2h 2nh2 − 2nh− 2h − − −
3 2-dimensional Painlevé equations: g = sl2, n = 2
In this case, a general element of gn is written by
Xλ = λ2
(
u0 v0
w0 −u0
)
+ λ
(
u1 v1
w1 −u1
)
+
(
u2 v2
w2 −u2
)
.
The Painlevé equation obtained by our method depends on a choice of x0 = (u0, v0, w0). We
consider the following two cases.
(I)
(
u0 v0
w0 −u0
)
=
(
1 0
0 −1
)
, (II)
(
u0 v0
w0 −u0
)
=
(
0 0
1 0
)
.
From the former case, we will obtain the second and fourth Painlevé equations PII, PIV, and
from the latter one, we will obtain the first and second Painlevé equations PI, PII.
3.1 Case (I)
In this case, the functions ψi,j = ψj (since h = rank(g) = 1, we omit the subscript i) are given by
ψ1 = 2u1, ψ2 = 2u2 + u21 + v1w1,
ψ3 = 2u1u2 + v2w1 + v1w2, ψ4 = u22 + v2w2.
Multi-Poisson Approach to the Painlevé Equations 15
(see Example 2.2). The differential equation (2.1) defining the distribution D is u′j = 0, v′j = 2vj ,
w′j = −2wj for j = 1, 2. This is solved as a function of w1 as
u1 = U1, u2 = U2, v1 = V2/w1, v2 = V3/w1, w2 = W1w1,
where U1, U2, V2, V3, W1 are integral constants (initial values at w1 = 1), for which the subscripts
are given so that they are consistent with the weighted degrees (for example, since wdeg(v2w1) =
2 + 1 = 3, the weighted degree of V3 is three). This relation defines a coordinate transformation
(u1, v1, w1, u2, v2, w2) 7→ (U1, V2, w1, U2, V3,W1).
In the new coordinates, integral manifolds of the distribution D are straight lines along w1-
axis. In particular, the subset {w1 = 1} ⊂ g0n gives the realization of the orbit space g0n/D as
a submanifold and (U1, V2, U2, V3,W1) provides a global coordinate system of g0n/D.
At this stage, we have on g0n/D
ψ̃1 = 2U1, ψ̃2 = 2U2 + U2
1 + V2,
ψ̃3 = 2U1U2 + V3 + V2W1, ψ̃4 = U2
2 + V3W1,
and three Poisson structures P̃1 (Casimirs are ψ̃1, ψ̃3, ψ̃4), P̃2 (Casimirs are ψ̃1, ψ̃2, ψ̃4),
P̃3 (Casimirs are ψ̃1, ψ̃2, ψ̃3), and vector fields P̃3dψ̃4 = P̃2dψ̃3 = P̃1dψ̃2 which are expressed as
the Lax equation dX̃λ/dt = [Aλ, X̃λ], where
X̃λ = λ2
(
1 0
0 −1
)
+ λ
(
U1 V2
1 −U1
)
+
(
U2 V3
W1 −U2
)
,
Aλ = λ
(
1 0
0 −1
)
+
(
U1 V2
1 −U1
)
−
(
W1 0
0 −W1
)
.
The next purpose is to restrict the vector fields on a symplectic leaf. We will consider
P̃3dψ̃4 and P̃2dψ̃3 separately (P̃1dψ̃2 will not be considered because there are no parameters
satisfying (2.3)).
(i) Consider the vector field P̃3dψ̃4. For the Poisson tensor P̃3, a symplectic leaf is defined
by the level surface {ψ̃j = const, j = 1, 2, 3}. In order for the condition
∂X̃λ
∂α
=
∂Aλ
∂λ
=
(
1 0
0 −1
)
(3.1)
to be satisfied, we find that U2 in X̃λ has to include a parameter α which will be replaced by t
later. For this purpose, we take the symplectic leaf
S =
{
2U1 = 0, 2U2 + U2
1 + V2 = 2α2, 2U1U2 + V3 + V2W1 = α3
}
.
Hence, we put U1 = 0, U2 = α2 − V2/2, V3 = α3 − V2W1, and (V2,W1) gives a global coordinate
system for the symplectic leaf. Then, it turns out that X̃λ satisfies the condition (3.1) with
α = α2 on the symplectic leaf. Finally, by replacing α2 by t, we obtain the isomonodromic
deformation equation (2.4) with
Lλ = λ2
(
1 0
0 −1
)
+ λ
(
0 V2
1 0
)
+
(
t− V2/2 α3 − V2W1
W1 −(t− V2/2)
)
,
Aλ = λ
(
1 0
0 −1
)
+
(
−W1 V2
1 W1
)
.
16 H. Chiba
The Poisson tensor P̃3 on the symplectic leaf with coordinates (V2,W1) is given by
P̃3 =
(
0 2
−2 0
)
.
To change to Darboux’s coordinates, we put V2 = −2p2, W1 = q1. Then, P̃3 becomes the canon-
ical symplectic matrix. In the coordinates (q1, p2), the isomonodromic deformation equation is
a Hamiltonian system. The Hamiltonian function ψ̃4 for the vector field P̃3dψ̃4 is written as
ψ̃4 = U2
2 + V3W1 = (t− V2/2)2 + (α3 − V2W1)W1
= p22 + 2q21p2 + 2tp2 + α3q1 + t2.
This is reduced to the Hamiltonian function (1.2) of the second Painlevé equation by a certain
coordinate change and the isomonodromic deformation equation (2.4) is equivalent to the second
Painlevé equation.
(ii) Consider the vector field P̃2dψ̃3. For the Poisson tensor P̃2, a symplectic leaf is defined
by the level surface {ψ̃j = const, j = 1, 2, 4}. In order for the condition
∂X̃λ
∂α
= λ
∂Aλ
∂λ
= λ
(
1 0
0 −1
)
(3.2)
to be satisfied, we find that U1 in X̃λ has to include a parameter α which will be replaced
by t later, and the other components of X̃λ cannot include α. For this purpose, we take the
symplectic leaf
S =
{
2U1 = 2α1, 2U2 + U2
1 + V2 = α2 + α2
1, U
2
2 + V3W1 = α4
}
.
This relation is rewritten as
U1 = α1, V2 = α2 − 2U2, V3 =
(
α4 − U2
2
)
/W1.
By substituting them, X̃λ satisfies the condition (3.2) with α = α1. Finally, by replacing α1
by t, we obtain the isomonodromic deformation equation (2.4).
The Poisson tensor P̃2 on the symplectic leaf with coordinates (U2,W1) is given by
P̃2 =
(
0 W1
−W1 0
)
.
For Darboux’s coordinates, we put U2 = p1q1 − β2 and W1 = q1, where β2 is an arbitrary
constant. Then, P̃2 is transformed to the canonical symplectic matrix. In the coordinates
(q1, p1), the isomonodromic deformation equation is a Hamiltonian system. The Hamiltonian
function ψ̃3 for the vector field P̃2dψ̃3 is written as
ψ̃3 = 2U1U2 + V3 + V2W1 = 2tU2 +
(
α4 − U2
2
)
/W1 + (α2 − 2U2)W1
= −p21q1 − 2p1q
2
1 + 2tp1q1 + 2β2p1 + (α2 + 2β2)q1 − 2β2t+
α4 − β22
q1
.
We choose the free parameter β2 to be α4 = β22 so that the Hamiltonian becomes a polynomial.
This is the Hamiltonian function (1.3) of the fourth Painlevé equation up to some scaling. The
isomonodromic deformation equation (2.4) is equivalent to the fourth Painlevé equation, where
Lλ = λ
(
1 0
0 −1
)
+
(
t α2 + 2β2 − 2p1q1
1 −t
)
+
1
λ
(
p1q1 − β2 −p21q1 + 2β2p1
q1 −(p1q1 − β2)
)
,
Aλ = λ
(
1 0
0 −1
)
+
(
t− q1 α2 + 2β2 − 2p1q1
1 −(t− q1)
)
.
Multi-Poisson Approach to the Painlevé Equations 17
3.2 Case (II)
In this case, the functions ψi,j = ψj are given by
ψ1 = v1, ψ2 = u21 + v2 + v1w1,
ψ3 = 2u1u2 + v2w1 + v1w2, ψ4 = u22 + v2w2.
The differential equation (2.1) defining the distribution D is u′j = −vj , v′j = 0, w′j = 2uj for
j = 1, 2. We can assume without loss of generality that v1 = 1 by a suitable scaling of variables
(indeed, v1 is a common Casimir of P1, P2, P3). Thus, the equations are solved as a function
of u1 as
v1 = 1, v2 = V2, u2 = V2u1 + U3,
w1 = −u21 +W2, w2 = −V2u21 − 2U3u1 +W4,
where V2, U3, W2, W4 are integral constants (initial values at u1 = 0). This relation defines
a coordinate transformation
(u1, w1, u2, v2, w2) 7→ (u1,W2, U3, V2,W4).
In the new coordinates, integral manifolds of the distribution D are straight lines along u1-
axis. In particular, the subset {u1 = 0} ⊂ g0n gives the realization of the orbit space g0n/D and
(W2, U3, V2,W4) provides a global coordinate system of g0n/D restricted to v1 = 1.
On g0n/D, we have functions
ψ̃1 = 1, ψ̃2 = V2 +W2, ψ̃3 = V2W2 +W4, ψ̃4 = U2
3 + V2W4,
and three Poisson structures P̃1 (Casimirs are ψ̃1, ψ̃3, ψ̃4), P̃2 (Casimirs are ψ̃1, ψ̃2, ψ̃4),
P̃3 (Casimirs are ψ̃1, ψ̃2, ψ̃3), and vector fields P̃3dψ̃4 = P̃2dψ̃3 = P̃1dψ̃2 which are expressed as
the Lax equation dX̃λ/dt = [Aλ, X̃λ], where
X̃λ = λ2
(
0 0
1 0
)
+ λ
(
0 1
W2 0
)
+
(
U3 V2
W4 −U3
)
,
Aλ = λ
(
0 0
1 0
)
+
(
0 1
W2 0
)
−
(
0 0
V2 0
)
.
The next purpose is to restrict the vector fields on a symplectic leaf.
(i) Consider the vector field P̃3dψ̃4. For the Poisson tensor P̃3, a symplectic leaf is defined
by the level surface {ψ̃j = const, j = 1, 2, 3}. For the condition
∂X̃λ
∂α
=
∂Aλ
∂λ
=
(
0 0
1 0
)
, (3.3)
we find that W4 in X̃λ has to include a parameter α. To this end, we take the symplectic leaf as
S =
{
ψ̃1 = 1, ψ̃2 = V2 +W2 = 0, ψ̃3 = V2W2 +W4 = α4
}
.
Hence, we put V2 = −W2, W4 = α4 +W 2
2 , so that (W2, U3) gives a global coordinate system on
the leaf. Then, X̃λ satisfies the condition (3.3) with α = α4. Finally, by replacing α4 by t, we
obtain the isomonodromic deformation equation (2.4) with
Lλ = λ2
(
0 0
1 0
)
+ λ
(
0 1
W2 0
)
+
(
U3 −W2
t+W 2
2 −U3
)
,
Aλ = λ
(
0 0
1 0
)
+
(
0 1
2W2 0
)
.
18 H. Chiba
The Poisson tensor P̃3 on the symplectic leaf is given by
P̃3 =
(
0 1
−1 0
)
,
which is already in canonical form. On the symplectic leaf, the function ψ̃4 is written as
ψ̃4 = U2
3 + V2W4 = U2
3 −W 3
2 − tW2.
This is the Hamiltonian function (1.1) of the first Painlevé equation (up to some scaling) and
the isomonodromic deformation equation (2.4) coincides with the first Painlevé equation.
(ii) For the vector field P̃2dψ̃3, we again obtain the second Painlevé equation and the detailed
calculation is omitted.
4 4-dimensional Painlevé equations: g = sl2, n = 3
In this case, a general element of gn is written as
Xλ = λ3
(
u0 v0
w0 −u0
)
+ λ2
(
u1 v1
w1 −u1
)
+ λ
(
u2 v2
w2 −u2
)
+
(
u3 v3
w3 −u3
)
.
For the definition of g0n, we again consider the following two cases.
(I)
(
u0 v0
w0 −u0
)
=
(
1 0
0 −1
)
, (II)
(
u0 v0
w0 −u0
)
=
(
0 0
1 0
)
.
From the former case, we will obtain Hamiltonian functions (1.9), (1.10), (1.11), and from the
latter one, we will obtain (1.7), (1.8), (1.12).
4.1 Case (I)
In this case, the functions ψi,j = ψj (since h = rank(g) = 1, we omit the subscript i) are given by
ψ1 = 2u1,
ψ2 = 2u2 + u21 + v1w1,
ψ3 = 2u1u2 + 2u3 + v2w1 + v1w2,
ψ4 = u22 + 2u1u3 + v3w1 + v2w2 + v1w3,
ψ5 = 2u2u3 + v3w2 + v2w3,
ψ6 = u23 + v3w3.
The differential equation (2.1) defining the distribution D is u′j = 0, v′j = 2vj , w
′
j = −2wj for
j = 1, 2, 3. This is solved as a function of w1 as
u1 = U1, u2 = U2, u3 = U3,
v1 = V2/w1, v2 = V3/w1, v3 = V4/w1, w2 = W1w1, w3 = W2w1,
where U1, U2, U3, V2, V3, V4, W1, W2 are integral constants (initial values at w1 = 1). This
relation defines a coordinate transformation
(u1, v1, w1, u2, v2, w2, u3, v3, w3) 7→ (U1, V2, w1, U2, V3,W1, U3, V4,W2).
Multi-Poisson Approach to the Painlevé Equations 19
In the new coordinates, integral manifolds of the distribution D are straight lines along w1-
axis. In particular, the subset {w1 = 1} ⊂ g0n gives the realization of the orbit space g0n/D as
a submanifold and (U1, V2, U2, V3,W1, U3, V4,W2) provides a global coordinate system of g0n/D.
At this stage, we have on g0n/D
ψ̃1 = 2U1,
ψ̃2 = 2U2 + U2
1 + V2,
ψ̃3 = 2U1U2 + 2U3 + V3 + V2W1,
ψ̃4 = 2U1U3 + U2
2 + V4 + V2W2 + V3W1,
ψ̃5 = 2U2U3 + V4W1 + V3W2,
ψ̃6 = U2
3 + V4W2,
and two vector fields
P̃2dψ̃3 = P̃3dψ̃4 = P̃4dψ̃5, (4.1)
P̃2dψ̃4 = P̃3dψ̃5 = P̃4dψ̃6. (4.2)
The differential equations of these vector fields are expressed in Lax form as
∂X̃λ
∂t1
=
[
A1, X̃λ
]
,
∂X̃λ
∂t2
=
[
A2, X̃λ
]
, (4.3)
respectively, where
X̃λ = λ3
(
1 0
0 −1
)
+ λ2
(
U1 V2
1 −U1
)
+ λ
(
U2 V3
W1 −U2
)
+
(
U3 V4
W2 −U3
)
,
A1 = λ
(
1 0
0 −1
)
+
(
U1 V2
1 −U1
)
−
(
W1 0
0 −W1
)
,
A2 = λ2
(
1 0
0 −1
)
+ λ
(
U1 V2
1 −U1
)
+
(
U2 V3
W1 −U2
)
−
(
W2 0
0 −W2
)
.
The next purpose is to restrict the vector fields on a symplectic leaf.
(i) Consider the pair of vector fields P̃4dψ̃5 and P̃4dψ̃6. For the Poisson tensor P̃4, a symplectic
leaf is defined by the level surface {ψ̃j = const, j = 1, 2, 3, 4}. In order for the two conditions
∂X̃λ
∂α
=
∂A1
∂λ
=
(
1 0
0 −1
)
,
∂X̃λ
∂α′
=
∂A2
∂λ
= 2λ
(
1 0
0 −1
)
+
(
U1 V2
1 −U1
)
(4.4)
to be satisfied, we find that U3 in X̃λ has to include a parameter α, which will be replaced by t1
later, and U2 and W2 have to include a parameter α′, which will be replaced by t2 later. For
this purpose, we take the symplectic leaf
S =
{
ψ̃1 = 0, ψ̃2 = 4α2, ψ̃3 = 2α3, ψ̃4 = α4 + 4α2
2
}
.
Further, we change the coordinate as W2 = W̃2 + α2 because W2 should include a parameter.
Then, the above relation for S is rearranged as
U1 = 0, U2 = 2α2 −
1
2
V2, U3 = α3 −
1
2
V3 −
1
2
V2W1,
V4 = α4 + α2V2 −
1
4
V 2
2 − V2W̃2 − V3W1.
20 H. Chiba
Substituting them into X̃λ, A1 and A2, we can verify the condition (4.4) with α = α3 and
α′ = α2. By replacing α3, α2 by t1, t2, respectively, we obtain the isomonodromic deformation
equations
∂Lλ
∂t1
= [A1, Lλ] +
∂A1
∂λ
,
∂Lλ
∂t2
= [A2, Lλ] +
∂A2
∂λ
, (4.5)
which are equations of (V2, V3,W1, W̃2) with two independent variables t1, t2, where
Lλ = λ3
(
1 0
0 −1
)
+ λ2
(
0 V2
1 0
)
+ λ
(
2t2 − 1
2V2 V3
W1 −
(
2t2 − 1
2V2
))
+
(
t1 − 1
2V3 −
1
2V2W1 α4 + t2V2 − 1
4V
2
2 − V2W̃2 − V3W1
W̃2 + t2 −
(
t1 − 1
2V3 −
1
2V2W1
) )
,
A1 = λ
(
1 0
0 −1
)
+
(
−W1 V2
1 W1
)
,
A2 = λ2
(
1 0
0 −1
)
+ λ
(
0 V2
1 0
)
+
(
t2 − 1
2V2 − W̃2 V3
W1 −
(
t2 − 1
2V2 − W̃2
)) .
The Poisson tensor P̃4 on the symplectic leaf written in the coordinates (V2, V3,W1, W̃2) is given
by
P̃4 =
0 0 0 2
0 0 2 0
0 −2 0 0
−2 0 0 0
.
The above two isomonodromic deformation equations can be written as Hamiltonian systems.
The Hamiltonian functions of these equations are obtained by deleting U1, U2, U3, V4 from ψ̃5
and ψ̃6 by using the above relations, and changing to Darboux’s coordinates by a scaling so
that the above P̃4 is transformed to the canonical symplectic matrix. In this manner, we obtain
Hamiltonian functions (1.9) of (PII-2)2 given in Section 1.
(ii) Consider the pair of vector fields P̃3dψ̃4 and P̃3dψ̃5. For the Poisson tensor P̃3, a sym-
plectic leaf is defined by the level surface {ψ̃j = const, j = 1, 2, 3, 6}. For the two conditions
∂X̃λ
∂α
= λ
∂A1
∂λ
= λ
(
1 0
0 −1
)
,
∂X̃λ
∂α′
= λ
∂A2
∂λ
= 2λ2
(
1 0
0 −1
)
+ λ
(
U1 V2
1 −U1
)
, (4.6)
we find that U2 in X̃λ has to include a parameter α, and U1 and W1 have to include a parame-
ter α′. For this purpose, we take the symplectic leaf
S =
{
ψ̃1 = 4α1, ψ̃2 = 2α2 + 6α2
1, ψ̃3 = α3 + 4α1(α2 + α2
1), ψ̃6 = α6
}
.
Further, we change the coordinate as W1 = W̃1 + α1. Then, the above relations for S yield
U1 = 2α1, U2 = α2 + α2
1 −
1
2
V2, V3 = α3 − 2U3 − V2W̃1 + α1V2,
V4 =
(
α6 − U2
3
)
/W2.
Substituting them into X̃λ, A1 and A2, we can verify the condition (4.6) with α = α2 and
α′ = α1. By replacing α2, α1 by t1, t2, respectively, we obtain the isomonodromic deformation
equations (4.5), which are equations of (V2, W̃1, U3,W2) with two independent variables t1, t2.
Multi-Poisson Approach to the Painlevé Equations 21
The Poisson tensor P̃3 on the symplectic leaf expressed in the coordinates (V2, W̃1, U3,W2) is
given by
P̃3 =
0 2 0 0
−2 0 0 0
0 0 0 W2
0 0 −W2 0
.
To change to Darboux’s coordinates, put(
V2, W̃1, U3,W2
)
= (2p1, q1, q2p2 − β3, p2),
where β3 is an arbitrary parameter. In the new coordinates, we obtain
P̃3 =
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
. (4.7)
Therefore, the two isomonodromic deformation equations (4.5) are Hamiltonian systems in this
coordinate system. The Hamiltonian functions are obtained by deleting (U1, U2, V3, V4) from ψ̃4
and ψ̃5 and by changing to the coordinates (p1, q1, p2, q2). It is easy to verify that if we set
β23 = α6, then two functions become polynomials, which give Hamiltonian functions (1.10)
of (PIV)2 given in Section 1.
(iii) Consider the pair of vector fields P̃2dψ̃3 and P̃2dψ̃4. For the Poisson tensor P̃2, a sym-
plectic leaf is defined by the level surface {ψ̃j = const, j = 1, 2, 5, 6}. In this case, we cannot
find an integer l and a parameter α′ such that the condition
∂X̃λ
∂α′
= λl
∂A2
∂λ
holds. Hence, we impose only one condition
∂X̃λ
∂α
= λ2
∂A1
∂λ
= λ2
(
1 0
0 −1
)
. (4.8)
For it, U1 in X̃λ has to include a parameter α. To this end, take the symplectic leaf
S =
{
ψ̃1 = 2α1, ψ̃2 = 2α2 + α2
1, ψ̃5 = α5, ψ̃6 = α6
}
.
This is rearranged as
U1 = α1, V2 = 2α2 − 2U2, V3 = (α5 − 2U2U3 − V4W1)/W2,
V4 =
(
α6 − U2
3
)
/W2.
Substituting them into X̃λ and A1, we can verify the condition (4.8) with α = α1. By repla-
cing α1 by t, we obtain the isomonodromic deformation equation
∂Lλ
∂t
= [A1, Lλ] +
∂A1
∂λ
.
The Poisson tensor P̃2 on the symplectic leaf with coordinates (U2,W1, U3,W2) is given by
P̃2 =
0 W1 0 W2
−W1 0 −W2 0
0 W2 0 0
−W2 0 0 0
.
22 H. Chiba
To change to Darboux’s coordinates, put
(U2,W1, U3,W2) = (p1q1 + p2q2 − β2, q1, p1q2 − β3, q2),
where β2 and β3 are arbitrary parameters. In the new coordinates (q1, p1, q2, p2), P̃2 is reduced
to the same form as (4.7). The isomonodromic deformation equation is a Hamiltonian system
whose Hamiltonian function is ψ̃3 written in this coordinate system. It is easy to verify that
if we set α6 = β23 and α5 = 2β2β3, then ψ̃3 written in the coordinates (q1, p1, q2, p2) becomes
a polynomial. In this manner, the Hamiltonian function (1.11) given in Section 1 is obtained.
4.2 Case (II)
In this case, the functions ψi,j = ψj are given by
ψ1 = v1,
ψ2 = u21 + v2 + v1w1,
ψ3 = 2u1u2 + v3 + v2w1 + v1w2,
ψ4 = u22 + 2u1u3 + v3w1 + v2w2 + v1w3,
ψ5 = 2u2u3 + v3w2 + v2w3,
ψ6 = u23 + v3w3.
The differential equation (2.1) defining the distribution D is u′j = −vj , v′j = 0, w′j = 2uj for
j = 1, 2, 3. We can assume without loss of generality that v1 = 1 by a suitable scaling of
variables. These equations are solved with respect to u1 as
v1 = 1, w1 = −u21 +W2,
u2 = V2u1 + U3, v2 = V2, w2 = −V2u21 − 2U3u1 +W4,
u3 = V4u1 + U5, v3 = V4, w3 = −V4u21 − 2U5u1 +W6,
where W2, U3, V2, W4, U5, V4, W6 are integral constants (initial values at u1 = 0). This relation
defines a coordinate transformation
(u1, w1, u2, v2, w2, u3, v3, w3) 7→ (u1,W2, U3, V2,W4, U5, V4,W6).
In the new coordinates, integral manifolds of the distribution D are straight lines along u1-
axis. In particular, the subset {u1 = 0} ⊂ g0n gives the realization of the orbit space g0n/D
as a submanifold and (W2, U3, V2,W4, U5, V4,W6) provides a global coordinate system of g0n/D
restricted to v1 = 1.
On g0n/D, we have functions
ψ̃1 = 1,
ψ̃2 = V2 +W2,
ψ̃3 = V4 +W4 + V2W2,
ψ̃4 = U2
3 +W6 + V2W4 + V4W2,
ψ̃5 = 2U3U5 + V2W6 + V4W4,
ψ̃6 = U2
5 + V4W6,
and two vector fields (4.1), (4.2) expressed in Lax form (4.3) with
X̃λ = λ3
(
0 0
1 0
)
+ λ2
(
0 1
W2 0
)
+ λ
(
U3 V2
W4 −U3
)
+
(
U5 V4
W6 −U5
)
,
Multi-Poisson Approach to the Painlevé Equations 23
A1 = λ
(
0 0
1 0
)
+
(
0 1
W2 0
)
−
(
0 0
V2 0
)
,
A2 = λ2
(
0 0
1 0
)
+ λ
(
0 1
W2 0
)
+
(
U3 V2
W4 −U3
)
−
(
0 0
V4 0
)
.
The next purpose is to restrict the vector fields on a symplectic leaf.
(i) Consider the pair of vector fields P̃4dψ̃5 and P̃4dψ̃6. For the Poisson tensor P̃4, a symplectic
leaf is defined by the level surface {ψ̃j = const, j = 1, 2, 3, 4}. In order for the two conditions
∂X̃λ
∂α
=
∂A1
∂λ
=
(
0 0
1 0
)
,
∂X̃λ
∂α′
=
∂A2
∂λ
= 2λ
(
0 0
1 0
)
+
(
0 1
W2 0
)
(4.9)
to be satisfied, we find that W6 in X̃λ has to include a parameter α, which will be replaced
by t1 later, and W4 and V4 have to include a parameter α′, which will be replaced by t2 later.
For this purpose, we take the symplectic leaf
S =
{
ψ̃1 = 1, ψ̃2 = 0, ψ̃3 = 3α4, ψ̃4 = α6
}
.
Further, we change the coordinate as W4 = W̃4 + 2α4. Then, the above relation for S is
rearranged as
V2 = −W2, V4 = α4 − W̃4 +W 2
2 , W6 = α6 − U2
3 + 2W2W̃4 −W 3
2 + α4W2.
Substituting them into X̃λ, A1 and A2, we can verify the condition (4.9) with α = α6 and
α′ = α4. By replacing α6, α4 by t1, t2, respectively, we obtain the isomonodromic deformation
equations (4.5), which are equations of (W2, U3, W̃4, U5) with two independent variables t1, t2.
The Poisson tensor P̃4 on the symplectic leaf written in this coordinate system is given by
P̃4 =
0 0 0 1
0 0 −1 0
0 1 0 W2
−1 0 −W2 0
.
To change to Darboux’s coordinates, put(
W2, U3, W̃4, U5
)
= (q1, p2, q2, p1 + p2q1).
Then, P̃4 is transformed to the canonical symplectic matrix. In the coordinates (q1, p1, q2, p2),
the above two isomonodromic deformation equations are Hamiltonian systems. The Hamiltonian
functions are obtained by deleting V2, V4, W6 from ψ̃5 and ψ̃6 by using the above relations, and
changing to Darboux’s coordinates. In this manner, we obtain Hamiltonian functions (1.7)
of (PI)2 given in Section 1.
(ii) Consider the pair of vector fields P̃3dψ̃4 and P̃3dψ̃5. For the Poisson tensor P̃3, a sym-
plectic leaf is defined by the level surface {ψ̃j = const, j = 1, 2, 3, 6}. For the two conditions
∂X̃λ
∂α
= λ
∂A1
∂λ
= λ
(
0 0
1 0
)
,
∂X̃λ
∂α′
= λ
∂A2
∂λ
= 2λ2
(
0 0
1 0
)
+ λ
(
0 1
W2 0
)
, (4.10)
we find that W4 in X̃λ has to include a parameter α, and W2 and V2 have to include a parame-
ter α′. For this purpose, we take the symplectic leaf
S =
{
ψ̃1 = 1, ψ̃2 = 3α2, ψ̃3 = α4 + 3α2
2, ψ̃6 = α10
}
.
24 H. Chiba
Further, we change the coordinate as V2 = Ṽ2+α2. Then, the above relation for S is rearranged as
W2 = 2α2 − Ṽ2, W4 = α4 + α2
2 − α2Ṽ2 − V4 + V 2
2 , W6 =
(
α10 − U2
5
)
/V4.
Substituting them into X̃λ, A1 and A2, we can verify the condition (4.10) with α = α4 and
α′ = α2. By replacing α4, α2 by t1, t2, respectively, we obtain the isomonodromic deforma-
tion equations (4.5). The Poisson tensor P̃3 on the symplectic leaf written in the coordinates
(Ṽ2, U3, V4, U5) is given by
P̃3 =
0 1 0 0
−1 0 0 0
0 0 0 −V4
0 0 V4 0
.
To change to Darboux’s coordinates, put(
Ṽ2, U3, V4, U5
)
= (p2, q2, p1, q1p1 − α5),
where α5 is an arbitrary parameter. Then, P̃3 is transformed to the canonical symplectic mat-
rix. In the coordinates (q1, p1, q2, p2), the above two isomonodromic deformation equations are
Hamiltonian systems. The Hamiltonian functions are obtained by deleting W2, W4, W6 from ψ̃4
and ψ̃5 by using the above relations, and by changing to Darboux’s coordinates. It is easy to
verify that if we set α10 = α2
5, then two functions become polynomials. In this manner, we
obtain Hamiltonian functions (1.8) of (PII-1)2 given in Section 1.
(iii) Consider the vector fields P̃2dψ̃3 and P̃2dψ̃4. For the Poisson tensor P̃2, a symplectic
leaf is defined by the level surface {ψ̃j = const, j = 1, 2, 5, 6}. In this case, we cannot find an
integer l and a parameter α′ such that the condition
∂X̃λ
∂α′
= λl
∂A2
∂λ
holds. Hence, we impose only one condition
∂X̃λ
∂α
= λ2
∂A1
∂λ
= λ2
(
0 0
1 0
)
. (4.11)
For it, W2 in X̃λ has to include a parameter α. To this end, take the symplectic leaf
S =
{
ψ̃1 = 1, ψ̃2 = α2, ψ̃5 = α8, ψ̃6 = α10
}
.
This is rearranged as
W2 = α2 − V2, W4 = (α8 − 2U3U5 − V2W6)/V4, W6 =
(
α10 − U2
5
)
/V4.
Substituting them into X̃λ and A1, we can verify the condition (4.11) with α = α2. By repla-
cing α2 by t, we obtain the isomonodromic deformation equation
∂Lλ
∂t
= [A1, Lλ] +
∂A1
∂λ
.
The Poisson tensor P̃2 on the symplectic leaf with coordinates (U3, V2, U5, V4) is given by
P̃2 =
0 −V2 0 −V4
V2 0 V4 0
0 −V4 0 0
V4 0 0 0
.
Multi-Poisson Approach to the Painlevé Equations 25
To change to Darboux’s coordinates, put
(U3, V2, U5, V4) = (p1q1 + p2q2 − β3, p2, p1q2 − β5, p1),
where β3 and β5 are arbitrary parameters. Then, P̃2 is transformed to the canonical symplectic
matrix. In the coordinates (q1, p1, q2, p2), the above isomonodromic deformation equation is
a Hamiltonian system. The Hamiltonian function is obtained by deleting W2, W4, W6 from ψ̃3
by using the above relations, and by changing to Darboux’s coordinates. It is easy to verify that
if we set α10 = β25 and α8 = 2β3β5, then ψ̃3 written in the coordinates (q1, p1, q2, p2) becomes
a polynomial. This procedure yields the Hamiltonian function (1.12) given in Section 1.
5 4-dimensional Painlevé equations: g = so5, n = 1
According to [8], we use the following representation for the Lie algebra g ' so5 of type B2
Xi =
pi qi ri si 0
ti ui vi 0 si
wi xi 0 vi −ri
yi 0 xi −ui qi
0 yi −wi ti −pi
.
Consider the Lie algebra g1 = {Xλ = λX0 + X1 |Xi ∈ g ' so5}. For the definition of g01, we
only consider the following case
X0 =
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
,
(i.e., x0 = t0 = 1 and zeros otherwise). The purpose in this section is to derive the Hamilto-
nian (1.13) for Cosgrove’s equation. The other choice of X0 may yield different Painlevé systems.
Note that n = 1, rank(g) = h = 2, m1 = 1 and m2 = 3. We have the following functions
ψ1,1 = −2q1 − 2v1,
ψ1,2 = −2q1t1 − u21 − 2v1x1 − 2s1y1,
ψ2,1 = −2s1,
ψ2,2 = q21 − 2s1t1 + 2q1v1 − 4s1x1,
ψ2,3 = 2q21t1 + 2q1t1v1 + 2s1u1w1 − 4s1t1x1 + 2q1v1x1 − 2s1x
2
1 − 2q1s1y1 + 2s1v1y1,
ψ2,4 = q21t
2
1 − 2q1u1v1w1 + 2q1s1w
2
1 + 2q1t1v1x1 + 2s1u1w1x1
− 2s1t1x
2
1 − 2q1s1t1y1 − 2q1v
2
1y1 + 2s1v1x1y1 + s21y
2
1,
which are coefficients of the characteristic polynomial det(µ−Xλ).
We solve the differential equations for the two dimensional distribution D as functions
of (p1, r1) with the initial condition (q1, s1, t1, u1, v1, w1, x1, y1) = (Q,S, T, U, V,W,X, Y ) at
(p1, r1) = (0, 0). The expressions of solutions are too long and omitted here. These solutions
define a coordinate transformation
(p1, r1, q1, s1, t1, u1, v1, w1, x1, y1) 7→ (p1, r1, Q, S, T, U, V,W,X, Y ).
26 H. Chiba
In the new coordinates, integral manifolds of the distribution D are plains which are parallel to
the (p1, r1)-plain. In particular, the subset {p1 = r1 = 0} ⊂ g01 gives the realization of the orbit
space g01/D as a submanifold and (Q,S, T, U, V,W,X, Y ) provides a global coordinate system
of g01/D.
At this stage, we have on g01/D
ψ̃1,1 = −2Q− 2V,
ψ̃1,2 = −2QT − U2 − 2V X − 2SY,
ψ̃2,1 = −2S,
ψ̃2,2 = Q2 − 2ST + 2QV − 4SX,
ψ̃2,3 = 2Q2T + 2QTV + 2SUW − 4STX + 2QVX − 2SX2 − 2QSY + 2SV Y,
and the vector field P̃2dψ̃2,3, whose Casimir functions are ψ̃1,1, ψ̃1,2, ψ̃2,1 and ψ̃2,2. The corre-
sponding differential equation is expressed in Lax form as dX̃λ/dt = [Aλ, X̃λ], where
Xλ = λ
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
+
0 Q 0 S 0
T U V 0 S
W X 0 V 0
Y 0 X −U Q
0 Y −W T 0
,
Aλ = λ∇1ψ̃2,1 +∇1ψ̃2,2 + (V −Q)∇1ψ̃1,1 +
2
S
(
Q2 − SX
)
∇1ψ̃2,1,
∇1ψ̃2,1 =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
−2 0 0 0 0
0 −2 0 0 0
, ∇1ψ̃1,1 =
0 0 0 0 0
−2 0 0 0 0
0 −2 0 0 0
0 0 −2 0 0
0 0 0 −2 0
,
∇1ψ̃2,2 =
0 −2S 0 0 0
2(Q+ V ) 0 −4S 0 0
−2U 2Q 0 −4S 0
−2(T + 2X) 0 2Q 0 −2S
0 −2(T + 2X) 2U 2(Q+ V ) 0
.
The next purpose is to restrict the vector field on a symplectic leaf. For the Poisson tensor P̃2,
a symplectic leaf is defined by the level surface {ψ̃i,j = const, i, j = 1, 2}. In order for the
condition
∂X̃λ
∂α
=
∂Aλ
∂λ
=
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
−2 0 0 0 0
0 −2 0 0 0
(5.1)
to be satisfied, we find that Y in X̃λ has to include a parameter α which will be replaced by t
later. For this purpose, we take the symplectic leaf
S =
{
ψ̃2,1 = −2, ψ̃1,1 = −2α2, ψ̃2,2 = −2α4 + α2
2, ψ̃1,2 = 4α6 − 2α2α4
}
.
This is rewritten as
S = 1, Q = α2 − V, T = α4 − 2X − 1
2
V 2,
Y = −2α6 −
1
2
U2 − 1
2
V 3 − 3V X +
α2
2
V 2 + 2α2X + α4V.
Multi-Poisson Approach to the Painlevé Equations 27
Substituting them into X̃λ and Aλ, it turns out that the condition (5.1) is satisfied with α = α6.
Finally, by replacing α6 by t, we obtain the isomonodromic deformation equation (2.4). The
Poisson tensor P̃2 written with respect to the coordinates (U, V,W,X) is already in the canonical
symplectic matrix. Thus, the isomonodromic deformation equation is a Hamiltonian system
with the Hamiltonian function ψ̃2,3 written in the coordinate system (U, V,W,X). Since this
expression is too complicated, we further introduce the symplectic transformation
(U, V,W,X) =
(
p2, q1 +
13
18
α2, p1 +
4
13
p2
(
q1 +
13
18
α2
)
,
q2 +
1
3
α4 +
7
108
α2
2 −
2
13
(
q1 +
13
18
α2
)2
)
.
Then, the Hamiltonian function (1.15), which is equivalent to (1.13), is obtained.
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1 Introduction
2 Settings
2.1 Lie–Poisson structure on gn
2.2 Multi-Poisson structure on g0n
2.3 Symplectic reduction
2.4 Isospectral deformation to isomonodromic deformation
3 2-dimensional Painlevé equations: g = sl2, n=2
3.1 Case (I)
3.2 Case (II)
4 4-dimensional Painlevé equations: g = sl2, n=3
4.1 Case (I)
4.2 Case (II)
5 4-dimensional Painlevé equations: g = so5, n=1
References
|
| id | nasplib_isofts_kiev_ua-123456789-148562 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:57:29Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chiba, H. 2019-02-18T15:51:51Z 2019-02-18T15:51:51Z 2017 Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation / H. Chiba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M35; 34M45; 34M55 DOI:10.3842/SIGMA.2017.025 https://nasplib.isofts.kiev.ua/handle/123456789/148562 A multi-Poisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form ∂Xλ/∂t=[Xλ,Aλ] in the sense of the isospectral deformation, where Xλ,Aλ∈g depend rationally on the indeterminate λ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation ∂Xλ/∂t=[Xλ,Aλ]+∂Aλ/∂λ in the sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed. This method gives a few new Painlevé systems of dimension four. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation Article published earlier |
| spellingShingle | Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation Chiba, H. |
| title | Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation |
| title_full | Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation |
| title_fullStr | Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation |
| title_full_unstemmed | Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation |
| title_short | Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation |
| title_sort | multi-poisson approach to the painlevé equations: from the isospectral deformation to the isomonodromic deformation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148562 |
| work_keys_str_mv | AT chibah multipoissonapproachtothepainleveequationsfromtheisospectraldeformationtotheisomonodromicdeformation |