Multi-Poisson approach to the Painlev´ e equations:
from the isospectral deformation to the isomonodromic deformation
Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan
Hayato CHIBA 1
Apr 26, 2016 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
∂X
λ/∂t = [X
λ, A
λ] in the sense of the isospectral deformation, where X
λ, A
λ∈ 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 ∂X
λ/∂t = [X
λ, A
λ] + ∂A
λ/∂λ in the sense of the isomonodromic deforma- tion, which exhibits the Painlev´ e property, is proposed. This method gives a few new Painlev´ e systems of dimension four.
Keywords: Painlev´ e equations; Lax equations; multi-Poisson structure
1 Introduction
A differential equation defined on a complex region is said to have the Painlev´ e property if any movable singularity of any solution is a pole. Painlev´ e and his group classified second order ODEs having the Painlev´ e property and found new six differential equations called the Painlev´ e equations. Nowadays, it is known that they are written in Hamiltonian forms
(P
J) : dq
dt = ∂H
J∂p , dp
dt = − ∂H
J∂q , J = I, · · · , VI. (1.1) Among six Painlev´ e equations, the Hamiltonian functions of the first, second and fourth Painlev´ e equations are polynomials in both of the independent variable t and the dependent variables (q, p). They are given by
H
I= 1
2 p
2− 2q
3− tq, (1.2)
H
II= 1 2 p
2− 1
2 q
4− 1
2 tq
2− αq, (1.3)
H
IV= − pq
2+ p
2q − 2pqt − αp + βq, (1.4)
1
E mail address : [email protected]
respectively, where α, β ∈ C are arbitrary parameters. Another important property of the Painlev´ e equations is that they are expressed as Lax equations. Let L
λand A
λbe square matrices which depend rationally on the indeterminate λ called the spectral parameter. The Painlev´ e equations are written in Lax form as
∂L
λ∂t = [L
λ, A
λ] + ∂A
λ∂λ , (1.5)
for some choice of L
λand A
λ. This equation arises from the compatibility condition of the two differential systems
∂Ψ
∂λ = L
λΨ, ∂ Ψ
∂t = A
λΨ. (1.6)
Since the monodromy of the former system ∂ Ψ/∂λ = L
λΨ is independent of t if the equation (1.5) is satisfied, (1.5) is called the isomonodromic deformation equation.
Another type of the Lax equation is of the form
∂X
λ∂t = [X
λ, A
λ], (1.7)
which is called the isospectral deformation equation because the eigenvalues of the matrix X
λis 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 → T g be the Poisson tensor and F : g → C a smooth function.
Then, the vector field P dF on g can be expressed as the Lax equation (1.7) with some X
λ, A
λ∈ g [1].
It is notable that the isospectral deformation equation (1.7) is completely inte- grable for most examples, although the isomonodromic deformation equation (1.5) is not in general; it is believed that solutions of an isomonodromic deformation equation define new functions called the Painlev´ e transcendents.
In Nakamura [16], a way to obtain the isospectral deformation equation (1.7) from the isomonodromic deformation equation (1.5) 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 Painlev´ e equations and 40-types of four dimen- sional Painlev´ e equations are completely integrable.
The purpose in the present paper is opposite; a way to construct the isomon- odromic deformation equation (1.5) from the isospectral deformation equation (1.7) will be proposed. Let g be a simple Lie algebra over C . Consider the set of g-valued polynomials of degree n
g
n:= { X
λ:= X
0λ
n+ X
1λ
n−1+ · · · + X
n| X
i∈ g } ,
with the indeterminate λ. This set g
nis equipped with a structure of a Lie algebra
by a certain Lie bracket. At first, the isospectral deformation equation (1.7) on
g
nis constructed with the aid of the bi-Poisson theory of Magri et. al [12, 13, 14, 15]. Isospectral deformation equations obtained in this method are shown to be completely integrable (Thm.2.4). Next, we restrict the equations onto a symplectic leaf. Let φ
1, · · · , φ
Nbe Casimir functions of an underlying Poisson structure on g
n. A symplectic leaf S is defined by the level surface of them as
S := { φ
i= α
i(constant) | i = 1, · · · , N } .
Restricted on the leaf S, the isospectral deformation equation (1.7) becomes an integrable Hamiltonian system. Since the matrix X
λ∈ g
ndepends on the parameters α := (α
1, · · · , α
N), it is denoted as X
λ= X
λ(t, α).
Now suppose that there exists a parameter, say α
j, such that the following condition holds
∂X
λ∂α
j(t, α) = ∂A
λ∂λ . (1.8)
Eq.(1.7) is put together with Eq.(1.8) to yield
∂X
λ∂t (t, α) + ∂X
λ∂α
j(t, α) = [X
λ, A
λ] + ∂A
λ∂λ . Define the Lax matrix L
λby
L
λ:= X
λ(t, α) |
αj=t,
where the parameter α
jsatisfying the condition (1.8) is replaced by t. Then, the above equation is rewritten as the isomonodromic deformation equation (1.5).
Remark that the isomonodromic deformation equation (1.5) is equivalent to the zero curvature condition of the connection 1 form L
λdλ + A
λdt on a vector bundle over the (t, λ)-space, while the condition (1.8) is the exactness condition of the connection 1 form X
λdλ + A
λdα
j.
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 Painlev´ e equations (1.2), (1.3), (1.4) will be obtained in Section 3.
More generally, for g = sl(2, C ) with general n, one can obtain several Painlev´ e hierarchies of dimension 2n − 2, including the first Painlev´ e hierarchy (P
I)
m[10, 11, 17], the second-first Painlev´ e hierarchy (P
II-1)
m[5, 6, 10, 11], the second-second Painlev´ e hierarchy (P
II-2)
mand the fourth Painlev´ e hierarchy (P
IV)
m[7, 10]. They are 2m-dimensional Hamiltonian PDEs of the form (m = n − 1)
∂q
j∂t
i= ∂H
i∂p
j, ∂p
j∂t
i= − ∂H
i∂q
j, j = 1, · · · , m; i = 1, · · · , m H
i= H
i(q
1, · · · , q
m, p
1, · · · , p
m, t
1, · · · , t
m)
(1.9)
consisting of m Hamiltonians H
1, · · · , H
mwith m independent variables t
1, · · · , t
m.
When m = 1 (the case (I)), (P
I)
1and (P
IV)
1are reduced to the first and fourth
Painlev´ e equations, respectively. Both of (P
II-1)
1and (P
II-2)
1coincide with the second Painlev´ e equation, while they are different systems for m ≥ 2. When m = 2 (the case (II)), Hamiltonians of (P
I)
2, (P
II-1)
2, (P
II-2)
2and (P
IV)
2are given by
(P
I)
2
H
1= 2p
2p
1+ 3p
22q
1+ q
14− q
21q
2− q
22− t
1q
1+ t
2(q
21− q
2), H
2= p
21+ 2p
2p
1q
1− q
15+ p
22q
2+ 3q
13q
2− 2q
1q
22+t
1(q
12− q
2) + t
2(t
2q
1+ q
1q
2− p
22),
(1.10)
(P
II-1)
2
H
1= 2p
1p
2− p
32− p
1q
12+ q
22− t
1p
2+ t
2p
1+ 2αq
1, H
2= − p
21+ p
1p
22+ p
1p
2q
12+ 2p
1q
1q
2+t
1p
1+ t
2(t
2p
1− p
1q
21+ p
1p
2) − α(2p
2q
1+ 2q
2+ 2t
2q
1),
(1.11)
(P
II-2)
2
H
1= p
1p
2− p
1q
21− 2p
1q
2+ p
2q
1q
2+ q
1q
22+ q
2t
1+ t
2(q
1q
2− p
1) + αq
1, H
2= p
21− p
1p
2q
1+ p
22q
2− 2p
1q
1q
2− p
2q
22+ q
12q
22+t
1(q
1q
2− p
1) − t
2(p
1q
1+ q
22+ q
2t
2) + αp
2,
(1.12) (P
IV)
2
H
1= p
21+ p
1p
2− p
1q
12+ p
2q
1q
2− p
2q
22− t
1p
1+ t
2p
2q
2+ αq
2+ βq
1, H
2= p
1p
2q
1− 2p
1p
2q
2− p
22q
2+ p
2q
1q
22+p
2q
2t
1+ t
2(p
1p
2− p
2q
22+ p
2q
2t
2) + (p
1− q
1q
2+ q
2t
2)α − βp
2, (1.13) 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 (1.7), and if there are several parameters satisfying (1.8); such parameters will be replaced by distinct times t
1, t
2, · · · .
We will find other 4-dimensional Painlev´ e systems with Hamiltonian functions H
(1,1,2,0)= − p
21q
1− 2p
1q
12+ 2p
1q
2− 2p
1p
2q
2− 2p
2q
1q
2+(2p
1q
1+ 2p
2q
2)t + (2α
2+ 2β
2)q
1+ 2β
2p
1+ 2β
3p
2, (1.14) H
(−1,4,1,2)= p
1− p
22− 2p
1q
1q
2− p
2q
22+ 2β
3q
2+ 2β
5q
1+ p
2t, (1.15)
H
Cosgrove= − 4p
1p
2− 2p
22q
1− 73
128 q
14+ 11
8 q
21q
2− 1 2 q
22− q
1t − α
248
(
q
1+ α
26
)
q
12, (1.16)
where α
i, β
i∈ C are arbitrary parameters (the subscripts for parameters are related to the weighted degrees so that the Hamiltonian functions become quasihomoge- neous, 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 H
Cosgrovewill 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 q
1= y, we obtain
y
′′′′= 18yy
′′+ 9(y
′)
2− 24y
3+ 16t + αy(y + 1
9 α). (1.17)
This equation was given in Cosgrove [8], denoted by F-VI, without a proof that
it has the Painlev´ e property. Since this system is obtained as the isomonodromic
deformation equation in this paper, this equation actually enjoys the Painlev´ e prop- erty. In his paper [8], it is conjectured that this equation defines a new Painlev´ e transcendent (i.e. it is not reduced to known equations). Another expression of the Hamiltonian function of the same system is
H e
Cosgrove= 2p
1p
2− 18
13 p
22q
1− 2
169 q
41− 180
13 q
12q
2+ 6q
22− 8q
1t + 8
9 α
2q
13+ 8
27 α
22q
12. (1.18) The corresponding Hamiltonian system is also reduced to (1.17).
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(x
1, · · · , x
n) is called a quasihomogeneous polynomial if there are integers a
1, · · · , a
nand h such that
H(λ
a1x
1, · · · , λ
anx
n) = λ
hH(x
1, · · · , x
n) (1.19) for any λ ∈ C . A polynomial H is called a semi-quasihomogeneous if H is decom- posed into two polynomials as H = H
P+ H
N, where H
Psatisfies (1.19) and H
Nsatisfies
H
N(λ
a1x
1, · · · , λ
anx
n) ∼ o(λ
h), | λ | → ∞ .
The integer wdeg(H) := h is called the weighted degree of H with respect to the weight wdeg(x
1, · · · , x
n) := (a
1, · · · , a
n). For example, if we define degrees of variables by wdeg(q, p, t) = (2, 3, 4) for H
I, wdeg(q, p, t) = (1, 2, 2) for H
IIand wdeg(q, p, t) = (1, 1, 1) for H
IV, then Hamiltonian functions have the weighted de- grees 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 X
λ∈ g
nand 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 Chiba [2, 3, 4]
for the detailed study of the weights of the Painlev´ e equations.
wdeg(q, p, t) wdeg(H)
P
I(2, 3, 4) 6
P
II(1, 2, 2) 4 P
IV(1, 1, 1) 3
Table 1: Weights for two dimensional Painlev´ e equations.
2 Settings
2.1 Lie-Poisson structure on g n
We define a multi-Poisson structure on a certain Lie algebra following Magri et.
al [12, 13, 14, 15]. Let (g, [ · , · ]) be a simple Lie algebra over C . Consider the set
wdeg(q
1, p
1, q
2, p
2) wdeg(t
1, t
2) wdeg(H
1, H
2)
(P
I)
2(2, 5, 4, 3) 6, 4 8, 10
(P
II-1)
2(1, 4, 3, 2) 4, 2 6, 8 (P
II-2)
2(1, 3, 2, 2) 3, 2 5, 6
(P
IV)
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
H
Cosgrove(2, 5, 4, 3) 6 8
Table 2: Weights for four dimensional Painlev´ e equations.
of g-valued polynomials of degree n
g
n:= { X
λ:= X
0λ
n+ X
1λ
n−1+ · · · + X
n| X
i∈ g } , (2.1) with the indeterminate λ. The bracket defined by
[X
λ, Y
λ]
n:= [X
n, Y
n] + λ([X
n, Y
n−1] + [X
n−1, Y
n]) + · · · +λ
n([X
0, Y
n] + [X
1, Y
n−1] + · · · + [X
n, Y
0])
introduces the structure of a Lie algebra on g
n. Note that [X
λ, Y
λ]
ncoincides with [X
λ, Y
λ] 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 Poisson 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 g
n, we define a nondegenerate symmetric bilinear form η on g
nby
η(X
λ, Y
λ) :=
∑
n i=0Tr(X
iY
n−i),
by which g
nis identified with its dual. For a smooth function F : g
n→ C , define the gradient ∇ F ∈ g
nthrough (dF )(Y
λ) = η( ∇ F, Y
λ), and define ∇
iF ∈ g by
∇ F = ( ∇
nF )λ
n+ ( ∇
n−1F )λ
n−1+ · · · + ∇
0F.
Using them, the Lie-Poisson bracket on g
nis given by { F, G }
0:= η(X
λ, [ ∇ F, ∇ G]
n)
= Tr(X
0· [ ∇
0F, ∇
0G]) + Tr (X
1· ([ ∇
0F, ∇
1G] + [ ∇
1F, ∇
0G])) + · · · +Tr(X
n· ([ ∇
0F, ∇
nG] + · · · + [ ∇
nF, ∇
0G]))
= − Tr( ∇
0F · ([X
0, ∇
0G] + [X
1, ∇
1G] + · · · + [X
n, ∇
nG])) − · · ·
− Tr( ∇
n−1F · ([X
n−1, ∇
0G] + [X
n, ∇
1G])) − Tr( ∇
nF · [X
n, ∇
0G]).
The Poisson tensor (bivector) P
0: T
∗g
n→ T g
nis defined so that { F, G }
0= dF (P
0dG) = η( ∇ F, P
0dG) =
∑
n i=0Tr( ∇
iF · (P
0dG)
i).
This implies
− (P
0dG)
0= [X
0, ∇
0G] + [X
1, ∇
1G] + · · · + [X
n, ∇
nG]
.. .
− (P
0dG)
n−1= [X
n−1, ∇
0G] + [X
n, ∇
1G]
− (P
0dG)
n= [X
n, ∇
0G].
The following expression is useful
P
0: dG 7→ −
[X
0, · ] [X
1, · ] · · · [X
n, · ]
.. . . . .
[X
n−1, · ] [X
n, · ] [X
n, · ]
∇
0G .. .
∇
n−1G
∇
nG
=
[ ∇
0G, X
0] + [ ∇
1G, X
1] + · · · + [ ∇
nG, X
n] .. .
[ ∇
0G, X
n−1] + [ ∇
1G, X
n] [ ∇
0G, X
n]
. (2.2)
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(X
λ) is expanded as A(X
λ) = λ
nA(X
0) + λ
n−1A(X
1) + · · · + A(X
n).
Putting A(X
j) = A
j, P
0is represented as an (n + 1)dim(g) × (n + 1)dim(g) matrix
P
0= −
A
0· · · A
n−1A
nA
1· · · A
n.. . . . . A
n
. (2.3)
In what follows, suppose dim(g) = d, rank(g) = h and let m
1, · · · , m
hbe ex-
ponents of g. Let (y
1, · · · , y
d) 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 φ
i(y
1, · · · , y
d), i =
1, · · · , h, and they satisfy deg(φ
i) = m
i+ 1.
Let x
j:= (x
j,1, · · · , x
j,d) be coordinates on the j-th copy of g (coordinate expres- sion for X
j) and (x
0, · · · , x
n) coordinates on g
n. We define the weighted degrees of variables to be
wdeg(x
j) = wdeg(x
j,α) = j, wdeg(λ) = 1. (2.4) Then, X
λis quasihomogeneous (homogeneous in the weighted sense) of wdeg(X
λ) = n. Substituting y
α= x
0,αλ
n+x
1,αλ
n−1+ · · · + x
n,αinto φ
i(y
1, · · · , y
d) and expanding it in λ provide
φ
i(y
1, · · · , y
d) = φ
i,0(x
0, · · · , x
n)λ
(mi+1)n+ φ
i,1(x
0, · · · , x
n)λ
(mi+1)n−1+
· · · + φ
i,(mi+1)n(x
0, · · · , x
n), i = 1, · · · , h, which defines polynomials φ
i,jon g
nsatisfying
deg(φ
i,j) = m
i+ 1, wdeg(φ
i,j) = j. (2.5) Proposition 2.1.
(i) φ
i,jdepends only on (x
0, · · · , x
j) for 0 ≤ j ≤ n − 1.
(ii) φ
i,j(x
0, x
1, · · · , x
n) = φ
i,(mi+1)n−j(x
n, · · · , x
1, x
0).
(iii) For each i, j, α, the derivative ∂φ
i,j+k/∂x
k,αis independent of k = 0, · · · , n.
(iv) For each i, j , the gradient ∇
kφ
i,j+kis independent of k = 0, · · · , n.
(v) For each i, j, k, the equality
∑
n l=0A
l∂φ
i,j+k−l∂x
k=
∑
n l=0A
l∂φ
i,j−l∂x
0= 0 (2.6)
holds.
(vi) The Casimir functions of the Lie-Poisson structure P
0on g
nare φ
i,(mi+1)n−j, i = 1, · · · , h; j = 0, · · · , n.
Proof. (i) and (ii) follow from the definition of φ
i,j. (iii) For y
α= ∑
nk=0
λ
n−kx
k,α, we have
∂φ
i∂y
α= ∂x
k,α∂y
α∂
∂x
k,α(m
∑
i+1)n j=0λ
(mi+1)n−jφ
i,j=
(m
∑
i+1)n j=0λ
min−j+k∂φ
i,j∂x
k,α=
m
∑
in+k j=kλ
min−j+k∂φ
i,j∂x
k,α.
For the last equality, we used Part (i) combined with Part (ii). Thus we obtain
∂φ
i∂y
α=
min
∑
j=0
λ
min−j∂φ
i,j+k∂x
k,α.
Since the left hand side is independent of k, so is each coefficient of λ
min−jin the right hand side. Part (iv) immediately follows from (iii).
(v) The first equality is a consequence of Part (iii). Since φ
i(y) is a Casimir function of the Lie-Poisson structure on g, Adφ
i= 0, where A is a matrix defined before. Substituting y = ∑
nk=0
λ
n−kx
kyields 0 = A ∂φ
i∂y = (λ
nA
0+ λ
n−1A
1+ · · · + A
n)
min
∑
j=0
λ
min−j∂φ
i,j+k∂x
k= ∑
j,l
λ
min−j+n−lA
l∂φ
i,j+k∂x
k=
m
∑
in+l j=lλ
min+n−j∑
n l=0A
l∂φ
i,j+k−l∂x
k. This proves the second equality of (v).
To prove (vi), it is sufficient to show
A
0· · · A
n−1A
nA
1· · · A
n.. . . . . A
n
∂φ
i,(mi+1)n−j/∂x
0∂φ
i,(mi+1)n−j/∂x
1.. .
∂φ
i,(mi+1)n−j/∂x
n
= 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 m
i= m
1= 1. Denote a general element X
λ∈ g
nas
X
λ= λ
nX
0+ λ
n−1X
1+ · · · + X
n= λ
n( u
0v
0w
0− u
0)
+ λ
n−1( u
1v
1w
1− u
1)
+ · · · +
( u
nv
nw
n− u
n)
.
Let (u
j, v
j, w
j) be coordinates on the j-th copy of g and (u
0, v
0, w
0, · · · , u
n, v
n, w
n) coordinates on g
n. Then,
∇
jF =
1 2
∂F
∂u
j∂F
∂w
j∂F
∂v
j− 1 2
∂F
∂u
j
, A
j=
0 v
j− w
j− v
j0 2u
jw
j− 2u
j0
.
The Casimir function on g is given by φ
i= φ = u
2+ vw. Then, the functions φ
i,j= φ
jare defined by expanding
(λ
nu
0+ · · · + u
n)
2+ (λ
nv
0+ · · · + v
n)(λ
nw
0+ · · · + w
n) in λ. This gives
φ
j= ∑
k+l=j
