Funkcialaj Ekvacioj, 41 (1998) 483-503
Higher Order Painleve Equations of Type
$A_{l}^{(1)}$
By
Masatoshi NOUMI and Yasuhiko YAMADA
(Kobe University, Japan)
Absffact.
of type
A series of systems of nonlinear equations with affine Weyl group symmetry
is studied. This series gives a generalization of Painleve equations
to higher orders.
$A_{l}^{(1)}$
and
1.
$P_{¥mathrm{V}}$
$P_{¥mathrm{I}¥mathrm{V}}$
Introduction
In this paper we propose a series of systems of nonlinear differential
equations which have symmetry under the affine Weyl groups of type
$(l ¥geq 2)$ .
These systems are thought of as higher order analogues of the Painleve
. We will study in particular their Hamiltonian structures
equations
and
properties
of their -functions.
and some basic
, we consider a differential system for $(f+1)$ unknown
For each $/=2,3$ ,
functions ,
, containing complex parameters , _’ which correspond
. In what follows, we
to the simple roots of the affine root system of type
$+a_{l}=k$ .
Our differential system is defined as follows according to
set $a_{0}+$
the parity of : For $=2n(n =1,2, ¥ldots)$
$A_{l}^{(1)}$
$P_{¥mathrm{V}}$
$P_{¥mathrm{I}¥mathrm{V}}$
$¥mathrm{r}$
$¥ldots$
$f_{0}$
$¥ldots,f_{l}$
$¥alpha_{0}$
$¥alpha_{l}$
$A_{I}^{(1)}$
?
$l$
$l$
(0. 1)
where
(0.1)
$A_{2n}^{(1)}$
$0¥leq j¥leq 2n$
$A_{2n+1}^{(1)}$
:
:
$f_{j}^{¥prime}=f_{j}(¥sum_{1¥leq r¥leq n}f_{j+2r-1}-¥sum_{1¥leq r¥leq n}f_{j+2r})+¥alpha_{j}$
, and for $f=2n+1(n =1,2, ¥ldots)$
$f_{j}^{¥prime}=f_{j}(¥sum_{1¥leq r¥leq s¥leq n}f_{j+2r-1}f_{j+2s}-¥sum_{1¥leq r¥leq s¥leq n}f_{j+2r}f_{j+2s+1)}$
$+(¥frac{k}{2}-¥sum_{1¥leq r¥leq n}¥alpha_{j+2r})f_{j}+¥alpha_{j}(¥sum_{1¥leq r¥leq n}f_{j+2r})$
,
where $0¥leq j¥leq 2n+1$ . In (0.1) and (0.2), ’ stands for the derivation $d/dt$ with
for small 1 is
respect to an independent variable . A table of formulas of
are obtained
given in Appendix for convenience. Formulas for the other
from those simply by the rotation of indices. It can be shown that the differential systems for $1=2$ and $=3$ are equivalent to the fourth and the fifth
Painleve equations, respectively (see [9]).
$t$
$f_{0}^{¥prime}$
$f_{j}^{¥prime}$
$l$
484
Masatoshi NOUMI and Yasuhiko YAMADA
This paper is organized as follows. In Section 1, we discuss symmetry of
our system under the affine Weyl group of type
, by describing explicit
Backlund transformations. After formulating a Poisson structure for our
system, we will construct in Section 3 certain canonical coordinates with respect
to the Poisson structure, and show that our system can be equivalently written
as a Hamiltonian system with a polynomial Hamiltonian. In the final
we introduce a family of -functions for our system and discuss their Backlund
transformations. In particular we will show that the variables
are expressed
$A_{l}^{(1)}$
$¥mathrm{s}¥mathrm{e}¥mathrm{c}¥mathrm{t}¥mathrm{i}¥mathrm{o}¥mathrm{n}_{>}$
$¥mathrm{r}$
$f_{j}$
multiplicatively in terms of the -functions and their Backlund transformations.
This shows that our -functions are consistent with those introduced in [8] from
the viewpoint of discrete dynamical systems.
$¥mathrm{r}$
$¥mathrm{r}$
1.
Affine Weyl group symmetry
be the field of rational functions in
and $f=$
. Then the differential system (0. 1) (resp. (0.2)) defines the structure
of a differential field on $C(¥alpha; f)$ . We say that an automorphism of $C(¥alpha; f)$ is
a Backfund transformation of the differential system if it commutes with the
derivation ’.
For each $i=0,1$ ,
of $C(¥alpha; f)$ as
, we define an automorphism
follows:
Let
$C(¥alpha; f)$
$¥alpha=$
$(¥alpha_{0}, ¥ldots, ¥alpha_{l})$
$(f_{0}, ¥ldots,f_{l})$
$f$
$s_{i}$
_’
$s_{i}(¥alpha_{i})=-¥alpha_{i}$
(1.1)
$s_{i}(f_{i})=f_{i}$
,
,
$s_{i}(¥alpha_{j})=¥alpha_{j}+¥alpha_{i}(j=i¥pm 1)$
$s_{i}(f_{j})=f_{j}¥pm¥frac{¥alpha_{i}}{f_{i}}(j=i¥pm 1)$
where the indices 0, 1, . . .
define an automorphism
(1.2)
$f$
’
$¥pi$
,
,
$s_{i}(¥alpha_{j})=¥alpha_{j}(j¥neq i, i¥pm 1)$
$s_{i}(f_{j})=f_{j}(j¥neq i, i¥pm 1)$
,
,
are understood as elements of $Z/(l+1)Z$ . We also
of
$C(¥alpha; f)$
$¥pi(¥alpha_{j})=¥alpha_{j+1}$
,
by
$¥pi(f_{j})=f_{j+1}$
.
Theorem 1.1. ([8]). The automorphisms
and
,
define a representation of the extended affine Weyl group
. Namely, they satisfy the commutation relations
type
$s_{0},s_{1}$
_’
$s_{l}$
$¥pi$
described above
$¥tilde{W}=¥langle s_{0}, ¥_’ s_{l}, ¥pi¥rangle$
of
$A_{l}^{(1)}$
(1.3)
for
$i,j$
(1.4)
$s_{i}^{2}=1$
$=0,1$ ,
,
$s_{i}s_{j}=s_{j}s_{i}(j¥neq i, i¥pm 1)$
$f$
_’
,
$s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j}(j=i¥pm 1)$
and
$¥pi^{l+1}=1$
,
$¥pi s_{i}=s_{i+1}¥pi(i=0,1, -’ l)$
.
Note that the extended affine Weyl group $¥tilde{W}=W¥mathrm{x}¥{1, ¥pi, ¥_’ ¥pi^{l}¥}$ is the
extension of the ordinary affine Weyl group
by a
of type
cyclic group of order 1+1 generated by the diagram rotation .
$ W=¥langle s_{0}, ¥ldots,s_{l}¥rangle$
$A_{l}^{(1)}$
$¥pi$
Higher Order Painleve Equations
of
Type
485
$A_{l}^{(1)}$
Theorem 1.2. The action of
defined as above commutes with the
$C(
¥
alpha;
f)$
.
derivation of the differential fiefd
$¥tilde{W}$
In this sense, our differential system (0.1) (resp. (0.2)) admits the action
of type
, as a group of Backlund
of the extended affine Weyl group
transformations. Theorem 1.2 can be checked essentially by direct computations. We will explain below how such computations can be carried out.
$¥tilde{W}$
Remark 1.3.
A
$l(1)$
Let
$A$
$=(a_{ij})_{0¥leq i,j¥leq l}$
$A_{l}^{(1)}$
be the generalized Cartan matrix of type
:
(1.5)
$a_{jj}=2$
Then the action of
(1.6)
,
$a_{ij}=-1(j =i¥pm 1)$ ,
$s_{0}$
,
_’
$s_{l}$
on the simple roots
We introduce an
$(l+1)¥times(l+1)$
(1.7)
$u_{ij}=¥pm 1(j =i¥pm 1)$ ,
matrix
Then the Backlund transformations
$s_{0}$
$s_{i}(f_{j})=f_{j}+¥frac{¥alpha_{i}}{f_{i}}u_{ij}$
$¥alpha_{0}$
,
_’
$(¥mathrm{i}¥mathrm{j} =0,1, ¥_’ ¥mathit{1})$
$s_{i}(¥alpha_{j})=¥alpha_{j}-¥alpha_{i}a_{ij}$
(1.8)
$a_{ij}=0(j ¥neq i, i¥pm 1)$ .
,
.
$U=(u_{ij})_{0¥leq i,j¥leq l}$
by setting
$u_{ij}=0(j ¥neq i¥pm 1)$
_’
$s_{l}$
is described as
$¥alpha_{l}$
.
are determined by the formulas
$(¥mathrm{i}¥mathrm{j} =0,1, ¥_’ I)$
.
For a general treatment of affine Weyl group symmetry in terms of these
data $A$ and $U$, as well as discrete dynamical systems arising from Backlund
transformations, we refer the reader to [8].
For practical computations of Backlund transformations, it is convenient to
$(i =0, ¥_’ f)$ defined by
use the Demazure operators
$¥Delta_{i}$
(1.9)
The action of
$¥Delta_{i}(¥varphi)=¥frac{1}{¥alpha_{i}}(s_{i}(¥varphi)-¥varphi)$
$¥Delta_{i}$
$(¥varphi¥in C(¥alpha;f))$
.
on a product can be determined by the twisted Leibniz rule
(1.10)
$¥Delta_{i}(¥varphi¥psi)=¥Delta_{i}(¥varphi)¥psi+s_{i}(¥varphi)¥Delta_{i}(¥psi)$
We remark that
$¥Delta_{i}(¥varphi)=0$
$¥Delta_{i}(¥varphi¥psi)=¥varphi¥Delta_{i}(¥psi)$
when
$¥varphi$
.
is -invariant and that one has
if and only if
is -invariant. Note that (1.6) and (1.8) can be
$¥varphi$
$¥mathrm{s}_{¥mathrm{i}}$
$¥mathrm{s}_{¥mathrm{i}}$
rewritten as
(1.11)
$¥Delta_{i}(¥alpha_{j})=-a_{ij}$
,
$¥Delta_{i}(f_{j})=¥frac{u_{ij}}{f_{i}}(i,j=0, -’ ¥mathit{1})$
,
Masatoshi NOUMI and Yasuhiko YAMADA
486
Namely one has
in terms of the Demazure operators.
$¥Delta_{i}(¥alpha_{i})=-2$
(1.12)
$¥Delta_{i}(f_{i})=0$
,
,
$¥Delta_{i}(¥alpha_{j})=1(j=i¥pm 1)$
,
$¥Delta_{i}(f_{j})=¥frac{¥pm 1}{f_{i}}(j=i¥pm 1)$
$¥Delta_{i}(¥alpha_{j})=0(j¥neq i, i¥pm 1)$
,
,
$¥Delta_{i}(f_{j})=0(j¥neq i, i¥pm 1)$
.
It is also well known that the relations (1.3) imply
(1.13)
for
$¥Delta_{i}^{2}=0$
$i,j$
$=0$ ,
$¥ldots$
,
$l$
,
,
$¥Delta_{i}¥Delta_{j}=¥Delta_{j}¥Delta_{i}(j¥neq i¥pm 1)$
$¥Delta_{i}¥Delta_{j}¥Delta_{i}=¥Delta_{j}¥Delta_{i}¥Delta_{j}(j=i¥pm 1)$
,
.
, , we denote by $F_{j}=F_{j}(¥alpha; f)$
Theorem 1.2. For each $j=0$ ,
the polynomial appearing on the right-hand side of (0.1) or (0.2). Then it is
if and only if
easy to see that
Proof of
$f$
$¥ldots$
$s_{i}(f_{j})^{¥prime}=s_{i}(f_{j}^{¥prime})$
(1.14)
$(i,j =0, ¥ldots, f)$
$F_{j}-¥frac{¥alpha_{i}}{f_{i}^{2}}F_{i}u_{ij}=s_{i}(F_{j})$
or equivalently,
(1.15)
$¥Delta_{i}(F_{j})=-¥frac{u_{ij}}{f_{i}^{2}}F_{i}$
in terms of the Demazure operators.
check (1.15) for $j=0$ :
(1.16)
$¥Delta_{1}(F_{0})=¥frac{F_{1}}{f_{1}^{2}}$
,
$(i,j =0, ¥ldots, l)$
By the rotation symmetry, it is enough to
$¥Delta_{l}(F_{0})=-¥frac{F_{1}}{f_{l}^{2}}$
,
$¥Delta_{i}(F_{0})=0(i¥neq 0, f)$
We will show for example the equality
$2n+1$ , separately. When $f=2n$ , we have
$¥Delta_{1}(F_{0})=F_{1}/f_{1}^{2}$
(1.17)
Since
(1.18)
.
for $f=2n$ and
$F_{0}=f_{0}(f_{1}-f_{2}+¥cdots-f_{2n})+¥alpha_{0}$ .
$¥Delta_{1}(f_{j})=0$
for
$j¥neq 0,2$
, we have
$¥Delta_{1}(F_{0})=¥Delta_{1}(f_{0})(¥sum_{i=1}^{2n}(-1)^{i-1}f_{i})-s_{1}(f_{0})¥Delta_{1}(f_{2})+¥Delta_{1}(¥alpha_{0})$
$=-¥frac{1}{f_{1}}(¥sum_{i=1}^{2n}(-1)^{i-1}f_{i})-(f_{0}-¥frac{¥alpha_{1}}{f_{1}})¥frac{1}{f_{1}}+1$
$=¥frac{1}{f_{1}}(¥sum_{i=2}^{2n}(-1)^{i}f_{i}-f_{0})+¥frac{¥alpha_{1}}{f_{1}^{2}}=¥frac{1}{f_{1}^{2}}F_{1}$
.
$l$
$=$
Higher Order Painleve Equations
When
(1.19)
$l$
of
Type
$=2n+1$ , we have
$F_{0}=f_{0}(f_{1}f_{2}+f_{1}f_{4}+¥cdots+f_{2n-1}f_{2n}-f_{2}f_{3}¥_¥cdots-f_{2n}f_{2n+1})$
$+(¥frac{k}{2}-¥alpha_{2}-¥alpha_{4}¥_¥cdots-¥alpha_{2n})f_{0}+¥alpha_{0}(f_{2}+f_{4}+¥cdots+f_{2n})$
Hence,
487
$A_{l}^{(1)}$
$¥Delta_{1}(F_{0})$
.
is computed as follows:
$¥Delta_{1}(f_{0})(¥sum_{1¥leq r¥leq s¥leq n}f_{2r-1}f_{2s}-¥sum_{1¥leq r¥leq s¥leq n}f_{2r}f_{2s+1})+s_{1}(f_{0})¥Delta_{1}(f_{2})(f_{1}-¥sum_{r=1}^{n}f_{2r+1})$
$-¥Delta_{1}(¥alpha_{2})f_{0}+s_{1}(¥frac{k}{2}-¥sum_{r=1}^{n}¥alpha_{2r})¥Delta_{1}(f_{0})+¥Delta_{1}(¥alpha_{0})(¥sum_{r=1}^{n}f_{2r})+s_{1}(¥alpha_{0})¥Delta_{1}(f_{2})$
$=-¥frac{1}{f_{1}}(¥sum_{1¥leq r¥leq s¥leq n}f_{2r-1}f_{2s}-¥sum_{1¥leq r¥leq s¥leq n}f_{2r}f_{2s+1})$
$+(f_{0}-¥frac{¥alpha_{1}}{f_{1}})¥frac{1}{f_{1}}(f_{1}-¥sum_{r=1}^{n}f_{2r+1})-f_{0}+(¥frac{k}{2}-¥sum_{r=1}^{n}¥alpha_{2r+1}-¥alpha_{0})¥frac{1}{f_{1}}$
$+(¥sum_{r=1}^{n}f_{2r})+(¥alpha_{0}+¥alpha_{1})¥frac{1}{f_{1}}$
$=¥frac{1}{f_{1}}(¥sum_{1¥leq r¥leq s¥leq n}f_{2r}f_{2s+1}-¥sum_{2¥leq r¥leq s¥leq n}f_{2r-1}f_{2s}-¥sum_{r=1}^{n}f_{2r+1}f_{0})$
$+(¥frac{k}{2}-¥sum_{r=1}^{n}¥alpha_{2r+1})¥frac{1}{f_{1}}+¥frac{¥alpha_{1}}{f_{1}^{2}}(¥sum_{r=1}^{n}f_{2r+1})=¥frac{1}{f_{1}^{2}}F_{1}$
.
The other formulas in (1.16) can be checked similarly.
2.
$¥square $
Poisson structure
$¥{, ¥}$
By using the matrix $U$ defined in (1.7), we introduce the Poisson bracket
on $C(¥alpha; f)$ as follows:
(2. 1)
Note that
(2.2)
$¥{¥varphi, ¥psi¥}=¥sum_{0¥leq i,j¥leq l}¥frac{¥partial¥varphi}{¥partial f_{i}}u_{ij}¥frac{¥partial¥psi}{¥partial f_{j}}$
$¥{, ¥}$
is a
$¥mathrm{C}(¥mathrm{a})$
.
-bilinear skewsymmetric form such that
$¥{f_{i},f_{j}¥}=u_{ij}$
$(¥mathrm{i},¥mathrm{j} =0,1, ¥_’ l)$
,
488
Masatoshi NOUMI and Yasuhiko YAMADA
namely,
(2.3)
for
$¥{f_{i},f_{j}¥}=¥pm 1(j=i¥pm 1)$
$i,j$
$=0,1$ ,
$¥_’$
/.
,
$¥{f_{i},f_{j}¥}=0(j¥neq i¥pm 1)$
By direct calculations, one can show
invarian
Proposition 2.1. The skewsymmetric form ¥ ¥ above defines a
$C(
¥
alpha;
f)$
. Namely, one has
Poisson structure on the differential fiefd
,
,
(1)
,
(2)
.
(3) $w(¥{¥varphi, ¥psi¥})=¥{w(¥varphi), w(¥psi)¥}$ for any , $¥psi¥in C(¥alpha; f)$ and
$ {,
$¥{¥varphi, ¥psi_{1}¥psi_{2}¥}=¥psi_{1}¥{¥varphi, ¥psi_{2}¥}+¥{¥varphi, ¥psi_{1}¥}¥psi_{2}$
$¥tilde{W}-$
}$
$¥tau$
$¥{¥varphi_{1}¥varphi_{2}, ¥psi¥}=¥varphi_{1}¥{¥varphi_{2}, ¥psi¥}+¥{¥varphi_{1}, ¥psi¥}¥varphi_{2}$
$¥{¥varphi_{1}, ¥{¥varphi_{2}, ¥varphi_{3}¥}¥}+¥{¥varphi_{2}, ¥{¥varphi_{3}, ¥varphi_{1}¥}¥}+¥{¥varphi_{3}, ¥{¥varphi_{1}, ¥varphi_{2}¥}¥}=0$
$w¥in¥tilde{W}$
$¥varphi$
We remark that our Poisson bracket has a nontrivial radical. Regard
-dimensional vector space $E=$
as a skewsymmetric form on the
. Then its radical is precisely the subspace of all linear combisuch that ¥
for
with coefficients in
nations
$i=0$ , _’ . It is a one-dimensional subspace generated by
$¥{, ¥}$
$(¥mathrm{f}+1)$
$¥oplus_{j=0}^{l}C(¥alpha)f_{j}$
$c_{¥mathit{0}}f_{¥mathit{0}}+$
?
$ sum_{j=0}^{l}u_{ij}c_{j}=0$
$C(¥alpha)$
$+c_{l}f_{l}$
$¥mathrm{a}1¥mathrm{I}$
$l$
(2.4)
if
$g=f_{0}+f_{1}+-+f_{2n}$
$=2n$ ,
$l$
and is a two-dimensional subspace generated by
(2.5)
if
$g_{0}$
$g_{0}=f_{0}+f_{2}+-+f_{2n}$
$¥mathit{1}=2n+1$
,
$g_{1}$
are
,
$g_{1}=f_{1}+f_{3}+-+f_{2n+1}$
, respectively. Note that is
-invariant, and that $g_{0}+g_{1}$ is
$g$
$¥tilde{W}-$
$¥mathrm{W}$
,
invariant when $f=2n$ , and that
invariant when $=2n+1$ .
$¥tilde{W}-$
$l$
We will describe below our differential systems (0.1) and (0.2) by means of
the Poisson structure introduced above. In order to define a“Hamiltonian” we
need to fix some notation.
the -th fundamental weight of the
For each $i=1$ ,
, , we denote by
finite root system of type
$l$
$¥mathrm{i}$
$¥varpi_{i}$
$¥ldots$
$A_{l}$
(2.6)
$¥varpi_{i}=¥frac{1}{l+1}¥{(l+1-i)¥sum_{r=1}^{i}r¥alpha_{r}+i¥sum_{r=i+1}^{l}(l+1-r)¥alpha_{r}¥}$
$=¥sum_{r=1}^{l}$
$¥min¥{i, r¥}$
$-¥frac{ir}{l+1}$
)
$¥alpha_{r}$
form the dual basis of the simple roots
Note that
, _’
such that
, , with respect to the symmetric bilinear form
$=1$ , _’ ).
are then expressed as
The simple affine roots ,
and set
$¥alpha_{1}$
(
$¥ldots$
$¥mathrm{i},¥mathrm{j}$
(2.7)
$¥varpi_{0}=0$
.
$¥varpi_{1}$
$¥varpi_{l}$
$ $
$¥langle¥alpha_{i}, ¥alpha_{j}¥rangle=a_{ij}$
$¥langle, ¥rangle$
$¥alpha_{l}$
$ f$
$¥alpha_{0}$
_’
$¥alpha_{j}=-¥varpi_{j-1}+2¥varpi_{j}-¥varpi_{j+1}+¥delta_{j,0}k$
$¥alpha_{f}$
$(j =1, ¥ldots, f)$
,
Higher Order Painleve Equations
,
in terms of the fundamental weights
as
weights
fundamental
act on the
so,
$¥varpi_{1}$
_’
$¥varpi_{l}$
of
Type
489
$A_{l}^{(1)}$
We remark that the reflections
.
$¥ldots,s_{l}$
(2.8)
$s_{0}(¥varpi_{j})=¥varpi_{j}+¥alpha_{0}$
,
$s_{i}(¥varpi_{j})=¥varpi_{j}-¥delta_{i,j}¥alpha_{i}(i=1, ¥ldots, l)$
,
,
acts on
,
. Notice that the diagram rotation
for each $j=1$ ,
nontrivially.
; it is a circle with $(l+1)$ nodes
be the Dynkin diagram of type
Let
labeled by the elements of $Z/(f+1)Z$ . For each chain $C$ of , consisting
of consecutive nodes $j,j+1$ , _’ $j+m-1$ $(m ¥leq f)$ , we denote by $¥chi(C)$ the
alternating sum of corresponding fundamental weights:
$f$
$¥pi$
_’
$¥varpi_{1}$
$¥ldots$
$¥varpi_{l}$
$A_{l}^{(1)}$
$¥Gamma$
$¥Gamma$
(2.9)
$¥chi(C)=¥varpi_{j}-¥varpi_{j+1}+¥cdots+(-1)^{m-1}¥varpi_{j+m-1}$
.
with
, we denote by $¥chi(C)=¥sum_{i}¥chi(C_{i})$ the
For each subdiagram $C$ of
over all connected components .
sum of
$f+1$ , we denote by
the set of all subsets
For each $d=1$ ,
$K¥subset¥{0,1, ¥_’ ¥mathit{1}¥}$ with cardinality
such that the connected components of the
diagram $ K^{c}=¥Gamma$ , obtained by removing the nodes of $K$, are all chains of even
¥
¥
is
. We remark that the set
nodes. For each $K¥in¥Psi_{d}$ , we set
$K$
with
a
of
nonempty if and only if $f+1-d$ is even, and that subset
if and only if it has an expression
$|K|=d$ then belongs to
$K=¥{k_{1}, k_{2}, ¥_’ k_{d}¥}$ with a sequence $0¥leq k_{1}<k_{2}<¥cdots<k_{d}¥leq f$ such that
$ C¥neq¥Gamma$
$¥Gamma$
$¥chi(C_{i})$
$C_{i}$
$¥Psi_{d}$
_’
$d$
$f_{K}= prod_{i in K}f_{i}$
$¥Psi_{d}$
$¥{0, ¥_’ ¥mathit{1}¥}$
$¥mathit{9}_{d}^{2}$
(2. 10)
$(k_{1}, k_{2}, -’ k_{d})¥equiv(0,1,0, -)$
With this expression,
(2. 11)
$¥chi(K^{c})$
or
$(1, 0, 1, ¥_)$
$¥mathrm{m}¥mathrm{o}¥mathrm{d}2$
.
can be written as follows:
$¥chi(K^{c})=¥sum_{i=0}^{d-1}¥sum_{r=1}^{k_{i+1}-k_{i}-1}(-1)^{r-1}¥varpi_{k_{¥mathrm{i}}+r}$
where $k_{0}=k_{d}-l-1$ .
With the notation as above, we define a Hamiltonian
(2. 12)
$h_{0}$
for our system by
$h_{0}=¥sum_{K¥in¥Psi_{3}}f_{K}+¥sum_{K¥in¥Psi 1}¥chi(K^{c})f_{K}$
$=¥sum f_{k_{1}}f_{k_{2}}f_{k_{3}}+¥sum_{i=0}^{2n}(¥sum_{r=1}^{2n}(-1)^{r-1}¥varpi_{i+r})f_{i}$
when
$l$
$(k_{1},k_{2},k_{3})$
(1, 0, 1)
(2. 13)
is taken over all triples
where the summation
such that $0¥leq k_{1}<k_{2}<k_{3}¥leq 2n$ and that $(k_{1},k_{2}, k_{3})¥equiv(0,1,0)$ or
by
. When $f=2n+1$ , we define
$=2n$ ,
$¥sum f_{k_{1}}f_{k_{2}}f_{k_{3}}$
$¥mathrm{m}¥mathrm{o}¥mathrm{d}2$
$h_{0}$
$h_{0}=¥sum_{K¥in¥Psi_{4}}f_{K}+¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{2}}¥chi(K^{c})f_{K}+(¥sum_{i=1}^{2n+1}(-1)^{i-1}¥varpi_{i})^{2}$
.
490
Masatoshi NOUMI and Yasuhiko YAMADA
We remark that, when $f=2n+1$ , the sum appearing in the constant term of
(2. 13) has an alternative expression
(2. 14)
$¥sum_{i=1}^{2n+1}(-1)^{i-1}¥varpi_{i}=¥frac{1}{2}¥sum_{r=0}^{n}¥alpha_{2r+1}$
by simple roots. We give in Appendix some explicit formulas of
for convenience.
Proposition 2.2.
follows:
(2. 15)
When
$I$
$=2n$ ,
the
as
$h_{0}$
differential system (0.1) can
$f_{j}^{¥prime}=¥{h_{0},f_{j}¥}+¥delta_{j,0}k(j=0, -’ 2n)$
for small
$l$
be expressed
.
Hence one has
(2. 16)
When
$¥varphi^{¥prime}=¥{h_{0}, ¥varphi¥}+k¥frac{¥partial¥varphi}{¥partial f_{0}}$
$l$
$=2n+1$ , the
(2. 17)
with
defined
$g_{0}$
.
(0.2) can be expressed as follows:
by (2.5).
,
Hence one has
$¥varphi^{¥prime}=¥{h_{0}, ¥varphi¥}-¥frac{k}{2}(¥sum_{i=0}^{2n+1}(-1)^{i}f_{i}¥frac{¥partial¥varphi}{¥partial f_{i}})+kg_{0}¥frac{¥partial¥varphi}{¥partial f_{0}}$
any
$¥varphi¥in C(¥alpha; f)$
Proof.
(2. 19)
system
$¥varphi¥in C(¥alpha; f)$
$f_{j}^{¥prime}=¥{h_{0},f_{j}¥}-(-1)^{j}¥frac{k}{2}f_{j}+¥delta_{j,0}kg_{0}(j=0, -’ 2n+1)$
(2. 18)
for
differential
for any
.
For each $j=0$ ,
$X_{j}(¥varphi)=$
$f$
_’
, we define the vector field
$¥{¥varphi,f_{j}¥}=(¥frac{¥partial}{¥partial f_{j-1}}-¥frac{¥partial}{¥partial f_{j+1}})¥varphi$
We now consider the case of
computed as follows:
$A_{2n}^{(1)}$
.
$X_{j}$
by
.
From the definition (2.12),
$X_{j}(h_{0})$
$¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{2}(¥Gamma¥backslash ¥{j-1¥})}f_{K}-¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{2}(¥Gamma¥backslash ¥{j+1¥})}f_{K}+¥chi(¥Gamma¥backslash ¥{j-1¥})-¥chi(¥Gamma¥backslash ¥{j+1¥})$
$=f_{j}¥sum_{r=1}^{n}f_{j+2r-1}-f_{j}¥sum_{r=1}^{n}f_{j+2r}-¥varpi_{j-1}+2¥varpi_{j}-¥varpi_{j+1}$
$=f_{j}(¥sum_{i=1}^{2n}(-1)^{i-1}f_{j+i})+¥alpha_{j}-¥delta_{j,0}k=F_{j}-¥delta_{j,0}k$
,
is
Higher Order Painleve Equations
with the notation of
of
Type
491
$A_{l}^{(1)}$
extended to subdiagrams of
. This
$G$
proves (2.15). (For a subdiagram
of ,
stands for the set of all
$K
¥
subset
G$
subsets
with $|K|=d$ such that the connected components of
are all
chains of even nodes.) Formulas (2. 17) for the case of
can be established
in a similar way. In fact
is given by
$¥Gamma=¥{0,1, ¥ldots, ¥mathit{1}¥}$
$¥Psi_{d}$
$¥Gamma$
$¥ovalbox{¥tt¥small REJECT}_{d}(G)$
$G¥backslash K$
$A_{2n+1}^{(1)}$
$X_{j}(h_{0})$
$¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{3}(¥Gamma¥backslash ¥{j-1¥})}f_{K}-¥sum_{K¥in¥Psi_{3}(¥Gamma¥backslash ¥{j+1¥})}f_{K}$
$+¥sum_{¥{i¥}¥in¥ovalbox{¥tt¥small REJECT}_{1}(¥Gamma¥backslash ¥{j-1¥})}¥chi(¥Gamma¥backslash ¥{j-1, i¥})f_{i}-¥sum_{¥{i¥}¥in¥ovalbox{¥tt¥small REJECT}_{1}(¥Gamma¥backslash ¥{j+1¥})}¥chi(¥Gamma¥backslash ¥{j+1, i¥})f_{i}$
$=f_{j}(¥sum_{K¥in¥Psi_{2}([j+1,j-2])}f_{K}-¥sum_{K¥in¥Psi_{2}([j+2,j-1])}f_{K})$
$+(-¥varpi_{j-1}+2¥varpi_{j}-¥varpi_{j+1}+(-1)^{j}2¥sum_{i=1}^{2n+1}(-1)^{i-1}¥varpi_{i})f_{j}$
$+(-¥varpi_{j-1}+2¥varpi_{j}-¥varpi_{j+1})¥sum_{r=1}^{n}f_{j+2r}$
$=f_{j}(¥sum_{K¥in¥Psi_{2}([j+1,j-2])}f_{K}-¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{2}([j+2,j-1])}f_{K})$
$+(¥alpha_{j}-¥delta_{j,0}k+(-1)^{j}¥sum_{r=0}^{n}¥alpha_{2r+1})f_{j}+(¥alpha_{j}-¥delta_{j,0}k)¥sum_{r=1}^{n}f_{j+2r}$
$=F_{j}+(-1)^{j}¥frac{k}{2}f_{j}-¥delta_{j,0}k(¥sum_{r=0}^{n}f_{2r})$
where
$[a, b]$
stands for the chain
,
$¥{a, a+1, ¥ldots, b-1, b¥}$ .
This proves (2. 17).
$¥square $
3.
Canonical coordinates and the Hamiltonian system
By using Proposition 2.2, we introduce canonical coordinates for our
differential system. We discuss the two cases of
and
separately.
$A_{2n}^{(1)}$
Case
(3.1)
$A_{2n}^{(1)}$
$A_{2n+1}^{(1)}$
: In view of (2.3), we define a new coordinate system
$(q;p;x)=(q_{1}, ¥ldots, q_{n};p_{1}, ¥ldots,p_{n};x)$
492
Masatoshi NOUMI and Yasuhiko YAMADA
for the affine space with coordinates
(3.2)
$q_{1}=f_{2}$
,
$q_{2}=f_{4}$
$p_{1}=f_{1}$
,
$p_{2}=f_{1}+f_{3}$
,
,
$¥ldots$
$f=(f_{0}, -’ f_{2n})$
as follows:
,
$q_{n}=f_{2n}$
$p_{n}=f_{1}+f_{3}+-+f_{2n-1}$ ,
,?’
$x=g=f_{0}+f_{1}+¥cdots+f_{2n}$ .
Note that the inverse of this coordinate transformation is given by
$f_{0}=x-q_{1}-q_{2}¥_--q_{n}-p_{n}$ ,
(3.3)
$f_{1}=p_{1}$
,
,
$f_{2}=q_{1}$
$f_{2n-1}=p_{n}-p_{n-1}$
$f_{3}=p_{2}-p_{1}$
,
$f_{2n}=q_{n}$
,
$f_{4}=q_{2}$
,?’
.
Then it is easy to show that
(3.4)
for
$i,j$
$¥{p_{i}, q_{j}¥}=¥delta_{i,j}$
$=1$
,
_’
(3.5)
$n$
,
$¥{q_{i}, q_{j}¥}=¥{p_{i},p_{j}¥}=¥{p_{i},x¥}=¥{q_{i}, x¥}=0$
Hence we have
.
$¥{¥varphi, ¥psi¥}=¥sum_{i,j=1}^{n}(¥frac{¥partial¥varphi}{¥partial p_{i}}¥frac{¥partial¥psi}{¥partial q_{i}}-¥frac{¥partial¥varphi}{¥partial q_{i}}¥frac{¥partial¥psi}{¥partial p_{i}})$
Noting that
$¥partial/¥partial f_{0}=¥partial/¥partial x$
(3.6)
$(¥varphi, ¥psi¥in C(¥alpha;f))$
.
, from (2.16) we obtain
$¥varphi^{¥prime}=¥{h_{0}, ¥varphi¥}+k¥frac{¥partial¥varphi}{¥partial x}$
for any
$¥varphi¥in C(¥alpha; f)$
.
Let us denote by $H=H(q;p;x)¥in C(¥alpha)[q;p;x]$ the polynomial in $(q;p)$ which
in the coordinates $(q;p;x)$ . Then we see that
represents our Hamiltonian
the differential system (0.1) is equivalent to the Hamiltonian system
$h_{0}$
(3.7)
where
(3.8)
$¥frac{dq_{i}}{dt}=¥{H, q_{i}¥}=¥frac{¥partial H}{¥partial p_{i}}$
$i=1$ ,
$¥ldots,n$
.
,
$¥frac{dp_{i}}{dt}=¥{H,p_{i}¥}=-¥frac{¥partial H}{¥partial q_{i}}$
The Hamiltonian
$H$
,
$¥frac{dx}{dt}=k$
,
is determined explicitly as follow:
$H=(x-¥sum_{i=1}^{n}q_{i})(¥sum_{i=1}^{n}q_{i}p_{i})-¥sum_{i=1}^{n}q_{i}p_{i}^{2}-¥sum_{1¥leq i<j¥leq n}q_{i}(p_{i}-p_{j})q_{j}$
,
$-¥sum_{i=1}^{n}(¥sum_{r=1}^{i}¥alpha_{2r-1})q_{i}+¥sum_{i=1}^{n}¥alpha_{2i}p_{i}+¥beta x$
Higher Order
Pa ι nleve
Equations
of
Type
493
$A_{l}^{(1)}$
where
(3.9)
$¥beta=¥sum_{i=1}^{2n}(-1)^{i-1}¥varpi_{i}=¥frac{1}{2n+1}¥sum_{r=1}^{n}((n+1-r)¥alpha_{2r-1}-r¥alpha_{2r})$
Case
$A_{2n+1}^{(1)}$
.
: Note first that (2.17) implies
(3. 10)
$g_{0}^{¥prime}=¥frac{k}{2}g_{0}$
,
$g_{1}^{¥prime}=¥frac{k}{2}g_{1}$
.
Hence, by setting
(3.11)
$¥tilde{f_{2r}}=g_{0}f_{2r}$
,
$¥tilde{f_{2r+1}}=g_{0}^{-1}f_{2r+1}(r=0,1, -’ n)$
,
we obtain
(3.12)
$(j =0,1, ¥_’ 2n+1)$ .
$¥tilde{f_{j}}^{¥prime}=¥{h_{0},¥tilde{f_{j}}¥}+¥delta_{j,0}kg_{0}^{2}$
We now introduce a new coordinate system
(3.13)
$(q;p;x)=(q_{1}, -’ q_{n}; p_{1}-’ p_{n}; x_{0},x_{1})$
as follows:
$q_{1}=g_{0}f_{2}$
(3.14)
,
$p_{1}=g_{0}^{-1}f_{1}$
$q_{2}=g_{0}f_{4}$
,
,
$¥ldots$
,
$q_{n}=g_{0}f_{2n}$
$p_{2}=g_{0}^{-1}(f_{1}+f_{3})$
$x_{0}=g_{0}=f_{0}+f_{2}+-+f_{2n}$ ,
,
, $¥ldots,p_{n}=g_{0}^{-1}(f_{1}+f_{3}+ --+f_{2n-1})$ ,
$x_{1}=g_{1}=f_{1}+f_{3}+¥cdots+f_{2n+1}$
.
The inverse transformation is then given by
$f_{0}=x_{0}-x_{0}^{-1}(q_{1}+¥cdots+q_{n})$
(3.15)
$f_{1}=x_{0}p_{1}$
,
$f_{2}=x_{0}^{-1}q_{1}$
$f_{2n-1}=x_{0}(p_{n}-p_{n-1})$
,
,
,
$f_{3}=x_{0}(p_{2}-p_{1})$
$f_{2n}=x_{0}^{-1}q_{n}$
,
,
$f_{4}=x_{0}^{-1}q_{2}$
$f_{2n+1}=x_{1}-x_{0}p_{n}$ .
Since
$¥{p_{i}, q_{j}¥}=¥delta_{i,j}$
(3.16)
,
$¥{q_{i}, q_{j}¥}=¥{p_{i}, p_{j}¥}=0$
,
$¥{p_{i},x_{0}¥}=¥{q_{i},x_{0}¥}=¥{p_{i}, x_{1}¥}=¥{q_{i},x_{1}¥}=¥{x_{0},x_{1}¥}=0$
for
$i,j$
(3.17)
$=1$
,
$¥ldots,n$
, we have the same formula as (3.5) and
$¥varphi^{¥prime}=¥{h_{0}, ¥varphi¥}+¥frac{k}{2}(x_{0}¥frac{¥partial¥varphi}{¥partial x_{0}}+x_{1}¥frac{¥partial¥varphi}{¥partial x_{1}})$
,?’
for any
$¥varphi¥in C(¥alpha; f)$
494
Masatoshi NOUMI and Yasuhiko YAMADA
by (2.18). Let us denote by $H=H(q;p;x)¥in C(¥alpha)[q;p;x^{¥pm 1}]$ the polynomial in
$(q;p)$ which represents our Hamiltonian
in the coordinates $(q;p;x)$ . Then
we see that the differential system (0.2) is equivalent to the Hamiltonian system
$h_{0}$
$¥frac{dq_{i}}{dt}=¥{H, q_{i}¥}=¥frac{¥partial H}{¥partial p_{i}}$
,
$¥frac{dp_{i}}{dt}=¥{H,p_{i}¥}=-¥frac{¥partial H}{¥partial q_{i}}$
,
(3.18)
$¥frac{dx_{0}}{dt}=¥frac{k}{2}x_{0}$
where
$i=1$ ,
(3.18)
_’
$n$
,
$¥frac{dx_{1}}{dt}=¥frac{k}{2}x_{1}$
The Hamiltonian
.
,
$H$
is given in the form
$H=(x_{0}^{2}-¥sum_{i=1}^{n}q_{i})(¥sum_{i=1}^{n}q_{i}p_{i}(¥frac{x_{1}}{x_{0}}-p_{i}))$
?
$¥sum_{1¥leq i<j¥leq n}q_{i}q_{j}(p_{i}-p_{j})(¥frac{x_{1}}{x_{0}}+p_{i}-p_{j})+2¥gamma¥sum_{i=1}^{n}q_{i}p_{i}$
$-¥frac{x_{1}}{x_{0}}¥sum_{i=1}^{n}¥beta_{i}q_{i}+¥sum_{i=1}^{n}¥alpha_{2i}p_{i}+(¥gamma-¥varpi_{2n+1})x_{0}x_{1}+¥gamma^{2}$
,
where
(3.20)
$¥beta_{i}=¥sum_{r=1}^{i}¥alpha_{2r1}¥_(i=1, -’ n)$
,
$¥gamma=¥sum_{i=1}^{2n+1}(-1)^{i-1}¥varpi_{i}=¥frac{1}{2}¥sum_{r=0}^{n}¥alpha_{2r+1}$
.
Summarizing the results of this section, we have
Theorem 3.1.
symmetry
(3.20)
of
type
(1) The differential system (0.1) with affine Weyl group
$A_{2n}^{(1)}$
is equivalent to the Hamiftonian system
$¥frac{dq_{i}}{dt}=¥frac{¥partial H}{¥partial p_{i}}$
,
$¥frac{dp_{i}}{dt}=-¥frac{¥partial H}{¥partial q_{i}}$
$(i =1, ¥ldots,n)$
,
with an auxiliary variable such that $dx/dt$ $=k$ , where the $H=H(q;p;x)$ is the
polynomial (3.8).
is equivalent to the
(2) The differential system (0.2) of type
Hamiltonian system
$x$
$A_{2n+1}^{(1)}$
(3.20)
$¥frac{dq_{i}}{dt}=¥frac{¥partial H}{¥partial p_{i}}$
,
$¥frac{dp_{i}}{dt}=-¥frac{¥partial H}{¥partial q_{i}}$
$(i =1, ¥ldots,n)$
,
Higher Order Painleve Equations
of
Type
495
$A_{l}^{(1)}$
such that $dx_{0}/dt=kx_{0}/2$ ,
with two auxiliary variables
where $H=H(q;p;x)$ is the polynomial (3.19).
$dx_{1}/dt=kx_{1}/2$
$x_{0},x_{1}$
See Appendix for explicit formulas of the Hamiltonians
$H$
,
for small /.
Remark 3.2. When we regard our systems as Hamiltonian systems, it
,
would be more convenient to use constant parameters as $x=kt+c$ for
$x_{1}=c_{1}e^{kt/2}$ for
$x_{0}=c_{0}e^{kt/2}$
, rather than the auxiliary variables as in
,
and
Theorem 3. 1.
$A_{2n}^{(1)}$
$A_{2n+1}^{(1)}$
4.
$¥tau$
-Functions
and
In the following, we introduce a family of Hamiltonians ,
of
Section
2, we
the
Hamiltonian
From
system.
for
our
,
functions ,
define the other Hamiltonians , _’ by the diagram rotation:
$h_{0}$
$¥tau_{0}$
$¥ldots$
$¥ldots,h_{l}$
$¥tau-$
$h_{0}$
$¥tau_{l}$
$h_{1}$
(4. 1)
$h_{l}$
$(i =1, ¥ldots, l)$
$h_{i}=¥pi(h_{i-1})$
.
is invariant under
of highest degree in ,
Note that the component of
degree comhighest
a
have
common
Hence
,
the diagram rotation.
_’
ponent, and have different coefficients in lower degrees. We also introduce the
-functions , _’ for our system to be the dependent variables such that
by the linear
,
, namely as functions determined from
differential equations
$h_{0}$
$f_{0}$
$h_{0}$
$¥tau$
$¥tau_{0}$
$¥ldots,f_{l}$
$h_{l}$
$¥tau_{l}$
$h_{j}=k(¥log¥tau_{j})^{¥prime}$
$f_{0}$
(4.2)
$k¥tau_{j}^{¥prime}=h_{j}¥tau_{j}$
$¥ldots,f_{l}$
$(j =0,1, --’ l)$ .
In this section, we show that the affine Weyl group symmetry lifts to the level of
$¥tau$
-functions as well.
We first remark that our Hamiltonians have some remarkable properties in
relation to the action of $W$.
Proposition 4.1. With respect to the action
Hamiltonians have the following invariance:
$A_{2n}^{(1)}$
:
$s_{i}(h_{j})=h_{j}+¥delta_{i,j}k¥frac{¥alpha_{j}}{f_{j}}$
$A_{2n+1}^{(1)}$
:
$s_{i}(h_{j})=h_{j}+¥delta_{i,j}k¥frac{¥alpha_{j}}{f_{j}}g_{j}$
of
the
Weyl group, the
affine
$(i,j =0, ¥_ , f)$
,
(4.3)
where
$g_{j}$
stands
for
$g_{0}$
or
$g_{1}$
according as
$j¥equiv 0$
$(i,j =0, ¥ldots, ¥mathit{1})$
or 1
$¥mathrm{m}¥mathrm{o}¥mathrm{d}2$
We have only to show the case where $j=0$ .
we compute
as follows:
Proof.
$¥Delta_{i}(h_{0})$
,
.
In the case of
$A_{2n}^{(1)}$
,
496
Masatoshi NOUMI and Yasuhiko YAMADA
$¥Delta_{i}(f_{i-1}f_{i}f_{i+1})+¥Delta_{i}(f_{i-1}f_{i})¥sum_{K¥in¥Psi_{1}([i+3,i-2])}f_{K}+¥Delta_{i}(f_{i}f_{i+1})¥sum_{K¥in ¥mathit{9}_{1}^{7}([i+2,i-3])}f_{K}$
$+¥Delta_{i}(f_{i-1})¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{1}([i+2,i-2])}f_{K}+¥Delta_{i}(f_{i+1})¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{1}([i+2,i-2])}f_{K}$
$+¥sum_{j=0}^{2n}¥Delta_{i}(¥chi(¥Gamma¥backslash ¥{j¥}))f_{j}+s_{i}(¥chi(¥Gamma¥backslash ¥{i-1¥}))¥Delta_{i}(f_{i-1})+s_{i}(¥chi(¥Gamma¥backslash ¥{i+1¥}))¥Delta_{i}(f_{+1})$
$=(f_{i-1}-f_{i+1}-¥frac{¥alpha_{i}}{f_{i}})-¥sum_{K¥in¥Psi_{1}([i+3,i-2])}f_{K}+¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{1}([i+2,i-3])}f_{K}$
$-¥frac{1}{f_{i}}(¥sum_{K¥in¥Psi_{1}([i+2,i-2])}f_{K})+¥frac{1}{f_{i}}(¥sum_{K¥in¥Psi_{1}([i+2,i-2])}f_{K})$
$+¥sum_{K¥in¥Psi_{1}([i+1,i-2])}f_{K}-¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{1}([i+2,i-1])}f_{K}$
?
$(¥chi(¥Gamma¥backslash ¥{i-1¥})-¥alpha_{i})¥frac{1}{f_{i}}+(¥chi(¥Gamma¥backslash ¥{i+1¥})+¥alpha_{i})¥frac{1}{f_{i}}$
$=(¥alpha_{i}+¥varpi_{i-1}-2¥varpi_{i}+¥varpi_{i+1})¥frac{1}{f_{i}}=¥delta_{i,0}¥frac{k}{f_{i}}$
which proves (4.3) for
$A_{2n}^{(1)}$
.
The case of
,
$A_{2n+1}^{(1)}$
can be verified similarly.
We need also to know how adjacent pairs of the Hamiltonians
related. We propose some lemmas for this purpose.
Lemma 4.2.
For each
$i=0,1$ ,
$¥ldots$
$h_{j}$
$¥square $
are
, 1, one has
$¥pi(¥chi([i-1, i]))-¥chi([i, i+1])=¥left¥{¥begin{array}{l}-¥frac{1}{f+1}k¥¥¥frac{f}{f+1}k¥end{array}¥right.$
$((i=0)i¥neq 0),$
.
(4.4)
$¥pi^{-1}(¥chi([i+1, i+2]))-¥chi([i, i+1])=¥left¥{¥begin{array}{l}¥frac{1}{l+1}k¥¥-¥frac{l}{l+1}k¥end{array}¥right.$
$((i¥neq l)i=f)’$
.
Let $L$ be a subset of $¥Gamma=¥{0,1, ¥ldots, f¥}$ with $|L|=2m$ and suppose that $L$ is
a disjoint union of chains of even nodes. We say that $L$ contains $[/, i+1]$
evenly, if $[i, i+1]¥subset L$ and the complement
splits into chains of even
nodes. With this terminology, we have
$L¥backslash [i, i+1]$
of
Higher Order Painleve Equations
Lemma 4.3.
(4.5)
Under the assumption on
$L$
$L$
(4.6)
contains [0,
$¥pi^{-1}$
according as
$L$
1] evenly, or
not.
or
$[l, 0]$
$-¥frac{m}{l+1}k$
,
Similarly,
$(¥chi(¥pi L))-¥chi(L)=-¥frac{l-m+1}{l+1}k$
contains
497
$A_{l}^{(1)}$
above, one has
$¥pi(¥chi(¥pi^{-1}L))-¥chi(L)=¥frac{l-m+1}{l+1}k$
according as
Type
or
$¥frac{m}{l+1}k$
,
evenly, or not.
We omit the proof of Lemmas 4.2 and 4.3 since they can be proved by
direct calculations from the definitions.
Proposition 4.4.
{
$¥mathrm{I})$
In the case
has
(4.7)
of
$¥mathrm{A}_{2n}^{(1)}(f =2n)$
, for each $j=0$ ,
$h_{j+1}-h_{j}=k¥sum_{r=1}^{n}f_{j+2r}-¥frac{nk}{2n+1}x$
where
$x=¥sum_{i=0}^{2n}f_{i}$
.
$¥mathit{2}n$
_’
,
Hence
$-h_{j-1}+2h_{j}-h_{j+1}=k¥sum_{r=1}^{n}(f_{j+2r-1}-f_{j+2r})$
,
(4.8)
$h_{j-1}-h_{j+1}=k(f_{j}-¥frac{1}{2n+1}x)$
.
(2) In the case of $A_{2n+1}^{(1)}(l =2n+1)$ , one has
(4.9)
$h_{j+1}-h_{j}=k¥sum_{1¥leq r¥leq s¥leq n}f_{j+2r}f_{j+2s+1}$
$-¥frac{nk}{2n+2}¥sum_{K¥in¥Psi_{2}}f_{K}+(-1)^{j}¥frac{k}{4}¥sum_{i=0}^{2n+1}(-1)^{i}¥alpha_{i}$
for
each $j=0$ ,
_’
$2n+1$ .
,
Hence
$-h_{j-1}+2h_{j}-h_{j+1}=k¥sum_{1¥leq r¥leq s¥leq n}(f_{j+2r-1}f_{j+2s}-f_{j+2r}f_{j+2s+1})$
(4. 10)
$+k(¥frac{k}{2}-¥sum_{r=0}^{n}¥alpha_{j+2r})$
,
$h_{j-1}-h_{j+1}=kx_{j+1}(f_{j}-¥frac{1}{n+1}x_{j})$
,
, one
498
Masatoshi NOUMI and Yasuhiko YAMADA
where
$i¥equiv 0$
$x_{0}=¥sum_{r=0}^{2n}f_{2r}$
or 1
Proof.
$¥mathrm{m}¥mathrm{o}¥mathrm{d}2$
and $x_{1}=¥sum_{r=0}^{2n}f_{2r+1}$ , and
$x_{i}$
stands for
$x_{0}$
or
$x_{1}$
according as
.
When
$¥mathit{1}=2n$
, we have
$h_{0}=¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{3}}f_{K}+¥sum_{i=0}^{2n}¥chi([i+1, i-1])f_{i}$
,
(4. 11)
$h_{1}=¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{3}}f_{K}+¥sum_{i=0}^{2n}¥pi(¥chi([i, i-2]))f_{i}$
,
hence
(4. 12)
$h_{1}-h_{0}=¥sum_{i=0}^{2n}(¥pi(¥chi([i, i-2]))-¥chi([i+1, i-1]))f_{i}$ .
By Lemma 4.3, we compute
(4. 13)
$h_{1}-h_{0}=-¥frac{nk}{2n+1}(f_{0}+¥sum_{r=1}^{n}f_{2r-1})+¥frac{(n+1)k}{2n+1}¥sum_{r=1}^{n}f_{2r}$
$=k¥sum_{r=1}^{n}f_{2r}-¥frac{nk}{2n+1}¥sum_{i=0}^{2n}f_{i}$
,
which gives (4.7) for $j=0$ . Formulas (4.7) for $j=1$ , _’
are obtained by
applying the diagram rotation . Formulas (4.8) follow directly from (4.7).
When $=2n+1$ , we have
$¥mathit{2}n$
$¥pi$
$l$
$h_{0}=¥sum_{K¥in¥Psi_{4}}f_{K}+¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{2}}¥chi(K^{c}))f_{K}+¥gamma_{1}^{2}$
,
(4. 14)
$h_{1}=¥sum_{K¥in¥Psi_{4}}f_{K}+¥sum_{K¥in¥ovalbox{¥tt¥small REJECT}_{2}}¥pi(¥chi(¥pi^{-1}K^{c}))f_{K}+¥gamma_{0}^{2}$
where
$¥gamma_{1}=(¥sum_{r=0}^{n}¥alpha_{2r+1})/2$
and
$¥gamma_{0}=¥pi(¥gamma_{1})=$
$(¥sum_{r=0}^{n}¥alpha_{2r})/2$
.
,
Hence we have
$h_{1}-h_{0}=¥sum_{K¥in¥Psi_{2}}(¥pi(¥chi(¥pi^{-1}K^{c}))-¥chi(K^{c}))f_{K}+¥frac{k}{2}(¥gamma_{0}-¥gamma_{1})$
.
¥
since ¥
. The coefficients of
are computed by Lemma 4.3 to
obtain (4.9) for $j=0$ . Formulas (4.9) for the other are obtained by the
diagram rotation, and formulas (4.10) follow directly from (4.9).
$ gamma_{0}+ gamma_{1}=k/2$
$f_{K}$
$j$
$¥square $
Higher Order Painleve Equations
of
Type
499
$A_{l}^{(1)}$
By combining Propositions 4.1 and 4.4, we obtain
Proposition 4.5. The differential system (0.1) (resp. (0.2)) is expressed as
:
follows in terms of Hamiltonians ,
$h_{0}$
(4. 15)
_’
$h_{l}$
$k¥frac{f_{j}^{¥prime}}{f_{j}}=s_{j}(h_{j})+h_{j}-h_{j1}¥_-h_{j+1}$
$(j =0, ¥_’ l)$ ,
or equivalently,
(4. 16)
where
$k¥frac{f_{j}^{¥prime}}{f_{j}}=s_{j}(h_{j})-h_{j}+¥sum_{i=0}^{l}h_{i}a_{ij}$
$A$
$=(a_{ij})_{0¥leq i,j¥leq l}$
is the generalized Carton matrix
of (1.5).
Formulas (4. 15) can be verified by rewriting the right-hand side as the sum
, and then by applying Proposition 4.1 and
$(s_{j}(h_{j})-h_{j})+(-h_{j+1}¥_+2h_{j}-h_{j+1})$
4.4.
¥
the field of rational functions in
In what follows, we denote by ¥
$C(
¥
alpha;
f)$
. We define the structure of differential
with coefficients in
,
¥
¥
and
extend the automorphisms ,
now
by
field of
(4.2). We
of $C(¥alpha; f)$ to $C(¥alpha,f;¥tau)$ by setting
$C( alpha; f; tau)$
$¥tau_{0}$
_’
$¥tau_{l}$
$C( alpha; f; tau)$
(4.17)
where
$s_{0}$
$s_{i}(¥tau_{j})=¥tau_{j}(i¥neq j)$
$i,j$
$=0$ ,
$¥ldots$
,
$f$
,
$s_{j}(¥tau_{j})=¥frac{¥tau_{j-1}¥tau_{j+1}}{¥tau_{j}}f_{j}$
,
$¥pi(¥tau_{j})=¥tau_{j+1}$
.
and
Theorem 4.6 ([8]). The automorphisms ,
of ¥ ¥
scribed above define a representation of the extended affine Weyl group
$s_{0}$
_’
$¥pi$
$s_{l}$
Proposition 4.7.
-functions
(4. 18)
The variables
$f_{j}$
$(j =0,1, ¥_’ l)$
as follows:
$f_{j}$
$¥tilde{W}$
.
by
$¥tau$
are expressed in terms of
$f_{j}=¥frac{1}{k}(h_{j1}¥_-h_{j+1})+¥frac{x}{2n+1}$
$=¥frac{¥tau_{j-1}^{¥prime}}{¥tau_{j-1}}-¥frac{¥tau_{j+1}^{¥prime}}{¥tau_{j+1}}+¥frac{x}{2n+1}$
de-
$C( alpha; f; tau)$
Note that, by Proposition 4.4, one obtains an expression of
functions.
$¥tau$
$¥pi$
$¥ldots,s_{l}$
$(j =0, ¥ldots, ¥mathit{2}n)$
500
Masatoshi NOUMI and Yasuhiko YAMADA
when $f=2n$ , and
(4. 19)
$f_{j}=¥frac{1}{kx_{j+1}}(h_{j-1}-h_{j+1})+¥frac{x_{j}}{n+1}$
$=¥frac{1}{x_{j+1}}(¥frac{¥tau_{j-1}^{¥prime}}{¥tau_{j-1}}-¥frac{¥tau_{j+1}^{¥prime}}{¥tau_{j+1}})+¥frac{x_{j}}{n+1}$
when
$l$
$(j =0, --’ 2n+1)$
$=2n+1$ .
Hence we have
Proposition 4.8. For each $j=0$ ,
1, the action of on the
is given by the following bilinear operators of Hirota type:
_’
(4.20)
$s_{j}$
$z$
-functions
$¥tau_{j}$
$s_{j}(¥tau_{j})=¥frac{1}{¥tau_{j}}(D_{t}+¥frac{x}{2n+1})¥tau_{j-1}¥cdot¥tau_{j+1}$
$=¥frac{1}{¥tau_{j}}(¥tau_{j-1}^{¥prime}¥tau_{j+1}-¥tau_{j-1}¥tau_{j+1}^{¥prime}+¥frac{x}{2n+1}¥tau_{j-1}¥tau_{j+1})$
when
$¥mathit{1}=2n$
, and
(4.21)
$s_{j}(¥tau_{j})=¥frac{1}{¥tau_{j}}(¥frac{1}{x_{j+1}}D_{t}+¥frac{x_{j}}{n+1})¥tau_{j-1}¥cdot¥tau_{j+1}$
$=¥frac{1}{¥tau_{j}}(¥frac{1}{x_{j+1}}(¥tau_{j-1}^{¥prime}¥tau_{j+1}-¥tau_{j-1}¥tau_{j+1}^{¥prime})+¥frac{x_{j}}{n+1}¥tau_{j-1}¥tau_{j+1})$
when
as ¥
$¥mathit{1}=2n+1$
$j equiv 0$
or 1
, where
$¥mathrm{m}¥mathrm{o}¥mathrm{d}2$
$x_{j}$
stands for
$x_{0}=¥sum_{r=0}^{n}f_{2r}$
or
$x_{1}=¥sum_{r=0}^{n}f_{2r+1}$
according
.
From Proposition 4.5, we obtain
Theorem 4.9. The action of the extended
with the derivation of the differential fiefd ¥
affine
$C( alpha; f;¥tau)$
Proof. Since
case when
(4.22)
$i=j$ .
Weyl group
$s_{i}(¥tau_{j})^{¥prime}=s_{i}(¥tau_{j}^{¥prime})$
$¥underline{s_{j}(¥tau_{j})^{¥prime}}=+-+¥underline{¥tau_{j-1}^{¥prime}}¥underline{¥tau_{j+1}^{¥prime}}¥underline{¥tau_{j}^{¥prime}}¥underline{f_{j}^{¥prime}}$
$¥tau_{j-1}$
$¥tau_{j+1}$
$¥tau_{j}$
commutes
.
the equality
is obvious if
By Proposition 4.5, we compute
$s_{j}(¥tau_{j})$
$¥tilde{W}$
$f_{j}$
$=¥frac{1}{k}(h_{j-1}+h_{j+1}-h_{j}+k¥frac{f_{j}^{¥prime}}{f_{j}})$
$=¥frac{1}{k}s_{j}(h_{j})=s_{j}(¥frac{¥tau_{j}^{¥prime}}{¥tau_{j}})=¥frac{s_{j}(¥tau_{j}^{¥prime})}{s_{j}(¥tau_{j})}$
,
$i¥neq j$
, we consider the
Higher Order Painleve Equations
which implies
commutes with ’.
$s_{j}(¥tau_{j})^{¥prime}=s_{j}(¥tau_{j}^{¥prime})$
of
Type
501
$A_{l}^{(1)}$
It is clear that the action of the diagram rotation
.
$¥square $
$¥pi$
Theorem 4.9 means that one can lift the Backlund transformations of our
are
to the level of -functions so that each
system for the variables
invariant with respect to the subgroup $ W_{j}=¥langle s0, ¥_’ s_{j1}¥_’ s_{j1,-}¥_’ s_{l}¥rangle$ of $W$
$(j =0, ¥ldots, f)$ , and that one has the multiplicative formulas
$¥mathrm{r}$
$f_{j}$
(4.23)
$¥tau_{j}$
$(j =0, ¥ldots, l)$
$f_{j}=¥frac{¥tau_{j}s_{j}(¥tau_{j})}{¥tau_{j-1}¥tau_{j+1}}$
in terms of -functions. We have thus
for the dependent variables , _’
Backlund
transformations of our differential
of
the
structure
guaranteed that
system is consistent with the general scheme of our previous paper [8]. As a
consequence we see that the Backlund transformations of our system provides
in the sense of [8].
the discrete dynamical systems of type
$f_{0}$
$f_{l}$
$¥mathrm{r}$
$A_{l}^{(1)}$
Appendix
A. 1.
Explicit
formulas of
$f_{0}^{¥prime}$
for small
$l$
.
,
$A_{2}^{(1)}$
:
$f_{0}^{¥prime}=f_{0}(f_{1}-f_{2})+¥alpha_{0}$
$A_{4}^{(1)}$
:
$f_{0}^{¥prime}=f_{0}(f_{1}-f_{2}+f_{3}-f_{4})+¥alpha_{0}$
$A_{3}^{(1)}$
:
$f_{0}^{¥prime}=f_{0}(f_{1}f_{2}-f_{2}f_{3})+(¥frac{k}{2}-¥alpha_{2})f_{0}+¥alpha_{0}f_{2}$
$A_{5}^{(1)}$
:
$f_{0}^{¥prime}=f_{0}(f_{1}f_{2}+f_{1}f_{4}+f_{3}f_{4}-f_{2}f_{3}-f_{2}f_{5}-f_{4}f_{5})$
,
$+(¥frac{k}{2}-¥alpha_{2}-¥alpha_{4})f_{0}+¥alpha_{0}(f_{2}+f_{4})$
A.2.
Explicit
formulas of
$h_{0}$
for
small
$f$
,
.
.
$A_{2}^{(1)}$
:
$h_{0}=f_{0}f_{1}f_{2}+¥frac{1}{3}(¥alpha_{1}-¥alpha_{2})f_{0}+¥frac{1}{3}(¥alpha_{1}+2¥alpha_{2})f_{1}-¥frac{1}{3}(2¥alpha_{1}+¥alpha_{2})f_{2}$
$A_{4}^{(1)}$
:
$h_{0}=f_{0}f_{1}f_{2}+f_{1}f_{2}f_{3}+f_{2}f_{3}f_{4}+f_{3}f_{4}f_{0}+f_{4}f_{0}f_{1}$
$+¥frac{1}{5}(2¥alpha_{1}-¥alpha_{2}+¥alpha_{3}-2¥alpha_{4})f_{0}+¥frac{1}{5}(2¥alpha_{1}+4¥alpha_{2}+¥alpha_{3}+3¥alpha_{4})f_{1}$
$-¥frac{1}{5}(3¥alpha_{1}+¥alpha_{2}-¥alpha_{3}+2¥alpha_{4})f_{2}+¥frac{1}{5}(2¥alpha_{1}-¥alpha_{2}+¥alpha_{3}+3¥alpha_{4})f_{3}$
$-¥frac{1}{5}(3¥alpha_{1}+¥alpha_{2}+4¥alpha_{3}+2¥alpha_{4})f_{4}$
502
Masatoshi NOUMI and Yasuhiko YAMADA
:
$A_{3}^{(1)}$
$h_{0}=f_{0}f_{1}f_{2}f_{3}+¥frac{1}{4}(¥alpha_{1}+2¥alpha_{2}-¥alpha_{3})f_{0}f_{1}+¥frac{1}{4}(¥alpha_{1}+2¥alpha_{2}+3¥alpha_{3})f_{1}f_{2}$
$-¥frac{1}{4}(3¥alpha_{1}+2¥alpha_{2}+¥alpha_{3})f_{2}f_{3}+¥frac{1}{4}(¥alpha_{1}-2¥alpha_{2}-¥alpha_{3})f_{3}f_{0}+¥frac{1}{4}(¥alpha_{1}+¥alpha_{3})^{2}$
:
$A_{5}^{(1)}$
$h_{0}=f_{0}f_{1}f_{2}f_{3}+f_{1}f_{2}f_{3}f_{4}+f_{2}f_{3}f_{4}f_{5}+f_{3}f_{4}f_{5}f_{0}+f_{4}f_{5}f_{0}f_{1}+f_{5}f_{0}f_{1}f_{2}$
$+¥frac{1}{3}(¥alpha_{1}+2¥alpha_{2}+¥alpha_{4}-¥alpha_{5})f_{0}f_{1}+¥frac{1}{3}(¥alpha_{1}+2¥alpha_{2}+3¥alpha_{3}+¥alpha_{4}+2¥alpha_{5})f_{1}f_{2}$
$-¥frac{1}{3}(2¥alpha_{1}+¥alpha_{2}-¥alpha_{4}+¥alpha_{5})f_{2}f_{3}+¥frac{1}{3}(¥alpha_{1}-¥alpha_{2}+¥alpha_{4}+2¥alpha_{5})f_{3}f_{4}$
$-¥frac{1}{3}(2¥alpha_{1}+¥alpha_{2}+3¥alpha_{3}+2¥alpha_{4}+¥alpha_{5})f_{4}f_{5}+¥frac{1}{3}(¥alpha_{1}-¥alpha_{2}-2¥alpha_{4}-¥alpha_{5})f_{5}f_{0}$
$+¥frac{1}{3}(¥alpha_{1}-¥alpha_{2}+¥alpha_{4}-¥alpha_{5})f_{0}f_{3}+¥frac{1}{3}(¥alpha_{1}+2¥alpha_{2}+¥alpha_{4}+2¥alpha_{5})f_{1}f_{4}$
$-¥frac{1}{3}(2¥alpha_{1}+¥alpha_{2}+2¥alpha_{4}+¥alpha_{5})f_{2}f_{5}+¥frac{1}{4}(¥alpha_{1}+¥alpha_{3}+-¥alpha_{5})^{2}$
A.3.
Explicit
formulas of $H$ for
small .
$l$
$A_{2}^{(1)}$
:
$H=(x-q_{1})q_{1}p_{1}-q_{1}p_{1}^{2}-¥alpha_{1}q_{1}+¥alpha_{2}p_{1}+¥frac{1}{3}(¥alpha_{1}-¥alpha_{2})x$
$A_{4}^{(1)}$
:
$H=(x-q_{1}-q_{2})(q_{1}p_{1}+q_{2}p_{2})-q_{1}p_{1}^{2}-q_{2}p_{2}^{2}-q_{1}(p_{1}-p_{2})q_{2}$
$-¥alpha_{1}q_{1}-(¥alpha_{1}+¥alpha_{3})q_{2}+¥alpha_{2}p_{1}+¥alpha_{4}p_{2}+¥frac{1}{5}(2¥alpha_{1}-¥alpha_{2}+¥alpha_{3}-2¥alpha_{4})x$
$A_{3}^{(1)}$
:
$H=(x_{0}^{2}-q_{1})p_{1}q_{1}(¥frac{x_{1}}{x_{0}}-p1)+(¥alpha_{1}+¥alpha_{3})q_{1}p_{1}-¥alpha_{1}¥frac{x_{1}}{x_{0}}q_{1}+¥alpha_{2}x_{0}^{2}p_{1}$
$+¥frac{1}{4}$
$A_{5}^{(1)}$
:
$(¥alpha_{1}-2¥alpha_{2}-¥alpha_{3})x_{0}x_{1}+¥frac{1}{4}$ $(¥alpha_{1}+¥alpha_{3})^{2}$
$H=(x_{0}^{2}-q_{1}-q_{2})(q_{1}p_{1}(¥frac{x_{1}}{x_{0}}-p_{1})+q_{2}p_{2}(¥frac{x_{1}}{x_{0}}-p_{2}))$
?
$q_{1}q_{2}(p_{1}-p_{2})(¥frac{x_{1}}{x_{0}}+p_{1}-p_{2})+(¥alpha_{1}+¥alpha_{3}+¥alpha_{5})(q_{1}p_{1}+q_{2}p_{2})$
$-¥frac{x_{1}}{x_{0}}(¥alpha 1q1+(¥alpha]+¥alpha 3)q2)+x^{2}¥mathrm{o}(¥alpha 2p1+¥alpha 4p2)$
$+¥frac{1}{3}(¥alpha_{1}-¥alpha_{2}-2¥alpha_{4}-¥alpha_{5})x_{0}x_{1}+¥frac{1}{4}(¥alpha_{1}+¥alpha_{3}+¥alpha_{5})^{2}$
References
Groupes et Algebres de Lie, Chapitres 4, 5 et 6, Elements de Mathematique,
Masson, Paris, 1981.
Kac, V. G., Infinite dimensional Lie algebras, Third edition, Cambridge University Press,
1990.
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[2]
Higher Order Painleve Equations
of
Type
$A_{l}^{(1)}$
503
[3] Macdonald, I. G., Affine root systems and Dedekind’s -function, Inv. Math, 15 (1972),
91-143.
Noumi,
M., Okada, S., Okamoto, K. and Umemura, H., Special polynomials associated with
[4]
the Painleve equations II, to appear in the Proceedings of the Taniguchi Symposium
$¥eta$
“Integrable Systems and Algebraic Geometry”, RIMS, Kyoto University, Japan, 1997.
[5] Noumi, M. and Okamoto, K., Irreducibility of the second and the fourth Painleve equations, Funkcial. Ekvac., 40 (1997), 139-163.
[6] Noumi, M. and Yamada, Y., Symmetries in the fourth Painleve equation and Okamoto
polynomials, to appear in Nagoya Math. J. (q-alg/9708018).
[7] Noumi, M. and Yamada, Y., Umemura polynomials for Painleve V equation, to appear in
Phys. Lett. A.
[8] Noumi, M. and Yamada, Y., Affine Weyl groups, discrete dynamical systems and Painleve
equations, to appear in Comm. Math. Phys. (math.QA/9804132).
[9] Noumi, M. and Yamada, Y., Symmetric forms of the Painleve equations, in preparation.
[10] Okamoto, K., Studies of the Painleve equations, I. Ann. Math. Pura Appl., 146 (1987),
337-381; II. Jap. J. Math., 13 (1987), 47-76; III. Math. Ann., 275 (1986), 221-255; IV.
Funkcial. Ekvac., 30 (1987), 305-332.
[11] Umemura, H., On the irreducibility of the first differential equation of Painleve, in
“Algebraic Geometry and Commutative Algebra in honor of Masayoshi Nagata”, pp. 101119, Kinokuniya-North-Holland, 1987.
[12] Umemura, H., Special polynomials associated with the Painleve equations I, to appear in
the Proceedings of the Workshop on “Painleve Transcendents”, CRM, Montreal, Canada,
1996.
nuna adreso:
Masatoshi Noumi
Department of Mathematics
Graduate School of Science and Technology
Kobe University
Rokko, Kobe 657-8501
Japan
E-mail: noumicmath.kobe-u.ac.jp
Yasuhiko Yamada
Department of Mathematics
Graduate School of Science and Technology
Kobe University
Rokko, Kobe 657-8501
Japan
E-mail: yamadaycmath.kobe-u.ac.jp
(Ricevita la 5-an de junio, 1998)