On Matrix Painleve Systems
Yoshihiro Murata
Nagasaki University
20 September 2006 Isaac
Newton Institute
2
1. Introduction
2. Dimensional Reductions of ASDYM eqns
and Matrix Painleve Systems (Reconstruction of the result of Mason & Woodhouse)
3. Degenerations of Painleve Equations and
Classification of Painleve Equations
4. Generalized Confluent Hypergeometric Systems included in MPS (joint work with Woodhouse)
Contents
3
1. Introduction
Basic Motivation Can we have new good expressions of Painleve eqns to achieve further developments? (Background) We often use various expressions to find features of Painleve: e.g. Painleve systems (Hamiltonian systems) 3-systems of order 1 (Noumi-Yamada system)
Answer:
New possively good expressions exist.
4
common framework
Overview 1
1993,1996 1980’ ~ 1990’ Mason&Woodhouse Gelfand et al, H.Kimura et al ASDYM eqs Theory of GCHS (Generalized ConfluentSymmetric Hypergeometric Systems) reduction
reduced eqs (Matrix ODEs)
Painleve eqns
Matrix Painleve Systems
Detailed investigation
New reduction
relationship
concepts
Jordan groupPainleve group=
Grassmann Var
5
Overview2 (Common framework)
MPS
GCHS
ASDYM
6
Overview3
Young diagram ASDYang-Mills eq + Constraints = symmetry & region Painleve eqns degenerated eqns
× 3 type constant matrix ⇒ 15 type MPS
Matrix Painleve Systems
7
2. Dimmensional Reductions of ASDYM eqns and Matrix Painleve Systems
2.1 PreliminariesPainleve III’ Third Painleve eqn has two expressions:
These are transformed by PIII’ is often better than PIII
xyqxt ,2
8
Young diagrams and Jordan groups (Basic concepts in the theory of GCHS)
We express a Young diagram by the symbol λIf λ consists l rows and boxes, we
write as
e.g.
For , we define Jordan group Hλ as follows:
)λ...λ(λ,...,λ 1l01l0 )λ,...,(λλ 1l0
)11|λ(|(6,3,2)λ
n|λ| where)λ,...,(λλ 1l0
1l0 λ...λ|λ|
9
e.g.
If , Jordan group H(2,1,1) is a group of all matrices of the form:
(2,1,1)λ
0
1
10
1210
00
0
h
h
hh
hhhh
h
k
kλJ
10
So, on the case , we have Jordan groups Hλ as :4|λ|
11
Subdiagrams and Generic stratum of M(r,n) (Basic concepts in the theory of GCHS)
e.g. If , then subdiagrams μof weight 2 are
(2,0,0), (1,1,0), (1,0,1), (0,1,1)
(2,1,1)λ
12
e.g. On the case M(2,4), λ = (2,1,1), μ= (1,0,1)
e.g.
13
03
12
02
11
01
10
00
z
z
z
z
z
z
z
zZ
13
03
10
00
z
z
z
zZ
(0,1,1)(1,0,1),(1,1,0),(2,0,0),for 0det|)4,2()1,1,2( ZMZZ
13
2.2 New Reduction ProcessWe consider the ASDYM eq defined on a Grassmann variety.Reduction Process(1) Take where Let
then
(2) Take a metric on Let sl(2,C) gauge potential satisfies ASD condition
)4,2(\)2()4,2( 0MGLGr }2|)4,2({)4,2(0 rankZMZM
14
ASDYM eqn
(3) We consider projective Jordan group
whereλare Young diagrams of weight 4. PHλacts
(4) Restrict ASDYM onto PHλinvarinat regions
))4( ofcenter :(/HPH λλ GLZZ
)4,2(Gr
15
(5) Let . Then
gives 3-dimensional fibration
16
(6) Change of variables. There exists a mapping
St : orbit of PHλ
t
17
(7) PHλ–invariant ASDYM eqn Lλ
(8) Calculations of three first integrals
18
2.3 Matrix Painleve Systems
We call the combined system (Lλ + 1st Int) Matrix Painleve Sytem Mλ
19
We obtained 5 types of Matrix Painleve Systems: By gage transformations, constant matrix P is classified into 3 case
s:
Thoerem1 By the reduction process (1),…,(8), we can obtain Matrix Painleve Systems Mλfrom ASDYM eqns. Mλare classified into 15cases by Young diagram λand constant matrix P.
A B C
(4)(2,2)(3,1)(2,1,1)(1,1,1,1) M,M,M,M,M
20
3.1 Degenerations and Classification of Painleve eqns
e.g. PVI
PV
PIV PIII’
PII
PI
3. Degenerations of Painleve Equations
and Classification of Painleve Equations
0ε and
,εδby δ
,εγεδ-by γ
,εt1by t
,Pin Replace
: PP
21
11
21
1
VI
VVI
21
PV is divided into two different classes: PV(δ≠0) can be transformed into Hamiltonian system SV
PV(δ = 0) can’t use SV ; and equivalent to PIII’(γδ≠0)
PIII’ is divided into four different classes: (by Ohyama-Kawamuko-Sakai-Okamoto) PIII’(γδ≠0) type D6 generic case of PIII’
can be transformed into Hamiltonian sytem SIII’
PIII’(γ = 0,αδ≠0) type D7
PIII’(γ = 0,δ=0, αβ≠0) type D8
PIII’(β = δ = 0) type Q solvable by quadrature
PI : If we change eq to , it is solvable
by Weierstrass
tqq 26'' 26'' qq
22
For special values of parameters, PJ (J=II…VI) have classical solutions. Equations which have classical sol can be regarded as degenerated ones. This type degeneration is related to transformation groups of solutions.
Question 1:
Can we systematically explain these degenerations? Question 2:
Can we classify Painleve eqns by intrinsic reason?
23
3.2 Correspondences between Matrix Painleve Systems and Painleve Equations
Theorem2 λ P Correspondences
(1,1,1,1)
A B
C
(2,1,1)
A
B
C under calculation
)1(SVI
)1(SVI
)1(SVI
VS
VS~
III'S
)2/1α(PVI
)2/1α(PVI
)2/1α(PVI
)0γδ(PIII'
)0δ(PV
)0δ(PV
24
(3,1)
A
B
C
(2,2)
A
B
C
(4)
A
B
C
),,,(N IV nmlk IVS
),,,(N~
II nml
),,0,(N~
II nm
IIS IIP
IVP
Riccati
III'S
III'S~
III'S
)0γδ(PIII'
)0αδ,0γ(PIII'
)0δβ(PIII'
),,,(N II nmlk
),,,(N I nml
),,0,(N I nm
IIS IIP
IS
I
0
S
S
IP)0( l
)0( l
linear
I
0
L
L
25
PVI(D4) (α≠1/2)
PVI(D4) (α = 1/2)
PVI(D4) (α = 1/2)
PV(D5)
PIII’(D6)
?
PIV(E6)
PII(E7)
Riccati
PII(E7)
PI (E8)
Linear
PIII’(D6)
PIII’(D7)
PIII’(Q)
26
(1) Nondeg cases of MPS correspond to PII,…,PVI.
(2) All cases of Painleve eqns are written by Hamiltonian systems.
(3) Degenerations of Painleve eqns are characterized by Young diagramλand constant matrix P.
Degenerations of Painleve eqn are classified into 3 levels:
1st level: depend on λonly
2nd level: depend on λand P
3rd level: depend on transformation group of sols
(4) Parameters of Painleve Systems are rational functions of parameters (k,)l,m,n.
(5) On , numbers of parameters are decreased at the steps of canonical transformations between NJ and SJ.
IIIIV P,P,P
27
4. Generalized Confluent Hypergeometric Systems included in MPS
(joint work with Woodhouse) Summary1 (general case)
Theory of GCHS is a general theory to extend classical hypergeometric and confluent hypergeometric systems to any dimension paying attention to symmetry of variables and algebraic structure.
Original GCHS is defined on the space .
Factoring out the effect of the group , , we obtain
GCHS on and GCHS on
Concrete formula of is obtained (with Woodhouse)
)(M0λ r,nZ λ,G
λ,C λ,H
)(rGL λPH
λ,C
λλ \)( ZrGLU λλλ /PHUD
28
Summary 2 (On the case of MPS)
Painleve System SJ (J=II ~ VI) contains Riccati eqn RJ.
RJ is transformed to linear 2-system LSλ contained in Mλ(k,l,m,n).
LSλhas 3-parameters.
Let denote lifted up systems of onto , then we have following diagram.
λU
(1,1,1,1) Gauss Hypergeometric
(2,1,1) Kummer
(3,1) Hermite
(2,2) Bessel
(4) Airy
λ,H
29
From these, Matrix Painleve Systems
may be good expressions of Painleve eqns.
30
References:H.Kimura, Y.Haraoka and K.Takano, The Generalized Confluent Hypergeo
metric Functions, Proc. Japan Acad., 69, Ser.A (1992) 290-295.Mason and Woodhouse, Integrability Self-Duality, and Twistor Theory, Lo
ndon Mathematical Society Monographs New Series 15, Oxford University Press, Oxford (1996).
Y.Murata, Painleve systems reduced from Anti-Self-Dual Yang-Mills equation, DISCUSSION PAPER SERIES No.2002-05, Faculty of Economics, Nagasaki University.
Y.Murata, Matrix Painleve Systems and Degenerations of Painleve Equations, in preparation.
Y.Murata and N.M.J.Woodhouse, Generalized Confluent Hypergeometric Systems on Grassmann Variety, DISCUSSION PAPER SERIES No.2005-10, Faculty of Economics, Nagasaki University.
Y.Murata and N.M.J.Woodhouse, Generalized Confluent Hypergeometric Systems included in Matrix Painleve Systems, in preparation.