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|

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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

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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

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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

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(5) Let . Then

gives 3-dimensional fibration

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(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

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2.3 Matrix Painleve Systems

We call the combined system (Lλ ＋ 1st Int) Matrix Painleve Sytem Mλ

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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

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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

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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

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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?

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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

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(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

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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)

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(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

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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

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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

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From these, Matrix Painleve Systems

may be good expressions of Painleve eqns.

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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.