Coupled Painlevé Vi Systems in Dimension Four with Affine Weyl Group Symmetry of Type D

We give a reformulation of a six-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 . We also study some Hamiltonian structures of this system. 0. Introduction In [4], we proposed a 6-parameter family of four-dimensional coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 . This system can be considered as a genelarization of the Painlevé VI system. In this paper, from the viewpoint of its symmetry and holomorphy properties we give a reformulation of this system explicitly given by dq1 dt = ∂H ∂p1 , dp1 dt = − ∂H ∂q1 , dq2 dt = ∂H ∂p2 , dp2 dt = − ∂H ∂q2 , H =HV I(q1, p1, η, t;α0, α1, α2, α3 + 2α4 + α5, α3 + α6) +HV I(q2, p2, η, t;α0 + 2α2 + α3, α1 + α3, α4, α5, α6) + 2(q1 − η)q2{(q1 − t)p1 + α2}{(q2 − 1)p2 + α4} t(t− 1)(t− η) (η ∈ C − {0, 1}). (1) Here q1, p1, q2, p2 denote unknown complex variables, and α0, α1, . . . , α6 are complex parameters satisfying the relation α0 + α1 + 2(α2 + α3 + α4) + α5 + α6 = 1, where the symbol HV I(q, p, η, t; β0, β1, β2, β3, β4) is given in Section 2. If we take the limit η → ∞, we obtain the Hamiltonian system with well-known Hamiltonian H̃ (see [4]) dq1 dt = ∂H̃ ∂p1 , dp1 dt = − ∂H̃ ∂q1 , dq2 dt = ∂H̃ ∂p2 , dp2 dt = − ∂H̃ ∂q2 , H̃ = H̃V I(q1, p1, t;α0, α1, α2, α3 + 2α4 + α5, α3 + α6) + H̃V I(q2, p2, t;α0 + α3, α1 + 2α2 + α3, α4, α5, α6) + 2(q1 − t)p1q2{(q2 − 1)p2 + α4} t(t− 1) , (2) where the symbol H̃V I is also given in Section 2. We can obtain the system (1) by the following steps: (1) We symmetrize the holomorphy conditions r i of the system (2) (see Section 5). 2000Mathematics Subject Classification Numbers. 34M55, 34M45, 58F05, 32S65, 14E05, 20F55. 1 2 YUSUKE SASANO (2) By using these conditions and polynomiality of the Hamiltonian, we easily obtain the polynomial Hamiltonian of the system (1). For this reformulation of the Hamiltonian of type D (1) 6 , we will obtain a clear description of invariant divisors, birational symmetries and holomorphy conditions for the system (1) (see in Section 3). This paper is organized as follows. In Section 2, we give a reformulation of Hamiltonian of PV I and its symmetry and holomorphy. In Section 3, we state our main results for the system of type D (1) 6 . After we review the notion of accessible singularity in Section 4, we will state the relation between some accessible singularities of the system (1) and the holomorphy conditions ri given in Section 3. 1. Reformulation of PV I-case The sixth Painlevé system can be written as the Hamiltonian system dq dt = ∂HV I ∂p , dp dt = − ∂HV I ∂q , t(t− 1)(t− η)HV I(q, p, t;α0, α1, α2, α3, α4) = q(q − 1)(q − η)(q − t)p + {α1(t− η)q(q − 1) + 2α2q(q − 1)(q − η) + α3(t− 1)q(q − η) + α4t(q − 1)(q − η)}p + α2{(α1 + α2)(t− η) + α2(q − 1) + α3(t− 1) + tα4}q (α0 + α1 + 2α2 + α3 + α4 = 1, η ∈ C − {0, 1}). (3) The equation for q is given by dq dt2 = 1 2 ( 1 q + 1 q − 1 + 1 q − t + 1 q − η )( dq dt )2 − ( 1 t + 1 t− 1 + 1 q − t + 1 t− η ) dq dt + q(q − 1)(q − t)(q − η) t2(t− 1)2(t− η)2 { α 1 2 η(η − 1)(t− η) (q − η)2 + α 4 2 ηt q2 + α 3 2 (η − 1)(1− t) (q − 1)2 + (1− α 0) 2 t(t− 1)(t− η) (q − t)2 }. (4) If we take the limit η → ∞, we obtain the sixth Painlevé system PV I : dq dt = ∂H̃V I ∂p , dp dt = − ∂H̃V I ∂q , H̃V I(q, p, t; δ0, δ1, δ2, δ3, δ4) = 1 t(t− 1) [p(q − t)(q − 1)q − {(δ0 − 1)(q − 1)q + δ3(q − t)q + δ4(q − t)(q − 1)}p+ δ2(δ1 + δ2)q] (δ0 + δ1 + 2δ2 + δ3 + δ4 = 1), (5) whose equation for q is given by dq dt2 = 1 2 ( 1 q + 1 q − 1 + 1 q − t )( dq dt )2 − ( 1 t + 1 t− 1 + 1 q − t ) dq dt + q(q − 1)(q − t) t2(t− 1)2 { α 1 2 − α 4 2 t q2 − α 3 2 (1− t) (q − 1)2 + (1− α 0) 2 t(t− 1) (q − t)2 }. (6) COUPLED PAINLEVÉ VI SYSTEMS 3 The system (3) has extended affine Weyl group symmetry of type D (1) 4 , whose generators si, πj are given by s0(q, p, t;α0, α1, . . . , α4) →(q, p− α0 q − t , p, t;−α0, α1, α2 + α0, α3, α4), s1(q, p, t;α0, α1, . . . , α4) →(q, p− α0 q − η , t;α0,−α1, α2 + α1, α3, α4), s2(q, p, t;α0, α1, . . . , α4) →(q + α2 p , p, t;α0 + α2, α1 + α2,−α2, α3 + α2, α4 + α2), s3(q, p, t;α0, α1, . . . , α4) →(q, p− α3 q − 1 , t;α0, α1, α2 + α3,−α3, α4), s4(q, p, t;α0, α1, . . . , α4) →(q, p− α4 q , t;α0, α1, α2 + α4, α3,−α4), π1(q, p, t;α0, α1, . . . , α4) →(1− q,−p, 1− η, 1− t;α0, α1, α2, α4, α3), π2(q, p, t;α0, α1, . . . , α4) →( η − q η − 1 , (1− η)p, η η − 1 , η − t η − 1 ;α0, α4, α2, α3, α1), (7) π3(q, p, t;α0, α1, . . . , α4) →( (η − 1)(q − t) {η(t− 2) + 1}q + (η − η2 − 1)t+ η2 , (1− t)p+ (q − 1){(q − 1)p+ α2}{η(t− 2) + 1} (η − 1)2(t− 1) + (q − t){(q − t)p+ α2}{η(t− 2) + 1} η(t− 1)(t− η) , 1− η, (η − 1)t t− ηt+ η2(t− 1) ;α4, α1, α2, α3, α0). The phase space of the system (3) (resp. (5)) can be characterized by the rational surface of type D (1) 4 . The below figure denotes the accessible singular points and the resolution process for each system. Let us consider a polynomial Hamiltonian system with HamiltonianH ∈ C(t)[q, p]. We assume that (A1) deg(H) = 6 with respect to q, p. (A2) This system becomes again a polynomial Hamiltonian system in each coordinate ri (j = 0, 1, . . . , 4): r0 : x0 = −((q − t)p− α0)p, y0 = 1 p , r1 : x1 = −((q − η)p− α1)p, y1 = 1 p , r2 : x2 = 1 q , y2 = −(qp+ α2)q, r3 : x3 = −((q − 1)p− α3)p, y3 = 1 p , r4 : x4 = −(qp− α4)p, y4 = 1 p . (8) Then such a system coincides with the system (3). We remark that the system (3) has the following invariant divisors: