Coupled Painlev'e III systems with affine Weyl group symmetry of types $B_5^{(1)},D_5^{(1)}

dy dτ
2
−
d 1 dy 1 + (ay 2 + b) + cy 3 + τ dτ τ y
a = −8,
b = 4(1 − α1 ),
(1)
c = 0,
d nt of symmetry, the Hamiltonian system (11)
7 ∂HIII dq = , dt ∂p
7 We remark that for y = q/τ, t = τ 2 the Hamiltonian system with HIII is the special case of the third Painlev´ e system (see [9]):
D
(1)
(9) with (10)
d2 y 1 = 2 dτ y
3
By Proposition 1.1, we see that each Hamiltonian Hi (i = 1, 2, 3, 4) is equivalent by the transformations (13). The Hamiltonian system with H1 has the first integral. Proposition 1.2. The system with the Hamiltonian H1 (15) has the first integral I : (16) I = q 2 p2 + qp2 − (α0 + α1 )qp − α0 p. dq ∂H1 = , dt ∂p dp ∂H1 =− dt ∂q
Abstract. We find and study four kinds of five-parameter family of six-dimensional (1) (1) coupled Painlev´ e III systems with affine Weyl group symmetry of types D5 , B5 and (2) D6 . We show that each system is equivalent by an explicit birational and symplectic transformation, respectively. We also show that we characterize each system from the viewpoint of holomorphy.
1. Introduction In [3, 4, 5, 6, 7], we presented some types of coupled Painlev´ e systems with various affine Weyl group symmetries. In this paper, we present a 5-parameter family of coupled (1) Painlev´ e III systems with affine Weyl group symmetry of type D5 , which is explicitly given by (1) ∂H dx = , dt ∂y dy ∂H =− , dt ∂x dz ∂H = , dt ∂w
(α0 − α1 + 2α2 = 0),
2000 Mathematics Subjet Classification. Primary 34M55; Secondary 34M45. Key words and phrases. Affine Weyl group, birational symmetry, coupled Painlev´ e system.
1
2
YUSUKE SASANO
(5) q 2 p2 + (1 − 2α2 )qp + tp H2 = H2 (q, p, t; α2) = (α0 − α1 + 2α2 = 0), t (6) q 2 p2 + 2α2 qp − q (α0 − α1 + 2α2 = 0), H3 = H3 (q, p, t; α2) = t (7) q 2 p2 − tq 2 p − (1 − α0 − α1 )qp − α0 tq H4 = H4 (q, p, t; α0, α1 ) = (α0 − α1 + 2α2 = 0), t (8) (1) (1) q 2 p2 + α1 qp + q + tp D7 D7 (α0 + α1 = 1). HIII = HIII (q, p, t; α1) = t
We see that the relation between the Hamiltonian H1 and the first integral I is explicitly given by (17) tH1 = I.
The B¨ acklund transformations of this system satisfy Noumi-Yamada’s universal de(1) scription for D5 root system (see [1]). Since these universal B¨ acklund transformations have Lie theoretic origin, similarity reduction of a Drinfeld-Sokolov hierarchy admits such (1) a B¨ acklund symmetry. The aim of this paper is to introduce the system of type D5 . Our discovery of the system (1) of type D5 W ( D5 ) Drinfeld-Sokolov hierarchy?
´ III SYSTEMS WITH AFFINE WEYL GROUP COUPLED PAINLEVE (1) (1) (2) SYMMETRY OF TYPES B5 , D5 AND D6
YUSUKE SASANO
arXiv:0704.2467v1 [math.AG] 19 Apr 2007
(1)
Noumi-Yamada’s universal description (1) for D5 root system Figure 1.
We also show that this system coincides with some types of 5-parameter family of sixdimensional coupled Painlev´ e III systems with extended affine Weyl group symmetry of (1) (2) types B5 and D6 . Moreover, we show the relationship between the system of type (1) (1) (2) D5 and the system of types B5 and D6 by an explicit birational and symplectic transformation, respectively. This paper is organized as follows. In Section 2, we will introduce the system of type (1) D5 and its B¨ acklund transformations. In Section 3, we will propose two types of a 5parameter family of coupled Painlev´ e III systems in dimension six with extended affine (1) Weyl group symmetry of type B5 . We also show that each of them is equivalent to the system (1) by a birational and symplectic transformation, respectively. In Section 4, we will propose a 5-parameter family of coupled Painlev´ e III systems in dimension six with (2) extended affine Weyl group symmetry of type D6 and its B¨ acklund transformations. We also show that this system is equivalent to the system (1) by a birational and symplectic transformation.
D
(1)
7 The symbols H1 , H2 , H3 , H4 and HIII denote the polynomial Hamiltonians explicitly given as follows:
(4)
H1 = H1 (q, p, t; α0, α1 ) =
q 2 p2 + qp2 − (α0 + α1 )qp − α0 p t
3
the Hamiltonians H1 , H2 , H3 and H4 satisfy the following relations: (14) tr1 (H1 ) = H2 , tr2 ◦ tr1 (H1 ) = H3 ,
tr3 (H1 ) = H4 .
´ III SYSTEMS COUPLED PAINLEVE
D
dp ∂H 7 = − III dt ∂q
(1)
D
(1)