Four - Dimensional Painlevé Systems of Types

We find and study a five-parameter family of four-dimensional coupled Painlevé V systems with affine Weyl group symmetry of type D (1) 5 . We then give an explicit description of a confluence from those systems to a four-parameter family of four-dimensional coupled Painlevé III systems with affine Weyl group symmetry of type B (1) 4 . 0. Introduction It is well-known that the Painlevé systems PII , PIII , PIV , PV , PV I admit the affine Weyl groups of type A (1) 1 , C (1) 2 , A (1) 2 , A (1) 3 andD (1) 4 , respectively, as groups of Bäcklund transformations. This suggests the following general problem (see [6]): Problem 0.1. For each affine root system A with affine Weyl group W (A), find a system of differential equations for which W (A) acts as its Bäcklund transformations. One could expect that such nonlinear differential systems with affine Weyl group symmetry should admit rich mathematical structures, comparable with those of the Painlevé equations. In the case of type A (1) l , such equations are proposed in [4]. They are considered to be higher order versions of PV (resp. PIV ) when l is odd (resp. even). These two examples by Noumi and Yamada motivated the author to find examples of higher order versions other than the systems of type A (1) l . We will complete the study of the above problem in a series of four papers, for which this paper is the second, resulting in a series of equations for the remaining affine root systems of types B (1) l , C (1) l and D (1) l (see [11, 12, 13]). This paper is the stage in this project where we find and study four-dimensional coupled Painlevé V systems with W (D (1) 5 )-symmetry explicitly given by dx dt = ∂H D (1) 5 ∂y , dy dt = − ∂H D (1) 5 ∂x , dz dt = ∂H D (1) 5 ∂w , dw dt = − ∂H D (1) 5 ∂z (1) with the Hamiltonian H D (1) 5 =HV (x, y, t;α2 + α5, α1, α2 + 2α3 + α4) +HV (z, w, t;α5, α3, α4) + 2yz{(z − 1)w + α3} t , (2) where the symbol HV (q, p, t; γ1, γ2, γ3) denotes the Hamiltonian of the second-order Painlevé V systems given by HV (q, p, t; γ1, γ2, γ3) = q(q − 1)p(p+ t)− (γ1 + γ3)qp+ γ1p+ γ2tq t .