Drinfeld-Sokolov Hierarchies of Type A and Fourth Order Painleve Systems

Abstract

Abstract. We study the Drinfeld-Sokolov hierarchies of type A ð1Þ n associated with the regular conjugacy classes of W ðA n Þ. A class of fourth order Painlevé systems is derived from them by similarity reductions.

Geometric aspects of Painlevé equations

“…We show such computations in case of the root system of type D whose Cartan matrix is (3.9) with l = 5. The Dynkin diagram and the null root are given by 20) respectively. The corresponding Weyl group W(D…”

“…intersect with C 0 at two points, respectively. Therefore, in terms of the coordinate u of the complex torus C/Ω, it is known that the coordinates ( f, g) of a point on C 0 can be parametrized by elliptic functions of order 2 [74,135]: 20) where σ(u) = σ(u; ω 1 , ω 2 ) is the Weierstrass sigma function or a theta function. From this we obtain the parametrization of the curve C 0 21) through the renormalization by the action of PGL (2) 2 and the translation of u.…”

“…There are two additional points ( f (z), g(z)) , ( f (w), g(w)) , (7.18) at which the curve P( f, g) = 0 intersects with the reference curve C 0 . Here, the parameters z and w should satisfy 20) due to Abel's Theorem, or the relation between roots and coefficients for the Laurent polynomial…”

Rational Solutions of the Sasano Systems of Types B4(1), D4(1)and D5(2)

“…In this paper, we completely classify the rational solutions of the Sasano system of types B (1) 4 , D (1) 4 and D (2) 5 , which are all given by coupled P III systems and have the affine Weyl group symmetries of types B (1) 4 , D (1) 4 and D (2) 5 . The rational solutions are classified as one type by the Bäcklund transformation group.…”

A <i>q</i>-Analogue of the Higher Order Painlev&eacute; Type Equations with the Affine Weyl Group Symmetry of Type <i>D</i>

“…In particular, the classification of the fourth order equations obtained by isomonodromic deformation of Fuchsian equations has been established [13]. We have four kinds of systems of the fourth order; the Garnier system [4], the systems that admit the a‰ne Weyl group symmetry of type A [3] and D [14,15], respectively, and so-called the matrix Painlevé system [13]. It is clarified that each system of equations can be formulated as a Hamiltonian system.…”