Coupled Painlevé VI system withE⁽¹⁾₆-symmetry

Abstract

We present an new system of ordinary differential equations with affine Weyl group symmetry of type E(1) 6 . This system is expressed as a Hamiltonian system of sixth order with a coupled Painlevé VI Hamiltonian.

From Kp/Uc Hierarchies to Painlevé Equations

“…The connection between P(A (1) ℓ−1 ) and the KP hierarchy was first pointed out by Schiff [42], and it was studied independently by Noumi and Yamada (see, e.g., [28]) from a group-theoretical point of view. Their theory still has been developed with involving the Drinfel'd-Sokolov hierarchy ( [7]) and achieved various higher order Painlevé equations; see [9,10,11,20,32,40]. It would be an interesting and important problem to examine their relevance to our present results.…”

Six-dimensional Painlevé systems and their particular solutions in terms of rigid systems

“…(1, 3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (4, 1), (4, 2), (5, 1), (5, 2), (6, 1), (6,2).…”

“…. , 7) satisfy a rational Hamiltonian system Furthermore, we have 8 relations which is derived from the matrix components (3,5), (3,6), (4,3), (4,6), (5,3), (5,4), (6,3), (6,4). 19 In order to derive the Hamiltonian system of sixth order, we use the first four relations, whose explicit formulas are given as…”

Virasoro Constraints for Drinfeld-Sokolov hierarchies and equations of Painlevé type

“…For example, the topological solution τ s 0 for g = A In contrast to the string equation, the equation (1.9) makes sense in both cases (I) and (II). We will call this equation the similarity equation, for it is related to the so-called similarity reductions of integrable hierarchies in the literature (see, for example, [6,17,18,19]).…”

“…For such ODEs of Painlevé type, by representing them into a certain symmetric form, Noumi and Yamada constructed a class of birational Bäcklund transformations, whose commutation relations admit the generating relations for affine Weyl groups [35,36]. This approach was developed by a series of work, for example, [17,18,19,26,31], most of which relied on matrix realizations of affine Kac-Moody algebras of some particular types. As it is hinted at the end of [36] (without a proof there), we obtain the second part of Theorem 1.2 aiming at a unified construction of the birational Bäcklund transformations related to the Drinfeld-Sokolov hierarchy associated to (g, s, 1).…”