The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI

Abstract

Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension µ, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give rational solutions of the Painleve VI equation for µ = 1, 2, . . ..

The “ghost” symmetry in the CKP hierarchy

“…where dλ ≡ ∞ dλ 2πi = Res λ=∞ . By now, the CKP hierarchy has attracted many researches [3][4][5][6][7][8][9][10][11][12][13]. In contrast to the KP and the BKP cases, there seems not a single tau function to describe the CKP hierarchy in the form of Hirota bilinear equations (that is, Hirota's equations are no longer of the type P (D)τ · τ = 0) [2].…”

“…The existence of this kind of tau function for the CKP hierarchy is a long-standing problem. Much work has been done in this field [3][4][5][6]. Since the existence of tau function of the CKP hierarchy is not proved, many important results on the Lax operator and dressing operator of the 2 CKP hierarchy can not be transferred to the tau function, until Chang and Wu [6] introduces a kind of tau function τ c (t) for the CKP case, which is related with the wave function in the following way…”

Solutions of the Painlevé VI Equation from Reduction of Integrable Hierarchy in a Grassmannian Approach

“…To set the work in context let us point out that the construction in this paper generalizes a simpler case of the three-dimensional Frobenius manifold of the twodimensional conformal field theory. In that special case it was shown in [3] that the above construction with a much simpler choice of µ = diag(−µ, 0, µ) and ν = 0 leads to the Painlevé VI equation :…”

“…Note that, in an equivalent parametrization of [9,3], the Painlevé parameters (α, β, γ, δ) take the following form :…”

Bäcklund transformations for certain rational solutions of Painlevé VI

“…In [1], which was a generalization of [2]( see also [8]), we showed that there is a connection between certain homogeneous solutions of the 3-component KP hierarchy and certain rational solutions (cf. [9]) of the Painlevé VI equation:…”