Inverse Problem for semisimple Frobenius Manifolds, Monodromy Data and the Painleve' VI Equation

On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy

A generalisation of the Stieltjes relations for the Painlevé-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher Painlevé-IV equation. The approach is based on the interpretation of the Stieltjes relations as local trivial monodromy conditions for certain Schrödinger equations in the complex domain. As a corollary a new class of the Schrödinger operators with trivial monodromy is constructed in terms of the Painlevé-IV transcendents…

Confluence on the Painlevé Monodromy Manifolds, their Poisson Structure and Quantisation

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlevé equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that quantisation and confluence commute. We also investigate the underlying cluster algebra structure and the relation to the versal deformations of singularities of type $D_4,A_3,A_2$, and $A_1$.

Cubic and quartic transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence

A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painleve VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painleve VI.

The sixth Painleve' equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlevé equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [44, 61]. We then express in terms of the Stokes matrices of the 3x3 irregular system the monodromy invariants p_{jk}=Tr(M_jM_k) of the 2-dimensional isomonodromic Fuchsian system with four singularit…

第一类 Painlevé 超越函数的单值数据的完全渐近展开：连接问题中的应用

Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems

本文通过改进复 WKB 方法，严格证明了第一类 Painlevé 超越函数在连接问题中单值数据的完全渐近展开，并给出了非线性特征值及极点参数的完整展开。

Apparent singularities of Fuchsian equations, and the Painlevé VI equation and Garnier systems

We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.