Related papers: Coupled Painleve VI system with E_6^{(1)} symmetry
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…
Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the…
We present a constructive procedure to obtain the critical behavior of Painleve' VI transcendents and solve the connection problem. This procedure yields two and one parameter families of solutions, including trigonometric and logarithmic…
A q-difference analogue of the Painlev\'e III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.
We describe all finite orbits of an action of the extended modular group $\bar{\Lambda}$ on conjugacy classes of SL(2,C)-triples. The result is used to classify all algebraic solutions of the general Painleve VI equation up to parameter…
Three $q$-Painlev\'e type equations are derived through degenerations of the $q$-Painlev\'e equation with affine Weyl group symmetry of type $q$-$E_8^{(1)}$. The three $q$-Painlev\'e type equations are associated with different realizations…
We show that the Garnier system in n-variables has affine Weyl group symmetry of type $B^{(1)}_{n+3}$. We also formulate the $\tau$ functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice $Q(C_{n+3})$ and…
The $q$-Painlev\'e equation of type $E^{(1)}_6$ is obtained by Pad\'e method. Special solutions in determinant formula to the $q$-Painlev\'e equation is presented. A relation between Pad\'e method and B\"acklund transformation of type…
A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the…
We show that the Painlev{\'e} VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in $C^3$ and include the additional constant gyrostat momentum. The…
A discretization of Painlev\'e VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point…
We find all non-abelian generalizations of $\text{P}_1 - \text{P}_6$ Painlev\'e systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax…
We exhibit a remarkable connection between sixth equation of Painleve list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and…
A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or B\"acklund transformations. We describe such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from \pvi itself, a…
In this paper we construct a new completely integrable system. This system is an instance of a master system of differential equations in $5$ unknowns having $3$ quartics constants of motion.We find via the Painlev\'e analysis the principal…
The sixth Painlev\'e equation is a basic equation among the non-linear differential equations with three fixed singularities, corresponding to Gauss's hypergeometric differential equation among the linear differential equations. It is known…
We discuss a family of Hamiltonians given by particular rational extensions of the singular oscillator in two-dimensions. The wave functions of these Hamiltonians can be expressed in terms of products of Laguerre and exceptional Jacobi…
Discrete Painlev\'e equations are integrable two-dimensional birational maps associated to a family of generalized Halphen surfaces. The latter can be seen either as $\mathbb P^2$ blown up at nine points or as $\mathbb P^1\times\mathbb P^1$…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
We prove that the holonomy group at infinity of the Painleve VI equation is virtually commutative.