Sixth Painleve Equation Research Articles

Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation

We consider the equivalence problem for projective-type scalar second-order ordinary differential equations with respect to invertible point changes of variables. Invariants of the equivalence transformation group of this family of equations are used to find necessary conditions for equivalence to the sixth Painleve equation.

Some Explicit Results for the Generalized Emptiness Formation Probability of the Six-Vertex Model

We study a multi-point correlation function of the six-vertex model on the square lattice with domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes the probability of observing ferroelectric order around all vertices of any Ferrers diagram λ at the top left corner of the lattice. For the free fermion model, we derive and compare explicit formulas for this correlation function in two cases: when the diagram λ has a square or a triangular shape. We find a connection between our formulas and the τ-function of the sixth Painleve equation. Bibliography: 25 titles.

Heuristic computational intelligence approach to solve nonlinear multiple singularity problem of sixth Painlev́e equation

The present study investigate the numerical solution of nonlinear singular system represented with sixth Painleve equation by the strength of artificial intelligence using feed-forward artificial neural networks (ANNs) optimized with genetic algorithms (GAs), interior point technique (IPT), sequential quadratic programming (SQP), and their hybrids. The ANN provided a compatible method for finding nature-inspired mathematical model based on unsupervised error for sixth Painleve equation and adaptation of weights of these networks is carried out globally by the competency of GA aided with IPT or SQP algorithms. Moreover, a hybrid approach has been adopted for better proposed numerical results. An extensive statistical analysis has been performed through several independent runs of algorithms to validate the accuracy, convergence, and exactness of the proposed scheme.

Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite’s two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitz determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Algebraic solutions of the sixth Painleve equation can be constructed with the help of RS-transformations of hypergeometric equations. Construction of these transformations includes specially ramified rational coverings of the Riemann sphere and the corresponding Schlesinger transformations (S-transformations). Some algebraic solutions can be constructed from rational coverings alone, without obtaining the corresponding pullbacked isomonodromy Fuchsian system, i.e., without the S part of the RS transformations. At the same time, one and the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. In the case of high degree coverings, construction of the S parts of the RS-transformations may represent some computational difficulties. This paper presents computation of explicit RS pullback transformations and derivation of algebraic Painleve VI solutions from them. As an example, we present a computation of all seed solutions for pullbacks of hyperbolic hypergeometric equations. Bibliography: 26 titles.

Local expansions of solutions to the fifth Painlevé equation

In this work, the methods of power geometry are used to find asymptotic expansions of solutions to the fifth Painleve equation as x → 0 for all values of its four complex parameters. We obtain 30 families of expan� sions, of which 22 are obtained from published expan� sions of solutions to the sixth Painleve equation. Among the other eight families, one was previously known and two can be obtained from the expansions of solutions to the third Painleve equation. Three fami�

HYPERGEOMETRIC SOLUTION OF A CERTAIN POLYNOMIAL HAMILTONIAN SYSTEM OF ISOMONODROMY TYPE

In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painleve equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.

Basic asymptotic expansions of solutions to the sixth Painlevé equation

Asymptotic expansions of solutions of three types are found for the sixth Painleve equation in its three singular points.

Middle Convolution and Heun's Equation

Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of ini- tial conditions of the sixth Painleve equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.

Local expansion of Painlevé VI transcendents around a fixed singularity

We study special solutions of the sixth Painleve equation with generic values of parameters whose linear monodromy can be calculated explicitly. We give two kinds of special solutions. One is the meromorphic solutions at a fixed singularity and the other is the solutions expressed in the Puiseux series at t=0. We give the respective character of these special solutions.

The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Several results including integral representation of solutions and Hermite– Krichever Ansatz on Heun’s equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painleve equation by monodromy preserving deformation and obtain solutions of the sixth Painleve equation which include Hitchin’s solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich–Verdier potential and additional apparent singularities of exponents − 1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schrodinger operator on our potential.

Higher Order Painlevé System of Type Arising from Integrable Hierarchy

A higher order Painleve system of type D2n+2(1) was introduced by Y. Sasano. It is an extension of the sixth Painleve equation (P VI ) for the affine Weyl group symmetry. It is also expressed as a Hamiltonian system of order 2n with a coupled Hamiltonian of P VI . In this article, we discuss a derivation of this system from a Drinfeld-Sokolov hierarchy.

Hamiltonians associated with the sixth Painlevé equation

We obtain the formula determining the general form of polynomial Hamiltonians associated with the sixth Painleve equation and prove its uniqueness. We prove the existence of nonpolynomial Hamiltonians associated with this equation. We identify the Hamiltonian class for which the defining differential equation coincides with the equation (h-equation) for the simplest polynomial Hamiltonian (the Okamoto Hamiltonian).

The sixth Painlevé equation arising fromD(1)4hierarchy

The sixth Painleve equation arises from a Drinfel'd-Sokolov hierarchy associated with the affine Lie algebra of type D_4 by similarity reduction.

A family of solutions of a higher order PVI equation near a regular singularity

Restriction of the N-dimensional Garnier system to a complex line yields a system of second-order nonlinear differential equations, which may be regarded as a higher order version of the sixth Painleve equation. Near a regular singularity of the system, we present a 2N-parameter family of solutions expanded into convergent series. These solutions are constructed by iteration, and their convergence is proved by using a kind of majorant series. For simplicity, we describe the proof in the case N = 2.

Q-Painlevé VI Equation Arising from q-UC Hierarchy

We study the q-difference analogue of the sixth Painleve equation (q-PVI) by means of tau functions associated with the affine Weyl group of type D5. We prove that a solution of q-PVI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-PVI in terms of the universal character which is a generalization of the Schur polynomial attached to a pair of partitions.

From Klein to Painlevé via Fourier, Laplace and Jimbo

We describe a method for constructing explicit algebraic solutions to the sixth Painleve equation, generalising that of Dubrovin and Mazzocco. There are basically two steps. First we explain how to construct finite braid group orbits of triples of elements of {\rm SL}_2({\rm C}) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier?Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painleve VI solutions. (In particular, this solves a Riemann?Hilbert problem explicitly.) Each step is illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We also prove that, unlike the algebraic solutions of Dubrovin and Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of {\rm SL}_2({\rm C}). The results of this paper also yield a simple proof of a recent theorem of Inaba, Iwasaki and Saito on the action of Okamoto's affine {\rm D}_4 symmetry group as well as the correct connection formulae for generic Painleve VI equations.

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ℙ1. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painleve equation.

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painleve equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painleve transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painleve equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painleve equation is obtained.

On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation

In this paper we find a class of solutions of the sixth Painleve equation appearing in the theory of WDVV equations. This class covers almost all the monodromy data associated to the equation, except one point in the space of the data. We describe the critical behavior close to the critical points in terms of two parameters and we find the relation among the parameters at the different critical points (connection problem). We also study the critical behavior of Painleve transcendents in the elliptic representation.