Robert Conte
Following the recent discovery of two new Schlesinger transformations (ST) for the sixth Painlevé equation, we give the interrelations between all the known STs. We thus isolate the unique one which at the same time conserves the independent variable and is not a product of other STs.
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Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painleve VI solutions without deriving full RS-pullback transformation…
The geometric model of the pathway linking Schlesinger and Garnier--Painlevé~6 systems based on an original orthonormalization of a set of elements in ${sl(2,\mathbb C)}$ is constructed. The explicit polynomial map of the Cartesian products of $n-2$ quadrics (the Zariski-topology chart of the phase space of the Garnier--Painlevé~6 system) into the phase space of the Schlesinger system and the rational inverse to this map are presented.
A local behavior of solutions of the Schlesinger equation is studied. We obtain expansions for this solutions, which converge in some neighborhood of a singular point. As a corollary the similar result for the sixth Painlevé equation was obtained. In our analysis, we use the isomonodromic approach to solve this problem.
We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlevé equations: difference Painlevé equation d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painlevé equation d-$P\left({A}_{1}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coor…
Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standa…
For the Schlesinger-type equation related to the fifth Painlevé equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.
We describe two algebraic solutions of the sixth Painlevé equation which are related to (isomonodromic) deformations of Picard-Fuchs equations of order two.
For the Painlevé 6 transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of the poles close to a critical point.
We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlevé equation with parameters $({1}/{8}, -{1}/{8}, {1}/{8}, {3}/{8})$ expressed in simple forms using periods of differentials on elliptic curves. Similarly for every integer $n$ different from $0$ and $-1$ we obtain one family of solutions to the sixth Painlevé equation with parameters $(\frac{9n^2+12n+4}{8}, -\frac{n^2}{8}, \frac{n^2}{8}, \frac{4-n^2}{8})$.
It is known, that among the formal solutions of the sixth Painlevé equation there met series with integer power exponents of the independent variable $x$ with coefficients in form of formal Laurent series (with finite main parts) in $\log^{-1} x$ (complicated expansions), or in $x^{{\rm i}\,θ}$, where ${\rm i}=\sqrt{-1},$ $θ\in\mathbb{R},$ $θ\neq 0$ (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic formal solutions of the sixth Painlevé equation.
We will study two types of special solutions of the sixth Painleve equation, which are invariant under the symmetries obtained from the Backlund transformations. In most cases, the fixed points of the Backlund transformations are classical solution, but our solutions are not classical for generic parameters. We will calculate the linear monodromy of these solutions exactly, and we will characterize them on Fricke's cubic surface of monodromy.
We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d'enfants related with the construction of the algebraic solutions of the sixth Painlevé equation via the method of $RS$-transformations.
The third, fifth and sixth Painlevé equations are studied by means of the weighted projective spaces ${\mathbb C}P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux's coordinates are systematically studied in a unified way with the aid of the orbifold structure of ${\mathbb C}P^3(p,q,r,s)$ and dynamical systems theory.
A study on the bilinear equation of the sixth Painlevé transcendents
本文研究如何由三个固定奇点附近的局部行为确定非线性微分方程，并通过要求齐次二次四阶方程在各奇点处为 (H) 型，推导出第六 Painlevé 方程。
In 1991, one of the authors showed the existence of quadratic transformations between the Painleve' VI equations with local monodromy differences $(1/2,a,b,\pm 1/2)$ and $(a,a,b,b)$. In the present paper we give concise forms of these transformations. %, up to fractional-linear transformations. They are related to the %better known quadratic transformations obtained by Manin and Ramani-Grammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particular…
A new method to construct algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of CP^1, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painleve VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. A generalization of this differential to hyperelliptic…
We consider connection between the Painleve-6 equation and explicitly uniformizable orbifolds
We consider an application of Grothendieck's dessins d'enfants to the theory of the sixth Painlevé and Gauss hypergeometric functions: two classical special functions of the isomonodromy type. It is shown that, higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related with some particular Belyi functions. Moreover, we introduce a notion of deformation of the dessins d'enfants and show that one dimensional deformations are a useful tool for construction of algebraic the sixth Painlevé functions.
Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation
本文研究第六 Painlevé 方程的 q-模拟，证明其可由 q-Schlesinger 方程导出并具有 q-Hamiltonian 结构，进而证明 Sakai 的 q-Okamoto 空间在自然极限下退化为微分 Okamoto 空间。
We study dynamics of solutions in the initial value space of the sixth Painlevé equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded, and that the complex limit set of each solution exists and is compact and connected.
We will study special solutions of the fourth, fifth and sixth Painlevé equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura's classical solutions and our solutions.
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 behaviors, and three classes of solutions with Taylor expansion at a critical point.
We derive a $q$-analogue of the matrix sixth Painlevé system via a connection-preserving deformation of a certain Fuchsian linear $q$-difference system. In specifying the linear $q$-difference system, we utilize the correspondence between linear differential systems and linear $q$-difference systems from the viewpoint of the spectral type. The system of non-linear $q$-difference equations thus obtained can also be regarded as a non-abelian analogue of Jimbo-Sakai's $q$-$P_{\mathrm{VI}}$.
We describe all finite orbits of an action of the extended modular group $\barΛ$ 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 equivalence.
After recalling some of the geometry of the sixth Painleve equation, we will describe how the Okamoto symmetries arise naturally from symmetries of Schlesinger's equations and summarise the classification of the Platonic Painleve six solutions. A key observation is that Painleve VI governs the isomonodromic deformations of certain Fuchsian systems on rank \emph{three} bundles.
The higher order Painleve system of type D^{(1)}_{2n+2} was proposed by Y. Sasano as an extension of the sixth Painleve equation for the affine Weyl group symmetry with the aid of algebraic geometry for Okamoto initial value space. In this article, we give it as the monodromy preserving deformation of a Fuchsian system.
We will explain how some new algebraic solutions of the sixth Painleve equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.
We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlevé equations and the differential equation is the $σ$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.
Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Lague…
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 differentials on corresponding Riemann surfaces, Abelian integrals, analytic connections (generalizations of Chazy's equations), and other attributes of uniformization can be obtained for these curves. As byproducts of the theory, we establish relations between Picard-Hitchin's curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous d…
We introduce and study isomonodromy transformations of the matrix linear difference equation Y(z+1)=A(z)Y(z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Z^{m(n+1)-1} on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Sch…
The critical behavior of a three real parameter class of solutions of the sixth Painlevé equation is computed, and parametrized in terms of monodromy data of the associated $2\times 2$ matrix linear Fuchsian system of ODE. The class may contain solutions with poles accumulating at the critical point. The study of this class closes a gap in the description of the transcendents in one to one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of…
Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked Fuchsian equation. But the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. This paper presents computations of explicit RS-pullback transformations, and derivati…
Krall-type orthogonal polynomials and integrable isomonodromic deformations
本文研究了Krall型正交多项式与可积等单调变形的关系，揭示了其与Painlevé方程及Schlesinger型偏微分方程的联系。
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 Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé 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 th…
The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painleve equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the cano…
We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ellitic form of the sixth Painlevé equation directly.
The paper concerns asymptotic studies for the sixth Painlevé transcendent as independent variable tends to infinity. The primary tool is averaging and the Whitham method. Elliptic ansatz, appropriate modulation equation and asymptotics for some partial solution to PVI are found.
The article studies the Fifth Painlevé equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity from the point of view of confluence from the Sixth Painlevé equation. This approach is developped separately on both sides of the Riemann-Hilbert correspondance. On the side of the nonlinear Painlevé-Okamoto foliation the relation between the nonlinear monodromy group of Painlevé VI and the "nonlinear wild monodromy pseudogroup" of Painlevé V (that is the pseudogroup generated by nonlinear monodromy, nonlinear Stokes operators and nonlinear exponential torus) is expl…
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-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain…
We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by …
A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be $τ$-functions for Painlevé systems, allowing for the former to be characterised as the solution of certain nonlinear equations. We consider the random matri…
We consider a degeneration of the $q$-matrix sixth Painlevé system. As a result, we obtain a system of non-linear $q$-difference equations, which describes a deformation of a certain non-Fuchsian linear $q$-difference system. We define the spectral type for non-Fuchsian $q$-difference systems and characterize the associated linear problem in terms of the spectral type. We also consider a continuous limit of the non-linear $q$-difference system and show that the resulting system of non-linear differential equations coincides with the matrix fifth Painlevé system.
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…
The critical and asymptotic behaviors of solutions of the sixth Painlevé equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.
Four 4-dimensional Painlevé-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlevé system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painlevé-type equations. If we only consider Painlevé-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painlevé-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equat…
In a recent work, we proposed the coupled Painlevé VI system with $A^{(1)}_{2n+1}$-symmetry, which is a higher order generalization of the sixth Painlevé equation ($P_{\rm VI}$). In this article, we present its particular solution expressed in terms of the hypergeometric function ${}_{n+1}F_n$. We also discuss a degeneration structure of the Painlevé system derived from the confluence of ${}_{n+1}F_n$.
This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin conn…
Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed.
Given the second order scalar Lax pair of the sixth Painlevé equation, we build a generalized heat equation with rational coefficients which does not depend any more on the Painlevé variable. Étant donné la paire de Lax scalaire de la sixième équation de Painlevé, nous donnons une construction directe de l'équation de la chaleur généralisée à coefficients rationnels qui ne dépend plus de la variable de Painlevé.
The critical and asymptotic behaviors of solutions of the sixth Painlevé equation, an their parametrization in terms of monodromy data, are synthetically reviewed. The explicit formulas are given. This paper has been withdrawn by the author himself, because some improvements are necessary.
The paper is an essentially extended version of the work math.CA/0601371, supplemented with an application. We present new results in the theory of classical $θ$-functions of Jacobi and $σ$-functions of Weierstrass: ordinary differential equations and series expansions. We also give the extension of canonical $θ$-functions and consider an application to the sixth Painlevé equation (P6). Picard--Hitchin's general solution of P6 is represented explicitly in a form of logarithmic derivative of a corresponding $τ$-function (Painlevé's form).
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting…
Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlevé equations, there are infinitely many non-equ…
In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{ρ_j} $ were derived.Here it is shown that in the case $ m=3 $ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlev…
In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrenc…
The diagonal spin-spin correlations $ \langle σ_{0,0}σ_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathb…
It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency (non-branching) of a singularity. For the elliptical repr…
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