The Painleve first equation can be represented as the equation of isomonodromic deformation of a Schrodinger equation with a cubic potential. We introduce a new algorithm for computing the direct monodromy problem for this Schrodinger equation. The algorithm is based on the geometric theory of Schrodinger equation due to Nevanlinna
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 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.
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 show that a 2-parameter family of $τ$-functions for the first Painlevé equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.
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.
The isomonodromy deformation equation for a 2x2 matrix linear ODE with a large parameter can be locally reduced to a (hyper)elliptic equation. To globalize this result, we apply the isomonodromy deformation method and obtain the modulation equations for the asymptotic algebraic curve. The method is applied to the degenerate solution of the Painleve first equation.
In this paper, we study the Painlevé VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left \{ 0,1\right \} $. (iii) If the monodromy group of the associated linear ODE of a solution $λ\left( t\right) $ is unitary, then $λ( t) $ has no poles in $\mathbb{R}% \backslash \{ 0,1\} $.
We study the global analytic properties of the solutions of a particular family of Painleve' VI equations with the parameters $β=γ=0$, $δ={1\over2}$ and $α$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of o…
We study the monodromy of Painlevé VI equation from a dynamical point of view. This is applied to the description of bounded orbits, and to a proof of the irreducibility of Painlevé VI equation in the sens of Casale and Malgrange. On our way, we compute the entropy of each element of the monodromy group, and we precise the dictionary between character varieties and Painlevé equations.
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…
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…
In the work we use integral formulas for calculating the monodromy data for the Painlevé-2 equation. The perturbation theory for the auxiliary linear system is constructed and formulas for the variation of the monodromy data are obtained. We also derive a formula for solving the linearized Painlevé-2 equation based on the Fourier-type integral of the squared solutions of the auxiliary linear system of equations.
In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlevé equations. The classification of ten families is given by considering the Riemann-Hilbert morphism from a moduli space of connections with certain type of regular and irregular singularities to a corresponding catetorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), which is called the monodromy space. Explicit equations of the monodromy spaces for ten families are calculated. Moreover, we obtain natural explicit f…
In this article, we propose a class of six-dimensional Painleve systems given as the monodromy preserving deformations of the Fuchsian systems. They are expressed as polynomial Hamiltonian systems of sixth order. We also discuss their particular solutions in terms of the hypergeometric functions defined by fourth order rigid systems.
We study the dependence of the tau function of Painlevé I equation on the generalized monodromy of the associated linear problem. In particular, we compute connection constants relating the tau function asymptotics on five canonical rays at infinity. The result is expressed in terms of dilogarithms of cluster type coordinates on the space of Stokes data.
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.
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 find the critical behavior close to the critical points in terms of two parameters and solve the connection problem. We also study the critical behavior of Painleve' transcendents in the Elliptic representation.
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.
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.
In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on $\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on $\mathbb{CP}^{1}$ to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodr…
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…
Matrix Szego biorthogonal polynomials for quasi-definite matrices of measures are studied. For matrices of Holder weights a Riemann-Hilbert problem is uniquely solved in terms of the matrix Szego polynomials and its Cauchy transforms. The Riemann-Hilbert problem is given as an appropriate framework for the discussion of the Szego matrix and the associated Szego recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations which a…
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…
Resurgent-transseries solutions to Painleve equations may be recursively constructed out of these nonlinear differential-equations -- but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painleve I and Painleve II contexts, nonlinear Stokes data g…
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…
We study monodromy reduction of Fuchsian connections from a sheave theoretic viewpoint, focusing on the case when a singularity of a special connection with four singularities has been resolved. The main tool of study is {based on} a bundle modification technique due to Drinfeld and Oblezin. This approach via invariant spaces and eigenvalue problems allows us not only to explain Erdélyi's classical infinite hypergeometric expansions of solutions to Heun equations, but also to obtain new expansions not found in his papers. As a consequence, a geometric proof of Takemura's eigenvalues inclusion …
The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions $u(t)$ as $t -> +/-\infty$ and $t -> +/-i\infty$ are derived and parametrized in terms of the monodromy data of the associated 2X2 linear auxiliary problem introduced in the first part of this work [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotical…
I study the solutions of a particular family of Painlevé VI equations with the parameters $β=γ=0, δ=1/2$ and $2α=(2μ-1)^2$, for $2μ\in\interi$. I show that the case of half-integer $μ$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$μ$ equation for any $μ$ such that $2μ\in\interi$. As an applicat…
The paper is about a Painlevé III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation, which we denote by P_{III}(0,0,4,-4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics. Complex multi-valued solutions on C^* are the natural context for most of the paper, but in the last …
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.
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…
Hypergeometric solutions to the q-Painlevé equations are constructed by direct linearization of disrcrete Riccati equations. The decoupling factors are explicitly determined so that the linear systems give rise to q-hypergeometric equations.
We study polynomials that are orthogonal with respect to a varying quartic weight \exp(-N(x^2/2+tx^4/4)) for t<0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painleve I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann-…
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 analyze the one parameter family of tronquée solutions of the Painlevé equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the paramet…
Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution
Université Paris-Saclay（巴黎萨克雷大学）；CEA Saclay（法国原子能和替代能源委员会萨克雷中心）；CNRS（法国国家科学研究中心）；Sorbonne université（索邦大学）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；The University of Hong Kong（香港大学）
We consider a second order reducible equation having non-resonant irregular singularities at $x=0$ and $x=\infty$. Both of them are of Poincaré rank 1. We introduce a small complex parameter $\varepsilon$ that splits together $x=0$ and $x=\infty$ into four different Fuchsian singularities $x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}$ and $x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$, respectively. The perturbed equation is a second order reducible Fuchsian equation with 4 different singularities, i.e. a Heun type equation. Then we prove that when the perturbed equation has ex…
