Painlevé II transcendents and their squares

Boutroux ansatz for the degenerate third Painlevé transcendents

For a general solution of the degenerate third Painlevé equation we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips along generic directions. The expression is obtained by using isomonodromy deformation of a linear system governed by the degenerate third Painlevé equation.

第五Painlevé超越函数的椭圆渐近的两个误差界

Two error bounds of the elliptic asymptotics for the fifth Painlevé transcendents

本文针对第五Painlevé方程在无穷远点的椭圆渐近展开，给出了误差项的显式渐近公式，并证明了其量级估计，同时推广到拉格朗日修正函数。

On WKB theoretic transformations for Painleve transcendents on degenerate Stokes segments

The WKB theoretic transformation theorem established in [KT2] implies that the first Painleve equation gives a normal form of Painleve equations with a large parameter near a simple P-turning point. In this paper we extend this result and show that the second Painleve equation (PII) and the third Painleve equation (PIII'(D7)) of type D7 give a normal form of Painleve equations on a degenerate P-Stokes segments connecting two different simple P-turning points and on a degenerate P-Stokes segment of loop-type, respectively. That is, any 2-parameter formal solution of a Painleve equation is reduc…

第一 Painlevé 超越函数的奇点

Singularities of the First Painlev{é} Transcendent

研究第一 Painlevé 超越函数的奇点位置和阶数，通过泰勒级数的比率测试方法进行分析，并探讨其在应用中的潜在价值。

An Asymptotic Reduction of a Painleve' VI equation to a Painleve' III (January 2011)

When the independent variable is close to a critical point, it is shown that PVI can be asymptotically reduced to PIII. In this way, it is possible to compute the leading term of the critical behaviors of PVI transcendents starting from the behaviors of PIII transcendents.

完全第三类 Painlevé 超越函数的椭圆渐近性

Elliptic asymptotics for the complete third Painlevé transcendents

本文利用线性系统的等单值形变与 WKB 分析，证明完全型第三 Painlevé 方程一般解在无穷远附近沿一般方向具有 Jacobi sn 函数形式的 Boutroux 椭圆渐近表示。

关于第六 Painlevé 超越函数双线性方程的研究

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。

The sixth Painleve transcendent and uniformizable orbifolds

We consider connection between the Painleve-6 equation and explicitly uniformizable orbifolds

Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $τ$-function of the Painlevé III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients…

Integral formulas for Painlevé-2 transcendent

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.

New Transformations for Painlevé's Third Transcendent

We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system $ B^{(1)}_2 $, to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the $τ$-functions with these parameter sets.

$q$-Painlevé 超越函数的椭圆渐近行为

Elliptic asymptotic behaviour of $q$-Painlevé transcendents

本文研究 $q$-Painlevé 超越函数在 $|q-1|\ll1$ 渐近区域中的椭圆函数极限行为，证明主导阶行为由椭圆函数描述，缓慢调制由完全椭圆积分近似。

On the algebraic independence of generic Painleve transcendents

We prove that if y" = f(y,y',t) is a generic Painleve equation from among the classes II to V then any collection of distinct solutions and their derivatives are algebraically independent over C(t). (Already proved by Nishioka for the single Painleve I equation). For generic Painleve VI we prove a slightly weaker statement.

Critical behaviours of the fifth Painlevé transcendents and the monodromy data

For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. It is also possible to compute the monodromy data in non-generic cases. Solutions of logarithmic type are derived from those of inverse logarithmic type through a Bäcklund transformation found by Gromak. In a special case the complex power type of solutions have relatively simple oscillatory expressions. For the comp…

Asymptotic behaviour of the third Painlevé transcendents in the space of initial values

We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlevé equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with essential singularity at infinity has an infinite number of poles and zeroes, and similarly at the origin.

Asymptotic behaviour of the fifth Painlevé transcendents in the space of initial values

We study the asymptotic behaviour of the solutions of the fifth Painlevé equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times.

Hirota bilinear equations for Painlevé transcendents

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

Asymptotic behaviour of the fourth Painlevé transcendents in the space of initial values

We study the asymptotic behaviour of solutions of the fourth Pain\-levé equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any non-special solution has an infinite number of poles and infinite number of zeroes.

Integrability, Random Matrices and Painlevé Transcendents

The probability that an interval $I$ is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval $I$ includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxilary quantities. We summarize and extend earlier work by expressing the probability and some of the auxilary quantities in terms of Painlevé transcendents.

A Review on The Sixth Painleve' Equation

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.

关于 Painlevé I 方程实解渐近性的研究

On the asymptotics of real solutions for the Painlevé I equation

本文利用 Riemann-Hilbert 方法重新研究实 Painlevé I 超越函数在自变量趋于负无穷时的渐近性，改进振荡型误差估计、给出奇异型精确误差估计，并建立相应 Hamilton 量的渐近性，同时修正文献中两处笔误。

椭圆渐近表示法中的第五型Painlevé超越函数

Elliptic asymptotic representation of the fifth Painlevé transcendents

本文通过Jacobi sn函数在类奶酪状带状区域内近似第五型Painlevé超越函数，其椭圆主部依赖于相位移作为单一积分常数，该常数由相关等蒙脱变形的单值数据参数化。

无界域上偏微分方程的数值方法

Numerical Approach to Painlevé Transcendents on Unbounded Domains

本文提出一种多域谱方法，用于研究无界域上偏微分方程的解，通过渐近级数近似解，无需在有限点截断级数。

Connection Formulae for Asymptotics of the Fifth Painlevé Transcendent on the Imaginary Axis: I

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as $t\to\imath\infty$ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE. $$ \frac{d}{dλ}Y= \left(\frac t2σ_3 + \frac{A_0}λ+\frac{A_1}{λ-1}\right)Y. $$ The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.

Tabulation of PVI Transcendents and Parametrization Formulas (August 17, 2011)

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.

Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

We modify the classical Paley-Wiener spaces $PW_x$ of entire functions of finite exponential type at most $x>0$, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points $z_1, ..., z_n$. We compute the reproducing kernels and relate their variations with respect to $x$ to a Krein differential system, whose coefficient (which we call the $μ$-function) and solutions have determinantal expressions. Arguments specific to the case where the "trivial zeros" $z_1, ..., z_n$ are in arithmetic progression on the imaginary axis allow us to e…

Monodromy of certain Painleve' VI transcendents and reflection groups

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…

通过零点与连接问题对实 Painlevé I 超越函数的分类：一项渐近研究

Classification of the real Painlevé I transcendents by zeros and connection problem: an asymptotic study

本文利用复 WKB 方法推导 Painlevé I 方程 Stokes 乘子的完整渐近展开，据此对实解按零点行为分类，并解决负向渐近与正零点位置之间的连接问题，从而在 $(r,b)$ 平面中建立精确相图。

On Painleve VI transcendents related to the Dirac operator on the hyperbolic disk

Dirac hamiltonian on the Poincare disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincare disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painleve VI equation with $γ=0$…

Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents

A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $τ$-function for certain Painlevé transcendents. The coefficients in this expansion are given in terms of conformal blocks of a two-dimensional conformal field theory, which can be written as infinite sums over pairs of partitions. In this note a discrete matrix model is proposed on a lattice whose partition function can be used to obtain a multiple integral representation for the length restricted partial sums of the Painlevé conformal blocks. This leads to expr…

Connection problem of the first Painlevé transcendent between poles and negative infinity

We consider a connection problem of the first Painlevé equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real $\mathrm{P_I}$ functions. We get a classification of the real $\mathrm{P_I}$ functions in terms of $(p,H)$ so that they behave differently at the negative infinity, where $p$ is the location of a pole and $H$ is the free parameter in the Laurent series. Some limiting-form connection formulas of $\mathrm{P_I}$ functions are obtained for large $H$. Specifically, for th…

Discrete Painlevé Equations

This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlevé equations, for which the developments of the last two decades have been rich and dynamic. These equations arise at the center of many viewpoints: random matrix theory, algebra, algebraic geometry, dynamical systems and the theory of transcendental functions. The purpose of this article is to reveal this confluence and modern perspectives on it.

The sixth Painleve transcendent and uniformization of algebraic curves

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…

Painlevé VI的Jimbo公式的一种新解释

A new interpretation of Jimbo's formula for Painlevé VI

该研究为Painlevé VI的Jimbo边界条件与单值公式提供新解释，前者对应秩3的Jimbo-Miwa-Mori-Sato方程长时间行为的收缩现象，后者可从非正则奇点的同单值形变角度理解。

Characteristic polynomials of complex random matrices and Painlevé transcendents

We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlevé transcendents, both at finite-$N$ and asymptotically as $N \to \infty$. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginib…

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。

Asymptotic Series for Some PainlevÉ VI Solutions

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.

关于第三Painlevé超越函数根的存在性与性质

On existence and properties of roots of third Painlevé' transcendents

本文研究第三Painlevé超越函数根的性质，通过哈密顿形式转化方程，证明非零点处零值解的存在性与解析性，构造根邻域近似解算法，推导根间及根与极点距离的下界。

Transition asymptotics for the Painlevé II transcendent

We consider real-valued solutions $u=u(x|s),x\in\mathbb{R}$ of the second Painlevé equation $u_{xx}=xu+2u^3$ which are parametrized in terms of the monodromy data $s\equiv(s_1,s_2,s_3)\subset\mathbb{C}^3$ of the associated Flaschka-Newell system of rational differential equations. Our analysis describes the transition, as $x\rightarrow-\infty$, between the oscillatory power-like decay asymptotics for $|s_1|<1$ (Ablowitz-Segur) to the power-like growth behavior for $|s_1|=1$ (Hastings-McLeod) and from the latter to the singular oscillatory power-like growth for $|s_1|>1$ (Kapaev). It is shown t…

Sub-normal Solutions to Painleve's Second Differential Equation

In a recent paper, Aimo Hinkkanen and Ilpo Laine proved that the transcendental solutions to Painleve's second differential equation w"=a+zw+w^3 have either order of growth 3 or else 3/2. We complete this result by proving that there exist no sub-normal solutions (order of growth 3/2) other than the so-called Airy solutions.

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…

Matching Procedure for the Sixth Painlevé Equation (May 2006)

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.

大初值下第一Painlevé超越函数的连接问题

Connection problem of the first Painlevé transcendents with large initial data

本文推广第一Painlevé方程初值问题研究，得到产生分界线解的初值曲线序列的渐近公式，用一致渐近方法推导其极限形式与非线性本征值，数值验证匹配效果优异。

Painlevé IV: roots and zeros

We consider the (real) fourth Painlevé equation in which both parameters vanish, analyzing the square-roots of its solutions and paying special attention to their zeros.

Projective structures, neighborhoods of rational curves and Painlev'e equations

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We deduce some transcendental result about Painlev'e equations.

Isomonodromy sets of accessory parameters for Heun class equations

In this paper, we consider the monodromy and, in particularly, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. While the isomonodromy sets of accessory parameters for the Heun class equations are described by the Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical v…

The Dirichlet Boundary Value Problem for Real Solutions of the first Painlevé Equation on Segments on Non-Positive Semi-Axis

We develop a qualitative theory for real solutions of the equation $y''=6y^2 -x$. In this work a restriction $x\leq0$ is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem $y(a)=y^0$, $y(b)=y_0$ for $a<b\leq0$.

霍尔曼德-伯恩哈德松常数的一个Painlevé方程

A Painlevé equation for the Hörmander-Bernhardsson constant

该研究确定霍尔曼德-伯恩哈德松常数与cosh-Gordon方程正则解的奇点相关，通过将多项式问题转化为Padé逼近问题，结合Deift-Zhou最速下降法与Painlevé超越函数完成分析。

Random Matrices with Merging Singularities and the Painlevé V Equation

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^α e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $α>-1/2$ and $t\in\mathbb R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of $α$, …

关于第四 $q$-Painlevé 方程的对称解

On symmetric solutions of the fourth $q$-Painlevé equation

本文研究第四 $q$-Painlevé 方程 $q\ extrm{P}_{\ extrm{IV}}$（即 $q{\ m P}(A_5^{(1)})$），证明其对称超越解的存在性，刻画对称性质并显式求解对应单值性问题。

A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential

We study the hermitian one matrix model with semi-classical potential. This is a general unitary invariant random matrix ensemble in which the potential has a derivative that is a rational function and the measure is supported on some collection of disjoint closed intervals. Such models have attracted much interest both due to their physical applications and relations to integrable systems. An object of central interest in random matrix theory is the correlation kernel, as this encodes the eigenvalue correlation functions. In recent years many results have been obtained proving that the correl…

Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of $P_I$

We show that the tritronquée solution of the Painlevé equation $\P1$, $ y"=6y^2+z$ which is analytic for large $z$ with $ \arg z \in (-\frac{3π}{5}, π)$ is pole-free in a region containing the full sector ${z \ne 0, \arg z \in [-\frac{3π}{5}, π]}$ and the disk ${z: |z| < 37/20}$. This proves in particular the Dubrovin conjecture, an open problem in the theory of Painlevé transcendents. The method, building on a technique developed in Costin, Huang, Schlag (2012), is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero within less than 1/10…