Some Rational Solutions to Painleve' VI
Gert Almkvist
从一个用超几何函数求解的Riccati方程出发，得到了Painleve' VI的有理解序列.
Starting with a Riccati equation solved by hypergeometric functions, some sequences of rational solutions to Painleve' VI are obtained.
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A Four-parametric Rational Solution to Painleve' VI
从Painleve' VI的一个有理解出发，利用Okamoto理论，得到了一个四参数有理解。
R. Fuchs' problem of the Painleve equations from the first to the fifth
我们将把把汇合型超几何方程变为Painleve型线性方程的所有有理变换从第一到第五类进行分类。我们表明，这种合理的转换对应于几乎所有的代数解的Painleve方程从第一至第五的Backlund变换。
Rational Solutions of the Painleve' VI Equation
本文对Painlevé VI方程的参数$α$，$β$，$γ$和$δ$的所有值进行分类，使得存在有理解.我们给出了一个公式，他们的双有理正则变换和对称的Painlevé VI方程。
Polynomial Solutions to Difference Equations Connected to Painleve' II-VI
与Painleve方程的有理解有关的多项式满足一定的差分方程。条件acertain所有的解决方案真的是多项式。
Computation of RS-pullback transformations for algebraic Painleve VI solutions
第六个Painleve方程的代数解可以使用超几何方程关于特殊分歧的有理覆盖的拉回变换来计算。特别是，正如第二作者和多兰所注意到的，一些代数解可以单独从有理覆盖中构造出来，而不需要计算拉回的Fuchsian方程。但是，相同的覆盖可以用来拉回不同的超几何方程，导致不同的代数Painleve VI解决方案。本文介绍了计算的显式RS-拉回变换，并推导出代数Painleve VI的解决方案。
Large-degree asymptotics of rational Painleve-II functions. II
本文是我们在arXiv：1310.2276中开始的关于非齐次Painleve-II方程的有理解和齐次耦合Painleve-II系统在大次极限下的相关有理解的分析的继续。在本文中，我们建立了渐近公式附近的某个曲线三角形在复平面，以前被证明是分开两种不同类型的渐近行为。我们的结果既显示三角退化的合理Painleve-II功能，也退化的tritronquee解决方案的Painleve-I方程。我们严格的分析是基于最速下降的方法适用于Riemann-Hilbert表示的理性Painleve-II功能，并提供领先的顺序公式以及误差估计。
Large-degree asymptotics of rational Painleve-II functions. I
非齐次Painleve-II方程和相关耦合Painleve-II系统的有理解最近出现在流体涡旋和sine-Gordon方程的研究中。对于正弦戈登应用程序，特别是它是感兴趣的理解大程度的渐近行为的理性Painleve-II功能。我们显式计算这两个家庭的有理函数有效的在整个复杂的平面，除了一个特定的分段光滑的封闭曲线的邻域的领导阶大的程度渐近。利用Deift-Zhou非线性最速下降法求解Riemann-Hilbert问题，得到了严格的误差界.
Painlevé Property and Generating Functions for Asymptotics
本文提出利用双变量形式级数分析Painlevé方程的渐近性，重点研究第三退化Painlevé方程的渐近行为。
The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann-Hilbert…
Remarks Towards the Classification of $RS_4^2(3)$-Transformations and Algebraic Solutions of the Sixth Painlevé Equation
我们引入了有理函数的除数型的概念，并通过$RS$ -变换的方法证明了它可以有效地用于与第六类Painlevé方程的代数解的构造有关的Dessins d 'enfants变形的分类.
All of the six Painlevé equations except the first have families of rational solutions, which are frequently important in applications. The third Painlevé equation in generic form depends on two parameters $m$ and $n$, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as $n\in\mathbb{Z}$ increases and how the patterns vary with $m\in\mathbb{C}$. This study suggests that it is reasonable to consider the rational solutions in the li…
The Painleve-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed n by m rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as m and n tend to infinity with r=m/n fixed. Generalizing a method of Bertola and Bothner used to study rational Painleve-II func…
Discrete equations from Bäcklund transformations of the fifth Painlevé equation
本文从第五Painlevé方程的Bäcklund变换导出离散方程，包括一个具有三元对称性的新离散方程，并利用广义Laguerre多项式和广义Umemura多项式推导出离散方程的有理解层次，同时利用非唯一有理解对得到满足同一离散方程的不同有理解层次。
A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation
本文扩展Taneda方法，为第三类Painlevé方程的Umemura多项式提供构造性代数证明，证实非线性递推生成的有理函数实为多项式，并揭示其根的信息。
Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlevé Equations
研究了黎曼球面上具有单极点的有理矩阵函数的分解问题。对于二次的情况下（两个极点），我们表明，使用乘法表示这样的矩阵函数，一个很好的坐标系在这个空间是由混合的剩余特征向量的矩阵和它的逆。我们的方法是由离散的isomonodromic变换的理论和它们与差分潘勒韦方程的关系。特别是，在这些坐标系中，基本的isomonodromic变换采用离散欧拉-拉格朗日方程的形式。其次，我们表明，dPV方程，以前得到的D。Arinkin和A. Borodin，可以理解为这种矩阵的剩余和它们的逆之间的简单关系。
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asym…
These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation PIV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlevé hierarchy. The construction is based on dressing ch…
Painleve I, Coverings of the Sphere and Belyi Functions
Painleve-I的解的极点理论等价于Nevanlinna问题，即构造一个在五个点上分歧的亚纯函数--计算多重性--并且没有临界点。我们构造了这样的亚纯函数作为有理函数的极限。在tritronquee解决方案的情况下，这些合理的功能是Belyi功能。
Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
本文研究第五潘勒韦方程的有理解，用广义拉盖尔多项式表示解，确定其性质、微分-差分方程、根结构与判别式，并给出朗斯基拉盖尔多项式系数的显式表达式。
Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them
研究了第二类Painlevé方程的有理解的特殊多项式--Yablonskiii-Vorob 'ev多项式。这些多项式的系数的整除性质，涉及4的权力，我们得到的Yablonskiii-Vorob 'ev多项式的非零根是无理的。进一步，通过对第二类Painlevé方程的有理解进行幂级数展开，得到了这些多项式的连续次数根之间的关系.
Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.
根据D. G.丘德诺夫斯基并给出了实例。例如，与{ \blackb R\/}上的权重$\exp(-x^4/4-tx^2)$相关的正交多项式的$a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$中的递归系数满足$4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$，并且$a_n^2$满足Painlevé ${\rm P}_{\rm IV}$方程。
In this paper we study different Hamiltonian systems with polynomial and rational Hamiltonians associated with the generic third Painlevé equation and present explicit birational transformations relating them.
It is shown how to calculate asymptotics of integrals over the positive semi-axis of two functions related to the Degenerate Third Painlevé Equation (dP3). As an example, the corresponding results for the meromorphic solution of the dP3 vanishing at the origin are presented.
As a new application of the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod, we provide a simpler and more rigorous proof of the connection formulas of some special solutions of the fifth Painlevé equation, which have been established earlier by Andreev and Kitaev.
On Algebraic Solutions to Painleve VI
我们宣布的一些结果，这可能会带来一个新的洞察力的分类代数解决方案的第六Painleve方程。主要结果包括参数的合理性，三角丢番图条件，以及作者所谓的四面体定理在某些情况下没有代数解。该方法是基于富有成效的互动之间的模量理论制定Painleve VI和动态字符品种通过黎曼-希尔伯特对应。
In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlevé equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for…
Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type
本文通过几何方法识别准庞加莱型哈密顿系统，利用双有理变换分析其全局结构。
The Painlevé-III equation with parameters $Θ_0=n+m$ and $Θ_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\to+\infty$ with $m\in\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is filled with …
The D7 degeneration of the Painleve-III equation has solutions that are rational functions of $x^{1/3}$ for certain parameter values. We apply the isomonodromy method to obtain a Riemann-Hilbert representation of these solutions. We demonstrate the utility of this representation by analyzing rigorously the behavior of the solutions in the large parameter limit.
Painleve Equations and Complex Reflections
我们将解释第六Painleve方程的一些新的代数解是如何从复反射群中产生的，从而推广了Hitchin和Dubrovin-Mazzocco关于实反射群的一些结果。找到这些解的显式公式的问题将在别处讨论。
We study some properties of tau-functions of an isomonodromic deformation leading to the fifth Painlevé equation. In particular, here is given an elementary proof of Miwa's formula for the logarithmic differential of a tau-function.
This paper first discusses irreducibility of a Painlevé equation $P$. We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian $\mathbb{H}$ to a Painlevé equation $P$. Complete integrability of $\mathbb{H}$ is shown to imply that all solutions to $P$ are classical (which includes algebraic), so in particular $P$ is solvable by ''quadratures''. Next, we show that the variational equation of $P$ at a given algebraic solution coincides with the normal variational equation …
Boundary conditions associated with the Painlevé III' and V evaluations of some random matrix averages
在以前的工作中的随机矩阵平均拉盖尔酉系综，概括的概率，间隔$ (0,s) $在硬边包含$ k $特征值的生成函数，进行了评估方面的Painlevé V超越$ σ$ -形式。然而，相应的微分方程的边界条件没有指定的完整的参数空间。在这里，这一任务是完成一般，并获得的功能形式进行比较，对最一般的小$ s $行为的Painlevé V方程的$ σ$ -形式已知的工作Jimbo。一个类似的研究进行了硬边缩放限制的随机矩阵的平均值，我们以前评估的Painlevé \IIId超越$ σ$形式。应用后者的结果给出了快速评估的汉克尔行列式出现在最近的工作康利，鲁宾斯坦和Snaith有关的衍生物的黎曼ζ函数。
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\} $.
Global Asymptotics of the Second Painlevé Equation in Okamoto's Space
我们研究了第二类潘勒韦方程在冈本首先构造的初始条件空间中的解，在自变量x趋于无穷大的极限中.同时，我们研究的相关方程的解决方案被称为第三十四潘勒韦方程。通过考虑退化的情况下，自治流，我们恢复已知的特殊解决方案，这是有理函数或艾里函数的表达。我们证明了在无穷远处不为零的解具有无穷多个极点。我们建筑的一个基本要素是证明特殊线条的结合是冈本空间动态的排斥剂。此外，我们还证明了解的极限集的存在性，并且是紧的和连通的。
Picard and Chazy solutions to the Painleve' VI equation
我研究了一个特殊的家庭的Painlevé VI方程的参数$β=γ=0, δ=1/2$和$2α=(2μ-1)^2$的解决方案，为$2μ\in\interi$。我表明，半整数$μ$的情况下是可积的，解决方案有两种类型：所谓的皮卡德解决方案和所谓的Chazy解决方案。给出了它们的显式公式，并完全确定了它们在奇点$0,1,\infty$附近的渐近行为和它们的非线性单值性。本文研究PVI $μ$方程解的解析延拓的结构，对于任何$μ$使得$2μ\in\interi$。作为一个应用，我对所有的代数解进行了分类。对于$μ$半整数，证明了它们与平面上的正多边形或星形多边形一一对应。对于$μ$整数，证明了所有的代数解都属于一个单参数有理解族。
Explicit solution of the problem of equivalence for some Painleve equations
对于任意二阶常微分方程，构造了一个检验，检验该方程在变量替换下是否等价于Painleve Ⅰ、Ⅱ或Painleve Ⅲ的三个零参数方程.如果这是真的，那么在Painleve方程I和II的情况下，给出了使用方程的微分不变式写出的变量的显式变化。
The Liénard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Liénard equation and some equations from the Painlevé--Gambier classification. We show that with the help of such connections one can construct general analytical solutions of the Liénard equation's subfamilies. In particular, we find three new integrable families of the Liénard equation. We also propose and discuss an approach for finding one--parameter families of closed--form analytical solutions of the Lié…
Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and N-recurrences for averages over U(N) -- \PIIIa and \PV $τ$-functions
在这项工作中，我们表明，$ N\times N $ Toeplitz行列式的符号$ z^μ\exp(-{1/2}\sqrt{t}(z+1/z)) $和$ (1+z)^μ(1+1/z)^ν\exp(tz) $ -已知的$τ$ -功能的\PIIIa和\PV系统-其特征在于非线性递归的反射系数的相应的正交多项式系统的单位圆。结果表明，这些递推关系完全等价于与有理曲面$ D^{(1)}_{6} \to E^{(1)}_{7} $（离散Painlevé { \rm II}）和$ D^{(1)}_{5} \to E^{(1)}_{6} $（离散Painlevé { \rm IV}）退化相关的离散潘勒韦方程.此外，它示出的差分方程推导的方法的基础上的Toeplitz格和Virasoro的约束，减少时，为了精确求和，相当于我们的递归。分别给出了反射系数的广义超几何函数$ {{}^{\vphantom{(1)}}_0}F^{(1)}_1, {{}^{\vphantom{(1)}}_1}F^{(1)}_1 $的表达式。
Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions
该研究通过Bäcklund变换生成Painlevé-III($D_6$)的解族，结合级数分析与Riemann-Hilbert方法证明其在大参数标度下收敛于Painlevé-III($D_8$)的解，并推导了相关有理解、Umemura多项式的渐近性质。
In this paper some open problems for Painlevé equations are discussed. In particular the following open problems are described: (i) the Painlevé equivalence problem; (ii) notation for solutions of the Painlevé equations; (iii) numerical solution of Painlevé equations; and (iv) the classification of properties of Painlevé equations.
A note on the R. Fuchs's problem for the Painlevé equations
本文考虑一阶完全可积系统偏微分方程组$\partial \Fi/partial x=A(x, t) \Fi, \partial \Fi/partial t=B(x, t) \Fi$与$\Fi=(\fi_1, \fi_2)^τ$，其中$A(x, t)$与$B(x, t)$是2 × 2全纯矩阵函数.在某些假设下，我们发现一个变量的变化，系统$\partial \Fi/\partial x=A(x, t) \Fi$减少到一个方程的变量$t$无关。作为一个应用，我们证明了R. Fuchs对Painlevé方程的猜想对某些代数解是正确的。
Ultradiscrete Painlevé VI with Parity Variables
我们引入了一个超离散化的奇偶变量的$q$ -差分Painlevé VI方程组。证明了$q$ -Painlevé VI方程的Riccati型解的超离散极限满足带奇偶变量的Painlevé VI方程组方程组，并证明了在奇偶变量的情况下，超离散极限是成立的.研究了超离散Riccati型方程和超离散Painlevé VI型方程的解.
Integrability, Random Matrices and Painlevé Transcendents
在具有酉对称性的矩阵系综中，区间$I$没有特征值的概率由Fredholm行列式给出。当矩阵系综中的权函数是经典权函数，且区间$I$包含支撑的端点时，Tracy和Widom给出了一种形式，给出了所需概率和一些辅助量的耦合微分方程。我们总结和扩展以前的工作表示的概率和一些辅助量的Painlevé超越。
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…
Six results on Painleve VI
在回顾了第六个Painleve方程的一些几何之后，我们将描述冈本对称性是如何从施莱辛格方程的对称性中自然产生的，并总结了柏拉图Painleve六解的分类。一个关键的观察是，Painleve VI管理的isomonodromic变形的某些Fuchsian系统的秩\emph{three}束。
It is known that the Painleve VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear differential equations. We inverstigate degenerations of the Ruijsenaars-van Diejen difference opearators and show difference analogues of the Painleve-Heun correspondence.
On real and imaginary roots of generalised Okamoto polynomials
本文推导了广义Okamoto多项式实根与虚根个数的精确公式，并据此给出Sasa-Satsuma方程的部分怪波列表及第四Painlevé方程在实轴或虚轴上无极点的有理解族，同时提出一种基于Bäcklund变换的奇点分布算法化程序。
We investigate the structure of $τ$-functions for the elliptic difference Painlevé equation of type $E_8$. Introducing the notion of ORG $τ$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $τ$-functions associated with the configuration of generic nine points in the projective plane.
Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answer…
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