Painlevé IV: roots and zeros

On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials

We study the real roots of the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation. It has been conjectured that the number of real roots of the nth Yablonskii-Vorob'ev polynomial equals [(n+1)/2]. We prove this conjecture using an interlacing property between the roots of the Yablonskii-Vorob'ev polynomials. Furthermore we determine precisely the number of negative and the number of positive real roots of the nth Yablonskii-Vorob'ev polynomial.

第三类Painlevé方程的Umemura多项式的构造性证明

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。

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。

Geometric Aspects of Painlevé Equations

In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on $\mathbb{P}^1\times\mathbb{P}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax p…

Existence of Nontrivial Negative Resonances for Polynomial Ordinary Differential Equations With Painlevé Property

The Painlevé classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification still remains an open problem especially for the higher-order equations. One of the main classical methods of Painlevé analysis is based on considering the resonance numbers corresponding to the possible indices of arbitrary coefficients in the Laurent expansion of the general solution in a neighborhood of a movable singularity. Complex and non-integer values of…

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.

Global asymptotics of the sixth Painlevé equation in Okamoto's space

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.

Simply-laced isomonodromy systems

A new class of isomonodromy equations will be introduced and shown to admit Kac-Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painleve equations, and shows where such Kac-Moody root systems occur "in nature". A key point is that one may go beyond the class of affine Kac-Moody root systems. As examples, by considering certain hyperbolic Kac-Moody Dynkin diagrams, we find there is a sequence of higher order Painleve systems lying over each of the classical Painleve equations. This leads to a conjecture about the Hilbert scheme of po…

Exceptional solutions to the Painlevé VI equation

We find all solutions of the Painlevé VI equations with the property that they have no zeros, no poles, no 1-points and no fixed points.

Exponentially-improved asymptotics and numerics for the (un)perturbed first Painlevé equation

The solutions of the perturbed first Painlevé equation $y"=6y^2-x^μ$, $μ>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on $C$, for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the …

Asymptotics of recurrence coefficients for the Laguerre weight with a singularity at the edge

In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the Deift-Zhou steepest descent method. The asymptotic formulas of the leading coefficients and the recurrence coefficients for large n are described in terms of a class of analytic solutions to the the σ-form of the Painlevé II equation and the Painlevé XXXIV equation.

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.

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.

Tronquée solutions of the Painlevé equation \P1

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…

第五潘勒韦方程的有理解。广义拉盖尔多项式

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。

星形Kac-Moody根系分类根的算法

Algorithm classifying roots of star-shaped Kac-Moody root systems

本文针对星形Kac-Moody根系提出一种有效算法，用于获取具有给定范数的根的Weyl群轨道代表元并实现为计算机程序，同时揭示了这些轨道与Fuchsian微分方程及高维Painleve型方程之间的关系。

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II

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…

Painlevé VI equations with algebraic solutions and family of curves

In families of Painlevé VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families of algebraic varieties. In our case, we have one family with zero dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers done by Herfurtner in 1992 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005.

关于广义Okamoto多项式的实根与虚根

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。

Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials

We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For $t>0,$ the factor $\rex^{-t/x}$ induces an infinitely strong zero at $x=0.$ With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a syst…

On the Increasing Tritronquée Solutions of the Painlevé-II Equation

The increasing tritronquée solutions of the Painlevé-II equation with parameter $α$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|<\tfrac{2}{3}π$ and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of $α$. Here these solutions are investigated from the point of view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex $α$, all such solutions are asymptotically p…

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.

Roots of generalised Hermite polynomials when both parameters are large

We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degenerations. Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.

Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics

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 Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type

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…

Large-degree asymptotics of rational Painleve-IV solutions by the isomonodromy method

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…

The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles

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 …

Painlevé II τ函数的Fredholm行列式表示

Fredholm determinant representation of the Painlevé II $τ$-function

本文通过将Painlevé II方程的通用τ函数表示为可积算子的Fredholm行列式，揭示了其零点轨迹（Malgrange除子）与Fredholm行列式零点之间的对应关系。

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…

Quantum curves and $q$-deformed Painlevé equations

We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve these are conjectured to be the q-difference Painlevé equations as in Sakai's classification. More precisely, we propose that the tau-functions of q-Painlevé equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau-functions in ter…

An Isomonodromy Cluster of Two Regular Singularities

We consider a linear $2\times2$ matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations. In our main example the latter limit reads as $P_6\to P_5$, where $P_n$ is the $n$-th Painlevé equatio…

Painlevé方程近似解的微分方程：在Painlevé-III $({\rm D}_7)$方程代数解中的应用

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

本文利用Alexander Its提出的方法，从外参数矩阵的简单条件直接导出极限微分方程，建立Painlevé-III (D$_7$)代数解的局部近似与Weierstrass方程的联系。

Painlev'e 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas

The goal of this article is to prove that the determinantal formulas of the Painlev'e 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary non-zero monodromy parameter. The result is established for two different Lax pairs associated to the Painlev'e 2 system, namely the Jimbo-Miwa Lax pair and the Harnad-Tracy-Widom Lax pair, whose spectral curves are not connected by any symplectic transformation. We provide a new method to prove the topological type property without using the insertion operators. In the process, ta…

Rational Solutions of the Painlevé-III Equation

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…

"Quantum" linearization of Painlevé equations as a component of their $L,A$ pairs

The procedure of the "quantum" linearization of the Hamiltonian ordinary differential equations with one degree of freedom is introduced. It is offered to be used for the classification of integrable equations of the Painleve type. By this procedure and all natural numbers $n$ we construct the solutions $Ψ(\hbar,t,x,n)$ to the non-stationary Shrödinger equation with the Hamiltonian $H = (p^2+q^2)/2$ which tend to zero as $x\to\pm\infty$. On the curves $x=q_n (\hbar, t) $ defined by the old Bohr-Sommerfeld quantization rule the solutions satisfy the relation $i\hbar Ψ'_x\equiv p_n (\hbar, t) Ψ$…

广义Gauss-Rys正交多项式

Generalized Gauss-Rys orthogonal polynomials

本研究针对对称线性泛函下的首一正交多项式序列$(P_n(x;z;λ))_{n\geq 0}$，结合其三项递推关系的参数特性，分析广义Gauss-Rys正交多项式的性质，给出递推系数渐近展开、关联的离散Painlevé与Painlevé方程，以及零点的静电诠释和参数动力学。

Painlevé-Kuratowski收敛的应用：具有收敛Clarke子微分的Lipschitz函数及由收敛方程定义的集合的收敛

Applications of the Painlevé-Kuratowski convergence: Lipschitz functions with converging Clarke subdifferentials and convergence of sets defined by converging equations

本文探讨了Painlevé-Kuratowski收敛在分析中的应用，包括Lipschitz函数的扩展、平方距离函数的逆定理以及函数纤维行为的研究，提供了实分析中Hurwitz定理的推广及代数集收敛性问题的部分结果。

The asymptotic zero distribution of Lommel polynomials as polynomials of the order with a variable complex argument

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved.

sinh-Gordon Painlevé III 方程实解的过渡渐近

Transition asymptotics for the real solutions of the sinh-Gordon Painlevé III equation

研究 sinh-Gordon Painlevé III 方程实解在参数和变量同时趋于无穷时的过渡行为，揭示奇异解与光滑解之间的渐近转换。

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…

实值零均值映射的根：线性泛函与等变映射的复合

Roots of real-valued zero mean maps: Compositions of linear functionals and equivariant maps

本文提出一个拓扑框架，用于约束零均值实值映射（由等变映射与线性泛函复合而成）的零点分布，建立三角多项式不变号集合系统的拓扑上界，并推导高效求积规则的存在性。

Zeros of a growing number of derivatives of random polynomials with independent roots

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $μ$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $μ$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.

Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity

In the present paper we derive a new Hankel determinant representation for the square of the Vorob'ev-Yablonski polynomial $\mathcal{Q}_n(x),x\in\mathbb{C}$. These polynomials are the major ingredients in the construction of rational solutions to the second Painlevé equation $u_{xx}=xu+2u^3+α$. As an application of the new identity, we study the zero distribution of $\mathcal{Q}_n(x)$ as $n\rightarrow\infty$ by asymptotically analyzing a certain collection of (pseudo) orthogonal polynomials connected to the aforementioned Hankel determinant. Our approach reproduces recently obtained results in…