不可积系统的Painleve分析与特解

固定参数下 Painlevé III 方程特殊函数解的渐近性质

Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters

本文利用 Bessel 函数的 Toeplitz 行列式表示与多重围道积分，计算了复平面上 Painlevé-III 方程特殊函数解的小 $x$ 和大 $x$ 渐近性质，并指出该表示可用于数值计算。

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlevé IV that admits special solutions. The BHC degenerates when either the Stokes' data for the irregular singularity at $\infty$ degenerates or the regular singular point at the origin becomes an apparent singularity. We show that if the BHC is written as isomonodromy family of biconfluent Heun equations (BHE), then the BHE degenerates precisely when it admits eigen-solutions of the biconfluent Heun operators, after choosing appropriate accessory paramet…

Painlevé-2方程的硬失稳

Hard loss of stability in Painlevé-2 equation

研究了带小参数的Painlevé-2方程的一个特殊渐近解.该解具有对应于分叉现象的临界点$t_*$。当$t<t_*$时，构造的解变化缓慢，而当$t>t_*$时，解振荡非常快。我们详细研究了过渡层，并获得了一个光滑的渐近解，使用一系列的缩放和匹配程序。

第三十四类Painleve方程特解的渐近性

Asymptotics for a special solution of the thirty fourth Painleve equation

在以前的文章中，我们研究了Z_{n,N} ^{-1}型酉随机矩阵系综的双尺度极限|\det M| ^{2α} e^{-N \Tr V（M）} dM，其中α> -1/2。因子|\det M| ^{2α}在原点附近诱导临界本征值行为。假设与V相关的极限平均特征值密度是正则的，并且原点是其支撑的右端点，我们计算了双标度极限中的极限特征值相关核为n，\to \infty使得n^{2/3}（n/N-1）= O（1），用Deift-Zhou最速下降方法求解Riemann-多项式在关于权|x| ^{2α} e^{-NV（x）}正交的直线上的Hilbert问题。我们的主要注意力是利用与Painleve XXXIV方程的一个特殊解u_α相联系的λ-函数构造原点附近的一个局部参数。这个解与Painleve II方程的一个特解有关，但它不同于通常的Hastings-McLeod解。本文将u_α（s）的渐近行为计算为s \to \pm \infty .我们猜想u_α的渐近性是由这一渐近性刻画的，并通过对包含u_α的Painleve XXXIV方程的一个单参数解族的渐近分析给出了支持性的论据.我们确定这个家庭作为家庭的tronquee解决方案的第三十四Painleve方程。

为什么我的有理 Painlevé V 解不是唯一的？

Why is my rational Painlevé V solution not unique?

本文研究 Painlevé V 方程在相同参数下出现多个有理解的非唯一性问题，基于 $A^{(1)}_{3}$ 仿射 Weyl 群平移算子形式体系推导发生条件，并识别种子解、给出系统构造闭式表达式的方法。

Classification of Real Solutions of the Fourth Painleve Equation

Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system associated with Painleve IV, as discussed in a recent paper), we give a number of results towards a classification of the real solutions of Painleve IV (or, more precisely, symmetric Painleve IV), according to their asymptotic behavior and singularities. We show the existence of globally nonsingular real solutions of symmetric Painleve IV for arbitrary nonzero v…

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real…

关于广义Freud权正交多项式的渐近性及其在Painlevé-IV特殊函数解中的应用

Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV

该研究推导了双切区域内广义Freud权正交多项式的渐近性，进而得到用抛物柱面函数表示的几类Painlevé-IV解的渐近公式，证明采用Riemann-Hilbert问题与Deift-Zhou非线性最速下降法。

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

We consider a family of tronquée solutions of the Painelvé II equation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2α+\frac12), \qquad α> -\frac12, \end{equation*} which is characterized by the Stokes multipliers $$s_1=-e^{-2απi },\quad s_2=ω, \quad s_3=-e^{2 απi} $$ with $ω$ being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if $ω=0$. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for $α> -1/2$ and $ω\geq 0$, with the constant terms evaluated explicitly. Our results …

Full-parameter discrete Painlevé systems from non-translational Cremona isometries

Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list. For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries. Translation elements of this symmetry group define discrete Painlevé equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlevé-t…