Lax forms of the $q$-Painlevé equations
Mikio Murata
由第六个潘勒韦方程的$q$ -类似物得到的所有$q$ -潘勒韦方程都用Lax形式表示。它们的特征在于相关的线性$q$差分方程的数据。文中还给出了$A_2$型潘勒韦方程的退化形式.
All $q$-Painlevé equations which are obtained from the $q$-analog of the sixth Painlevé equation are expressed in a Lax formalism. They are characterized by the data of the associated linear $q$-difference equations. The degeneration pattern from the $q$-Painlevé equation of type $A_2$ is also presented.
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We present a Lax pair for the sixth Painlevé equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlevé equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlevé equation and the various degenerate versions of the third Painlevé equation.
A $3 \times 3$ Lax Form for the $q$-Painlevé Equation of Type $E_6$
本文为 $E_6^{(1)}$ 型 $q$-Painlevé 方程构造了新的 $3\ imes 3$ 矩阵 Lax 形式及三阶标量方程，并探讨了连续极限。
On an integrable system of q-difference equations satisfied by the universal characters: its Lax formalism and an application to q-Painleve equations
泛特征标是Schur函数附加到一对划分上的推广。研究了一个由泛特征标所满足的q-差分方程的可积系统，它是q-KP族的推广，称为格q-UC族.我们描述格Q-UC层次作为其相关的线性系统（拉克斯形式主义）的兼容性条件，并探讨通过相似性降低的应用程序的Q-Painleve方程。特别是高阶模拟的q-Painleve VI方程。
On the Lax pairs of the sixth Painleve' equation
Painleve'的第六个方程对它的四个参数$(2 α,-2 β,2 γ,1-2 δ) =(θ_{\infty}^2,θ_{0}^2,θ_{1}^2,θ_{x}^2)$的依赖性是全纯的，因此人们期望它的所有拉克斯对都表现出这样的依赖性。这确实是Fuchs的二阶标量"Lax“对的情况，但是Jimbo和Miwa的二阶矩阵拉克斯对呈现出对$θ_\infty$的亚纯依赖（以及对其他三个$θ_j$的全纯依赖）。我们分析了此功能的原因并提出了抑制它的建议。
The coupled hirota equation with a 3*3 lax pair: painleve-type asymptotics in transition zone
针对具有3×3 Lax对的耦合Hirota方程，本文用Riemann-Hilbert方法和非线性最速下降分析证明其过渡区渐近首阶由耦合Painlevé II方程描述。
Construction of the Lax pairs for the delay Lotka-Volterra and delay Toda lattice equations and their reductions to delay Painleve equations
本文构造延迟Lotka-Volterra和延迟Toda格方程的Lax对及守恒量，通过空间2-周期约化推导延迟Painleve方程，并给出其自洽版本的行列式解。
For all non-equivalent matrix systems of Painlevé-4 type found by authors in arXiv:2107.11680, isomonodromic Lax pairs are presented. Limiting transitions from these systems to matrix Painlevé-2 equations are found.
It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of differential and difference systems. The characteristic equations in the case of the associated linear problems for various discrete Painlevé equations is biquadratic in the Painlevé variables. We show that the discrete isomonodromic deformations that define the discrete Painlevé equations may be succinctly described in terms of the characteristic equation of their La…
In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the $A_5^{(1)}$-surface $q$-Painlevé system which has the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$. Two new Lax pairs are found.
Towards second order Lax pairs to discrete Painlevé equations of first degree
本文研究了六个离散潘勒韦方程（Pn）的离散拉克斯对的求解问题.我们所做的选择是离散的卡尼尔对，一旦转换为矩阵形式。
A Lax pair for the additive difference Painlevé equation of type $E_7^{(1)}$ is explicitly obtained as certain linear difference equations of scalar form. The compatibility of the Lax pair is proved by using certain characterization of the coefficients in the Lax equation. Some Lax pairs for types $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$ are also given by the degeneration.
We wish to explore a link between the Lax integrability of the $q$-Painlevé equations and the symmetries of the $q$-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the $q$-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a Bäcklund transformation of the $q$-Painlevé equation. Each translational Bäcklund transformation may be lifted to the level of the associated linea…
Lax pairs, Painlevé properties and exact solutions of the alogero Korteweg-de Vries equation and a new (2+1)-dimensional equation
我们证明了Calogero Korteweg-de Vries（CKdV）方程的Lax对的存在性。此外，在（2+1）维情形下，我们对CKdV方程的Lax对中的T算子进行了修正，从而提出了一个新的（2+1）维方程.我们称之为（2+1）维CKdV方程。我们证明了CKdV方程和（2+1）维CKdV方程在具有Painlevé性质的意义下是可积的.构造了一些精确解。
The 8-parameter elliptic Sakai difference Painlevé equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrödinger equation for the $BC_1$ 8-parameter 'relativistic' Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlevé-Calogero correspondence to the highest level in the two hierarchies.
This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this article we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the non-magnetic represent…
Quantum Painlevé II Lax Pair and Quantum (Matrix) Analogues of Classical Painlevé II equation
提出显含 $\hbar$ 与任意场变量的量子 Painlevé II Lax 对，其相容性导出量子方程及对易关系，并可约化到经典与矩阵形式。
A discretization of Painlevé VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point of view. It turns out that the second approach is also successful and gives the same quantization as in the first approach.
We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions. A main result is, as an example of this framework, how we may obtain the q-Painlevé equation whose group of Bäcklund transformations is an affine Weyl group of type E_6^{(1)} as a similarity reduction of the discrete Schwarzian Korteweg-de Vries equation.
Holomorphic, rational Lax pairs of a $q$-discrete Painlevé VI equation
针对 $q$-离散 Painlevé VI 方程，本文构造若干在单值矩阵无穷远留数特征值相等时仍有效的全纯有理矩阵 Lax 对。
On difference-differential Lax pairs and integrals of Painlevé equations in finite characteristic
该研究收集经典Painlevé方程的二阶差分-微分Lax对并转化为特定矩阵形式，通过通用方法从这些Lax对获取特征p下的运动积分，为每个相关Painlevé方程生成可数积分列表。
We consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix valued version of the Lenard operators. In particular, the Riemann-Hilbert technique used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitely written in terms of these matrix valued Lenard operators and some solution of the hierarchy a…
Painleve analysis, Backlund transformation, Lax pair and periodic wave solutions for a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation in fluid mechanics
本文针对流体力学中的广义(2+1)维HSI方程，通过Painleve分析证实其特定条件下的可积性，构造双线性形式等关键数学结构，还推导并展示了周期波解。
Two-component soliton systems and the Painlevé equations
通过考虑额外的时间变量，我们给出了两个组件的KP层次的扩展。我们得到的线性$2\times 2$系统，考虑到通过减少过程的层次。由这个线性系统给出了Schlesinger系统的拉克斯对和第六类潘勒韦方程。还考虑了从通常的两分量KP族中处理其他潘勒韦方程的统一方法。
We consider two Lax systems for the homogeneous Painlevé II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and one of size $4\times 4$ introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in the early 2010's. We prove that solutions to the $4\times 4$ system can be derived from those to the $2\times 2$ system via an integral transform, and consequently relate the Stokes multipliers for the two systems. As corollaries we are able to express two kernels for determinantal processes as contour integrals involving the Flaschka-Newell Lax system: the tacnode…
Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation
由二阶双线性微分差分方程定义的Yablonskiii-Vorob 'ev多项式$y_{n}(t)$提供Toda格的有理解。它们也是第二类Painlevé方程（$P_{II}$）的有理解的多项式τ函数。在这里，我们定义了两个变量的多项式$Y_{n}(t,h)$上的晶格间距$h$，考虑合理的解决方案的离散时间Toda格介绍了苏里斯。这些多项式被证明有许多属性是类似的Yablonskiii-Vorob 'ev多项式，他们减少时，$h=0$。他们还提供了合理的解决方案，为一个特定的离散$P_{II}$，即所谓的{ \it交替离散} $P_{II}$，这种连接导致一个表达式的Umemura多项式的第三Painlevé方程（$P_{III}$）。证明了交替离散Painlevé方程的贝克隆变换是一个辛映射，时间平移也是辛的.最后，我们提出了交替离散$P_{II}$，恢复Jimbo和Miwa的Lax对$P_{II}$在连续极限$h\to 0$的Lax对。
Given the second order scalar Lax pair of the sixth Painlevé equation, we build a generalized heat equation with rational coefficients which does not depend any more on the Painlevé variable. Étant donné la paire de Lax scalaire de la sixième équation de Painlevé, nous donnons une construction directe de l'équation de la chaleur généralisée à coefficients rationnels qui ne dépend plus de la variable de Painlevé.
Painleve equations from Darboux chains - Part 1: P3-P5
我们证明了Painleve方程P3-P5可以从具有二次特征值依赖性的薛定谔问题的周期性达布变换序列中（以统一的方式）导出。一般问题自然分为三个不同的分支，每个分支都由一个无限的方程链描述。Painleve方程是通过在最低非平凡水平上周期性地闭合链而获得的。链提供“对称形式”的Painleve方程，从广田双线性形式和拉克斯对的派生。在本文（第一部分）中，我们详细分析了P3-P5的情况，而P6将在第二部分中研究。
Starting from the diffusion equation at beta random matrix hard edge obtained by Ramirez and Rider (2008), we study the question of its relation with Lax pairs for Painleve III. The results are in many respects similar to the ones found for soft edge by Bloemendal and Virag (2010). In particular, the values beta = 2 and 4 (but not beta = 1) allow for a simple connection with Painlevé III solutions and Lax pairs. However, there is an additional surprise for a special relation of parameters where a simple solution of the diffusion equation can be obtained, which is a one-parameter generalization…
Quivers and difference Painleve equations
我们将描述具有E6、E7和E8型仿射Weyl对称群的差分Painleve方程的自然“Lax对”，表明它们确实是作为某些Fuchsian微分方程系统的对称性而出现的。
Nous montrons que les équations du repère mobile des surfaces de Bonnet conduisent à une paire de Lax matricielle isomonodromique d'ordre deux pour la sixième équation de Painlevé. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painlevé equation.
This Letter presents a reduction of the lattice modified Korteweg-de-Vries equation that gives rise to a $q$-analogue of the sixth Painlevé equation. This new approach allows us to give the first ultradiscrete Lax representation of an ultradiscrete analogue of the sixth Painlevé equation.
Gauge Transformations and Weak Lax Equation
本文从SL（2，R）不变量规范理论的观点出发，考虑几个可积系统。在Drinfeld-Sokorov规范中，我们从零曲率条件得到了一个单参数非线性方程族。对于每个值的参数方程描述的弱Lax方程。它被转化为一组耦合方程，通过Painlevé测试，是可积的参数的任何整数值。对方程组进行连续的规范变换（Miura变换），我们得到一系列非线性方程。
In this article we prove that Lax pairs associated with $\hbar$-dependent six Painlevé equations satisfy the topological type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hbar$-expansion of the isomonodromic $τ$-function and of the determinantal formulas by applying the so-called topological recursion to the spectral curve attached to the Lax pair in all six Painlevé cases. Finally we illustrate the former results with the explicit computations of the first orders of the six $τ$-fun…
A non-commutative discrete first Painlevé hierarchy: the Lax pair approach
利用与离散第一Painlevé层级相关的等单子问题的非交换类比，构造了该层级的非交换版本d-PI_m^{nc}，并证明了两个层级均可由Svinin多项式S_s^k(n)表示，同时推导了非交换Volterra晶格层级到d-PI_m^{nc}层级的约化，给出了前三个成员的显式连续极限。
In this paper, we construct a new relation between ABS equations and Painlevé equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painlevé equations.
Finite-dimensional reductions of the discrete Toda chain
考虑离散Toda链的可积边界条件的构造问题。适当选择的封闭条件的限制链减少到众所周知的离散潘勒韦方程$dP_{III}$，$dP_{V}$，$dP_{VI}$。找到了这些离散潘勒韦方程的Lax表示。
The Painlevé--Kovalevskaya test is applied to find three matrix versions of the Painlevé II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to construct isomonodromic Lax pairs for them.
Nonisospectral deformations of noncommutative Laurent biorthogonal polynomials and matrix discrete Painlevé-type equations
本文通过非交换洛朗双正交多项式的非等谱变形建立其与矩阵离散潘勒韦方程的联系，并导出矩阵离散潘勒韦型方程。
Linearization And Solutions Of The Discrete Painlevé-III Equation
给出了有理特殊函数（Bessel）型离散Painlevé III（d-P $\rm_{III}$）方程的特解.这些解决方案使我们能够建立一个密切的平行之间的离散方程和连续的副本。此外，我们提出了d-P $\rm_{III}$的另一种形式，并通过明确推导其拉克斯对来证实其可积性。
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…
In this paper, we construct two lattices from the $τ$ functions of $A_4^{(1)}$-surface $q$-Painlevé equations, on which quad-equations of ABS type appear. Moreover, using the reduced hypercube structure, we obtain the Lax pairs of the $A_4^{(1)}$-surface $q$-Painlevé equations.
Principal models on a solvable group with nonconstant metric
导出了具有非常数度规的广义主模型的场方程，并给出了它们的拉克斯对的解析表达式.定义拉克斯对的方程求解最简单的可解群。该解决方案是依赖于一个自由变量，可以作为光谱参数。对所得到的模型进行了Painleve分析，并找到了其特解
We establish interpolation problems related to all the $q$-Painlevé equations of types from $E_7^{(1)}$ to $(A_2+A_1)^{(1)}$. By solving those problems, we can derive the evolution equations, the scalar Lax pairs and the determinant formulae of special solutions for the corresponding $q$-Painlevé equations.
Rules of discretization for Painlevé equations
离散Painlevé属性被精确定义，并说明了保持它的基本离散规则。离散Painlevé检验是丰富的一种新的方法，扰动的连续极限，并产生无限多的非对数条件。一个一般的，直接的方法来搜索离散的拉克斯对。
We propose a modification of the AKNS hierarchy that includes the "modified" Pohlmeyer-Lund-Regge (mPLR) equation. Similarity reductions of this hierarchy give the second, third, and fourth Painlevé equations. Especially, we present a new Lax representation and a complete description of the symmetry of the third Painlevé equation through the similarity reduction. We also show the relation between the tau-function of the mPLR hierarchy and Painlevé equations.
Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlevé equations
本文提出非交换非等谱Toda与Lotka-Volterra格，通过非等谱变形导出矩阵离散Painlevé I方程，并给出拟行列式解以证明平稳约化的合理性。
Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P$_{34}$ and P$_3$
本研究对KdV方程进行非自治约化，得到多分量二阶方程系统，构造了其等单值Lax对与构成特定群的Bäcklund变换，推广了Painlevé方程P$_{34}$和P$_3$。
Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy
本文提出了一种基于扭曲连接的哈密顿系统和Lax对，用于研究Painlevé 1级数的不规则单向变形。
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…
We study Padé interpolation problems on an additive grid, related to additive difference ($d$-) Painlevé equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By choosing suitable Padé problems, we can derive time evolution equations, scalar Lax pairs of contiguous type and determinant formulae of special solutions given in terms of hypergeometric functions, for the corresponding $d$-Painlevé equations.
Under the Flaschka-Newell Lax pair, the Darboux transformation for the Painlevé-II equation is constructed by the limiting technique. With the aid of the Darboux transformation, the rational solutions are represented by the Gram determinant, and then we give the large $y$ asymptotics of the determinant and the rational solutions. Finally, the solution of the corresponding Riemann-Hilbert problem is obtained from the Darboux matrices.
Darboux-Backlund Derivation of Rational Solutions of the Painleve IV Equation
该研究在带非等谱Virasoro约束的AKNS层级Lax框架下，利用逐次Darboux-Bäcklund变换构造Painlevé IV方程的有理解，并给出其Wronskian表示。
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…
Symbolic-computation study of integrable properties for the (2+1)-dimensional Gardner equation with the two-singular-manifold method
基于Painleve分析的奇异流形方法可以用来研究非线性偏微分方程的许多重要的可积性质，本文将双奇异流形方法应用于具有两个Painleve展开分支的（2+1）维Gardner方程，确定了Hirota双线性型、Backlund变换、拉克斯对和达布变换.基于所得到的拉克斯对，构造了二元达布变换，并通过N次符号计算迭代，得到了N-N-Grammian解.
One of the authors has recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation $h=H(p,q,t),$ where $H$ is a given Hamiltonian containing $t$ explicitly, yields the function $t=T(p,q,h)$, which defines a new Hamiltonian system with Hamiltonian $T$ and independent variable $h.$ By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for the…
In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $β$. Using this general result, the case $β=6$ is further considered here. This is the smallest even $β$, when the corresponding Lax pair and its relation to Painlevé II (PII) have not been known before, unlike cases $β=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $β=6$ involving the PII function in the coeffici…
Higher-order generalizations of the $A_6^{(1)}$- and $A_4^{(1)}$-surface type $q$-Painlevé equations
本文通过具有立方体周围一致性性质的偏差分方程组进行周期约化，构造了$A_6^{(1)}$型和$A_4^{(1)}$面型$q$-Painlevé方程的高阶推广，并展示了其扩展仿射Weyl群对称性和Lax对。
Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
本文研究带跳跃间断的拉盖尔权重生成的 Hankel 行列式，通过阶梯算子与 Riemann-Hilbert 方法，证明其对数导数满足推广的 Painlevé V 方程，并与耦合 Painlevé V 系统及广义 Painlevé III 方程建立联系。
In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associative Yang-Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of ${\mathbb Z}_2$ reduced elliptic t…
Vector systems of Painlevé type
本文通过群约简方法研究NLS、mKdV和KdV方程的向量泛化，得到具有等单调Lax表示的ODE系统，是P₁、P₂、P₃₄和P₄方程的多组分扩展，部分可视为非自治可积系统。
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