Yousuke Ohyama, Shoji Okumura
We will classify all rational transformations which change the confluent hypergeometric equations to linear equations of the Painleve type from the first to the fifth. We show such rational transformations correspond to almost all of algebraic solutions of the Painleve equations from the first to fifth up to the Backlund transformations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We describe two algebraic solutions of the sixth Painlevé equation which are related to (isomonodromic) deformations of Picard-Fuchs equations of order two.
We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by …
We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform to derive an invertible integral transformation relating two its Fuchs--Garnier pairs in 2X2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier pairs for the second Pain…
This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto-Painlevé space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlevé equation, for the Bäcklund transformations.
The question under consideration is Gevrey summability of power expansions of solutions to the third and fifth Painlevé equations near infinity. Methods of French and Japaneese schools are used to analyse these properties of formal power-series solutions. The results obtained are compared with the ones obtained by means of Power Geometry.
We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.
We apply the uniform asymptotics method proposed by Bassom, Clarkson, Law and McLeod to a special Painlevé V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation, which have been established earlier by McCoy and Tang via the isomonodromy and WKB methods.
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。
We investigate generalizations of the Charlier polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-β$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-β)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to solutions of the fifth Painlevé equation PV (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters.
We will consider four hierarchies of higher order analogues of the fourth (P4) and fifth (P5) Painleve equations. The necessary and sufficient conditions for having rational solutions will be presented. Further we well consider two more hierarchies of the (P4). We well show that both of them admit the solution in the form of $1/x$. The last result shows that all the considered P4-hierarchies have the common property: they assume the solution $1/x$ as also the original P4 does.
The third, fifth and sixth Painlevé equations are studied by means of the weighted projective spaces ${\mathbb C}P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux's coordinates are systematically studied in a unified way with the aid of the orbifold structure of ${\mathbb C}P^3(p,q,r,s)$ and dynamical systems theory.
The article studies the Fifth Painlevé equation and of the nonlinear Stokes phenomenon at its irregular singularity at infinity from the point of view of confluence from the Sixth Painlevé equation. This approach is developped separately on both sides of the Riemann-Hilbert correspondance. On the side of the nonlinear Painlevé-Okamoto foliation the relation between the nonlinear monodromy group of Painlevé VI and the "nonlinear wild monodromy pseudogroup" of Painlevé V (that is the pseudogroup generated by nonlinear monodromy, nonlinear Stokes operators and nonlinear exponential torus) is expl…
In this article we consider a first-order completely integrable system of partial differential equations $\partial \Fi/partial x=A(x, t) \Fi, \partial \Fi/partial t=B(x, t) \Fi$ with $\Fi=(\fi_1, \fi_2)^τ$ where $A(x, t)$ and $B(x, t)$ are 2 by 2 holomorphic matrices functions. Under some assumptions we find a variable change by which the system $\partial \Fi/\partial x=A(x, t) \Fi$ is reduced to an equation independent on the variable $t$. As an application we show that the R. Fuchs's conjecture for the Painlevé equations is true for some algebraic solutions.
The higher order Painleve system of type D^{(1)}_{2n+2} was proposed by Y. Sasano as an extension of the sixth Painleve equation for the affine Weyl group symmetry with the aid of algebraic geometry for Okamoto initial value space. In this article, we give it as the monodromy preserving deformation of a Fuchsian system.
We consider a degeneration of the $q$-matrix sixth Painlevé system. As a result, we obtain a system of non-linear $q$-difference equations, which describes a deformation of a certain non-Fuchsian linear $q$-difference system. We define the spectral type for non-Fuchsian $q$-difference systems and characterize the associated linear problem in terms of the spectral type. We also consider a continuous limit of the non-linear $q$-difference system and show that the resulting system of non-linear differential equations coincides with the matrix fifth Painlevé system.
Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painleve VI solutions without deriving full RS-pullback transformation…
In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to expressible in terms of the third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.
We will study special solutions of the fourth, fifth and sixth Painlevé equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura's classical solutions and our solutions.
Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked Fuchsian equation. But the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. This paper presents computations of explicit RS-pullback transformations, and derivati…
Moduli Spaces for the Fifth Painlevé Equation
本文利用联络、单值群与 Okamoto–Painlevé 模空间研究 P_5 的等单值性，证明其秩 4 Lax 对源于自然精细模空间，并导出等价于 Okamoto 形式的多项式 Hamilton 量。
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。
We consider the $n{\times}n$ matrix linear differential systems in the complex plane. We find necessary and sufficient conditions under which these systems have meromorphic fundamental solutions. Using the operator identity method we construct a set of systems which have meromorphic solutions. We prove that the well known operator with the sine kernel generates a class of meromorphic Painleve type functions. The fifth Painleve function belongs to this class. Hence we obtain a new and simple proof that the fifth Painleve function is meromorphic.
In 1991, one of the authors showed the existence of quadratic transformations between the Painleve' VI equations with local monodromy differences $(1/2,a,b,\pm 1/2)$ and $(a,a,b,b)$. In the present paper we give concise forms of these transformations. %, up to fractional-linear transformations. They are related to the %better known quadratic transformations obtained by Manin and Ramani-Grammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particular…
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。
We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conform…
Matrix Szego biorthogonal polynomials for quasi-definite matrices of measures are studied. For matrices of Holder weights a Riemann-Hilbert problem is uniquely solved in terms of the matrix Szego polynomials and its Cauchy transforms. The Riemann-Hilbert problem is given as an appropriate framework for the discussion of the Szego matrix and the associated Szego recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations which a…
After recalling some of the geometry of the sixth Painleve equation, we will describe how the Okamoto symmetries arise naturally from symmetries of Schlesinger's equations and summarise the classification of the Platonic Painleve six solutions. A key observation is that Painleve VI governs the isomonodromic deformations of certain Fuchsian systems on rank \emph{three} bundles.
On finding formal power-logarithmic expansions of solutions to $q$-difference equations
本文提出了一种确定 $q$-差分方程解正式幂-对数展开存在性的充分条件，并通过实例展示了不同 $q$ 值对解渐近展开的影响。
Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standa…
We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlevé equations: difference Painlevé equation d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painlevé equation d-$P\left({A}_{1}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coor…
The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painleve transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painleve equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories…
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…
In this paper, we study the second member of the second Painlevé hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system can be considered as a 1-parameter family of coupled Painlevé systems. This Hamiltonian is new. We also show that this system admits extended affine Weyl group symmetry of type $A_1^{(1)}$, and can be recovered by its holomorphy conditions. We also study a fifth-order ordinary differential equation satisfied by this Hamiltonian. After we transform this equatio…
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…
We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrence relations of the discrete Painlevé type, for example in the case with one free deformation variable the system was found to be characterised by a solution to the discrete fifth Painlevé equation. Here …
The critical behavior of a three real parameter class of solutions of the sixth Painlevé equation is computed, and parametrized in terms of monodromy data of the associated $2\times 2$ matrix linear Fuchsian system of ODE. The class may contain solutions with poles accumulating at the critical point. The study of this class closes a gap in the description of the transcendents in one to one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of…
This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin conn…
It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency (non-branching) of a singularity. For the elliptical repr…
We completely classify all of rational solutions of the Noumi and Yamada system of type $A_5^{(1)}$, which is a generalization of the fifth Painlevé equation. The rational solutions are classified to five classes by the Bäcklund transformation group.
We consider the associated linear problem for a q-analogue of the fifth Painleve equation (qPV). We identify a lattice of connection preserving deformations in the space of the connection data for the linear problem with the lattice of translational Backlund transformations for qPV, hence, show all translational Backlund transformations possess a Lax pair. We shall show that the big q-Laguerre polynomials, and a suitable generalization, solve a special case of the linear problem, hence, find solutions to qPV in terms of determinants of Hankel matrices with entries consisting of rational or hyp…
We consider the connection problem of the second nonlinear differential equation \begin{equation} \label{eq:1} Φ''(x)=(Φ'^2(x)-1)\cotΦ(x)+ \frac{1}{x}(1-Φ'(x)) \end{equation} subject to the boundary condition $Φ(x)=x-ax^2+O(x^3)$ ($a\geq0$) as $x\to0$. In view of that equation (1) is equivalent to the fifth Painlevé (PV) equation after a Möbius transformation, we are able to study the connection problem of equation (1) by investigating the corresponding connection problem of PV. Our research technique is based on the method of uniform asymptotics presented by Bassom el at. The monotonically so…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。