Yousuke Ohyama
We give a basic theory on monodromy evolving deformations. proposed by Chakravarty and Ablowitz in 1996. We show that Halphen's second quadratic system can be described by monodromy evolving deformations. Our result is a generalization of the work on the DH-V system by Chakravarty and Ablowitz.
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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.
The isomonodromy deformation equation for a 2x2 matrix linear ODE with a large parameter can be locally reduced to a (hyper)elliptic equation. To globalize this result, we apply the isomonodromy deformation method and obtain the modulation equations for the asymptotic algebraic curve. The method is applied to the degenerate solution of the Painleve first equation.
The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painleve equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the cano…
We will study a monodromy preserving deformation with an irregular singular point and determine the $τ$ function of the monodromy preserving deformation by the elliptic $θ$ function moving the argument $z$ and the period $Ω$.
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…
Monodromy and Isomonodromy for Linear $q$-Difference Systems with Coefficient Matrix $Ax+B$
本文针对系数矩阵为$Ax+B$的线性$q$-差分系统，构造典范基本解、建立等单值性理论，推导单值性数据显式公式，证明其$q\ o1$的微分极限对应微分系统的相关结构。
The Direct Monodromy Problem of Painleve-I
针对 Painlevé 第一方程对应的立方势薛定谔方程，提出一种基于 Nevanlinna 几何理论的新算法以计算其直接单值问题。
The isomonodromy deformation method is applied to the scaling limits in the linear NxN matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves which describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated to the n-large asymptotics in the sequence of the bi-orthogonal polynomials with cubic potentials.
Asymptotic and monodromy problems for higher-order Painlevé III equations
本文研究高阶III型Painlevé方程的渐近行为与monodromy变形问题，推导了Stokes矩阵和连接矩阵的显式公式，并应用于$tt^{*}$方程分析。
For the Schlesinger-type equation related to the fifth Painlevé equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.
In this paper, we extend the result of Kitaev and Korotkin to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $τ$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $τ$-function is expressed by the hyperelliptic $Θ$ function moving the argument $\z$ and the period $\B,$ where $t$ and the positions of regular singularities move $z$ and $\B,$ respectively.
We study the monodromy of meromorphic cyclic $\mathrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of M. Jimbo, T. Miwa, and K. Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo-Miwa-Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,…
We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlev{é} equation. We use the generalised monodromy map for this equation to give solutions to the infinite-dimensional Riemann-Hilbert problems arising from the Donaldson-Thomas theory of the A2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.
The critical and asymptotic behaviors of solutions of the sixth Painlevé equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.
The critical and asymptotic behaviors of solutions of the sixth Painlevé equation, an their parametrization in terms of monodromy data, are synthetically reviewed. The explicit formulas are given. This paper has been withdrawn by the author himself, because some improvements are necessary.
Deformations of differential equations
本研究针对线性微分方程的微扰问题，采用戴森型展开推导显式级数解，分析形变解的单值性与上同调上闭链的关联，还得到超几何方程无穷小形变下的一阶本征值修正及问题的自然界。
Nilpotence of Orbits under Monodromy and the Length of Melnikov Functions
本文研究了多项式变形$dF+\varepsilon\omega$中与$(F,\gamma)$相关的轨道深度、幂零类和导数长度等不变量，证明它们界定了首个非零Melnikov函数的长度，并通过四直线乘积实例揭示了这些不变量与Godbillon-Vey序列长度的关联。
Componentwise Geometry and Monodromy of Generalized Lamé Equations
本研究针对带奇点的广义拉梅方程建立分量几何与单值群理论，构造非偶分量的相关结构，证明其与经典拉梅方程的单值群等价性，实现谱几何等理论的分量转移。
We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain…
For linear Hamiltonian $2n\times 2n$ systems $J y'(x) = (λW(x)+H(x))y(x)$ we investigate the problem how the eigenvalues $λ$ depend on the entries of the coefficient matrix $H$. This question turns into a deformation equation for $H$ and a partial differential equation for the eigenvalues $λ$. We apply our results to various examples, including generalizations of the confluent Heun equation and the Chandrasekhar-Page angular equation. We are mainly concerned with the $2\times 2$ case, and in order to reduce the degrees of freedom in $H$ as much as possible, we will first convert such systems i…
We consider in the complex field the differential equation $\displaystyle \frac{d^2}{d x^2} Φ(x) = \frac{P_m(x,\a)}{x^2}Φ(x)$, where $P_m$ is a monic polynomial function of order $m$ with coefficients $\a=(a_1, ..., a_m)$. We investigate the asymptotic, resurgent, properties of the solutions at infinity, focusing in particular on the analytic dependence on $\a$ of the Stokes-Sibuya multipliers. Taking into account the non trivial monodromy at the origin, we derive a set of functional equations for the Stokes-Sibuya multipliers. We show how these functional relations can be used to compute the …
We give a geometric proof to the classical fact that the dimension of the deformations of a given generic Fuchsian equation without changing the semi-simple conjugacy class of its local monodromies (``number of accessory parameters'') is equal to half the dimension of the moduli space of deformations of the associated local system. We do this by constructing a weight 1 Hodge structure on the infinitesimal deformations of logarithmic connections, such that deformations as an equation correspond to the $(1,0)$-part. This answers a question of Nicholas Katz, who noticed the dimension doubling men…
We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.
Deformations of singularities of meromorphic $\mathfrak{sl}_2(\mathbb{C})$-connections and meromorphic quadratic differentials
本文研究无迹 $2\times2$ 亚纯线性 ODE 奇点的形变与汇合，建立形式不变量与二次微分射流的对应，并刻画等单值形变及其退化。
In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on $\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on $\mathbb{CP}^{1}$ to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodr…
The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of ``more complicated'' Schlesinger equations $S_{(n,m)}$ to ``simpler'' $S_{(n',m')}$ having $n'< n$ or $m' < m$.
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