We find and study a two-parameter family of coupled Painlevé II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables $t,s$. Setting $s=0$, we can obtain an autonomous version of the coupled Painlevé II systems. We also show its symmetry and holomorphy conditions.
We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three variables. We give a relation between it and the second member of the second Painlevé hierarchy. Moreover, we give some relations between an autonomous version of its polynomial Hamiltonian system in two variables and the mKdV hierarchies.
We find a two-parameter family of coupled Painlevé systems in dimension four with affine Weyl group symmetry of type $A_4^{(2)}$. For a degenerate system of $A_4^{(2)}$ system, we also find a one-parameter family of coupled Painlevé systems in dimension four with affine Weyl group symmetry of type $A_1^{(1)}$. We show that for each system, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new. Moreover, we find a one-parameter family of partial differential systems in three variables with $W(A_1^{(1)})$-symmetry. We show the rela…
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…
We find a four-parameter family of coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$. This is the first example which gave higher order Painlevé type systems of type $E_{6}^{(2)}$. We study its symmetry and holomorphy conditions.
We present a simple heuristic method to derive the Painlevé differential equations from the corresponding geometry of rational surafces. We also give a direct relationship between the cubic pencils and Seiberg-Witten curves.
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…
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…
A Frobenius manifold has tri-hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We focus on the case of dimension four and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painlevé VI$μ$ equation. This yields an explicit procedure associating to any semisimple Frobenius manifold of dimension three a tri-hamiltonian Frobenius manifold of dimension four. We carry out explicit examples for the case of Frobenius structures on Hurwitz spaces.
We find a four-parameter family of coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $A_7^{(2)}$. This is the first example which gave higher-order Painlevé equations of type $A_{2l+5}^{(2)}$. We then give an explicit description of a confluence process from this system to a 3-parameter family of coupled Painlevé V and III systems in dimension four with $W(A_5^{(2)})$-symmetry. For a degenerate system of $A_5^{(2)}$ system, we also find a two-parameter family of ordinary differential systems in dimension four with affine Weyl group symmetry of type $D_4^{(3)…
We find and study coupled Painlevé II systems in dimension 4, which can be obtained by a degeneration from the systems of type ${A_4}^{(1)}$. We compare these systems with other types of coupled Painlevé II systems from the viewpoint of the local index. We also give the phase spaces for these systems.
We give a reformulation of a six-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of type $D_6^{(1)}$ from the viewpoint of its symmetry and holomorphy properties.
We find and study a six-parameter family of coupled Painlevé III systems in dimension six with affine Weyl group symmetry of type $D_6^{(1)}$. We also find and study its degenerate systems with affine Weyl group symmetry of types $B_5^{(1)}$ and $D_5^{(2)}$.
In this paper, we introduce the notion of generalized rational Okamoto-Painlevé pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlevé equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlevé differential equations from the deformation of Okamoto-Painlevé pairs by using the local cohomology groups. Moreover the reason why the Painlevé equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S - Y. Hamiltonian structures for Okamoto-Painlevé pairs of ty…
We find and study four kinds of a 4-parameter family of four-dimensional coupled Painlevé III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$. We also show that these systems are equivalent by an explicit birational and symplectic transformation, respectively.
In this paper, we find a two-parameter family of coupled Painlevé systems in dimension four with affine Weyl group symmetry of type $D_3^{(2)}$. We also find a four-parameter family of 2-coupled $D_3^{(2)}$-systems in dimension eight with affine Weyl group symmetry of type $D_5^{(2)}$. We show that for each system, we give its symmetry and holomorphy conditions, respectively. These symmetries, holomorphy conditions and invariant divisors are new.
We find and study four kinds of five-parameter family of six-dimensional coupled Painlevé III systems with affine Weyl group symmetry of types $D_5^{(1)},B_5^{(1)}$ and $D_6^{(2)}$. We show that each system is equivalent by an explicit birational and symplectic transformation, respectively. We also show that we characterize each system from the viewpoint of holomorphy.
We find four kinds of six-parameter family of coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order Painlevé equations of types $B_l^{(1)},D_l^{(1)}$ and $D_l^{(2)}$, respectively. Each system can be expressed as a polynomial Hamiltonian system. We show that these systems are equivalent by an explicit birational and symplectic transformation, respectively. By giving each holomorphy condition, we can recover each system. These symmetries, holomorphy conditions and …
We find and study a five-parameter family of four-dimensional coupled Painlevé V systems with affine Weyl group symmetry of type $D_5^{(1)}$. We then give an explicit description of a confluence from those systems to a four-parameter family of four-dimensional coupled Painlevé III systems with affine Weyl group symmetry of type $B_4^{(1)}$.
In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painlevé I equation. We will show that this system passes the Painlevé $α$-test for all accessible singular points $P_i \ (i=1,2,3)$. We note its holomorphy condition of the first Painlevé system.
We present {\it symmetric Hamiltonians} for the degenerate Garnier systems in two variables. For these symmetric Hamiltonians, we make the symmetry and holomorphy conditions, and we also make a generalization of these systems involving symmetry and holomorphy conditions inductively. We also show the confluence process among each system by taking the coupling confluence process of the Painlevé systems.
An algebro-geometric setting for the study of the Painlevé VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two. Relations with the theory of elliptic functions and the quantum cohomology of projective plane are discussed.
It is proved that the Painlevé VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{ν^2}{4},\be=-\frac{ν^2}{4}, \ga=\frac{ν^2}{4},\de=\f1{2}-\frac{ν^2}{4})$ is a reduced hamiltonian system. Its phase space is the set of flat $SL(2,C)$ connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach…
We find a two-parameter family of ordinary differential systems in dimension five with the affine Weyl group symmetry of type $D_3^{(2)}$. We show its symmetry and holomorphy conditions. This is the second example which gave higher order Painlevé type systems of type $D_{3}^{(2)}$. By obtaining its first integrals of polynomial type, we can obtain a two-parameter family of coupled Hamiltonian systems in dimension four with the polynomial Hamiltonian.
