Yusuke Sasano
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.
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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 $D_5^{(2)}$.
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 $B_4^{(1)}$.
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 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 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 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 the theory of deformation of Okamoto-Painlevé pair (S,Y), a local cohomology group $H^1_D(Θ_S(-\log D))$ plays an important role. In this paper, we estimate the local cohomology group of pair (S,Y) for several types, and obtain the following results. For a pair (S,Y) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group $H^1_D(Θ_S(-\log D))$ is at least 1 dimensional. This fact is the key to understand Painlevé equation related to (S,Y). Moreover we show that, for the pairs (S,Y) of type $\tilde{A_8}$, the local cohomology group …
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 prove that for any autonomous 4-dimensional integral system of Painlevé type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli's theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus two component of the boundary of an affine patch of the Liouville torus.
In families of Painlevé VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families of algebraic varieties. In our case, we have one family with zero dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers done by Herfurtner in 1992 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005.
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.
On the geometry of a 4-dimensional extension of a $q$-Painlevé I equation with symmetry type $A_1^{(1)}$
本文对一个扩展了具有 $A_1^{(1)}$ 对称类型的 $q$-Painlevé I 方程的四维可积离散动力系统进行几何研究，通过奇点分解证明其可积性，并构造了两个四维 $q$-Painlevé 方程类似物。
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