arXiv subjects
Publications and source records attributed to Yusuke Sasano.
In this note, we give some holomorphy conditions of Fuji-Suzuki coupled Painlevé VI system. We also give two translation operators acting on the constant parameter $η$. We note a confluence process from the Fuji-Suzuki system to the Noumi-Yamada system of type $A_5^{(1)}$.
In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables $q_1,q_2$, however, has same five holomorphy conditions in the variables $p_1,p_2$.
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 remark on the Garnier system in two variables.
In this note, we review the notion of Painlevé scheme of the sixth Painlevé equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${Σ_2}$. The key method is Painlevé $α$-method for each accessible singular point. Giving a Painlevé scheme in the differential system satisfying certain conditions, we can recover the Painlevé VI system with the polynomial Hamiltonian. We also consider the case of the Painlevé V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular $(n+2)$-points in the Hirzebruch surface of degree $n$; ${Σ_n}$. This equation has symmetry of symmetric group of degree $n+2$.
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 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.
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 relation between its polynomial Hamiltonian system and an autonomous version of the system of type $A_1^{(1)}$.
We study a one-parameter family of the fourth-order ordinary differential equations obtained by similarity reduction of the modifed Sawada-Kotera equation. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four. We make this polynomial Hamiltonian from the viewpoint of accessible singularity and local index. We also give its symmetry and holomorphy conditions. These properties are new. Moreover, we introduce a symmetric form in dimension five for this Hamiltonian system by taking the two invariant divisors as the dependent variables. Thanks to the symmetric form, we show that this system admits the affine Weyl group symmetry of type $A_2^{(2)}$ as the group of its B{ä}cklund transformations.
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.
In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by $tanh(t)$. For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational B{ä}cklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two birational B{ä}cklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system.
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 invariant divisors are new. We also give an explicit description of a confluence process from the system of type $D_6^{(1)}$ to the system of type $A_5^{(1)}$ by taking the coupling confluence process from the Painlevé VI system to the Painlevé V system.
We present a four-parameter family of ordinary differential systems in dimension three with affine Weyl group symmetry of type $D_4^{(1)}$. By obtaining its first integral, we can reduce this system to the second-order non-linear ordinary differential equations of Painlevé type. We also study this system restricted its parameters. Each system can be obtained by connecting some invariant divisors in the system of type $D_4^{(1)}$. Each system admits affine Weyl group symmetry of types $B_3^{(1)},G_2^{(1)},D_3^{(2)}$ and $A_2^{(2)}$, respectively. These symmetries, holomorphy conditions and invariant divisors are new.
We find a five-parameter family of partial differential systems in two variables with two polynomial Hamiltonians. We give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new.
We study the phase space of the equations of Ince's table from the viewpoint of its accessible singularities and local index.
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.
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 equation into a system of the first-order ordinary differential equations of polynomial type in dimension five by birational transformations, we give its symmetry and holomorphy conditions.