On Some Hamiltonian Structures of Painleve Systems, III

Abstract

This is the second part of the series of our papers. In the preceding paper([11]), we studied a Hamiltonian structure of the sixth Painlevé system (H V I) equivalent to the sixth Painlevé equation P V I. In this paper, we continue the study for Painlevé systems (H J) or Painlevé equations P J for are the equations given by P V : d 2 x dt 2 = 1 2x + 1 x − 1 dx dt 2 − 1 t dx dt + (x − 1) 2 t 2 αx + β x +γ x t + δ x(x + 1) x − 1 , P IV : d 2 x dt 2 = 1 2x dx dt 2 + 3 2 x 3 + 4tx 2 + 2(t 2 − α)x + β x P III : d 2 x dt 2 = 1 x dx dt 2 − 1 t dx dt + 1 t (αx 2 + β) + γx 3 + δ x , P II : d 2 x dt 2 =2x 3 + tx + α, where x and t are complex variables, α, β, γ, and δ are complex constants([4]). It is known that each P J is equivalent to a Hamiltonian system (H J) : dx/dt = ∂H J /∂y, dy/dt = −∂H J /dx, where H V (x, y, t) = 1 t [x(x − 1) 2 y 2 − {κ 0 (x − 1) 2 + κ t x(x − 1) − ηtx}y + κ(x − 1)] (κ := 1 4 {(κ 0 + κ t) 2 − κ 2 ∞ }), H IV (x, y, t) =2xy 2 − {x 2 + 2tx + 2κ 0 }y + κ ∞ x, H III (x, y, t) = 1 t [2x 2 y 2 − {2η ∞ tx 2 + (2κ 0 + 1)x − 2η 0 t}y + η ∞ (κ 0 + κ ∞)tx], H II (x, y, t) = 1 2 y 2 − (x 2 + t 2)y − (α + 1 2)x. Here the relations between the constants in the equations P J and those in the Hamiltonians are given by α = κ ∞ 2 /2, β = −κ 0 2 /2, γ = −η(1 + κ t), δ = −η 2 /2