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32 Painlevé TranscendentsProperties

§32.6 Hamiltonian Structure

Contents

§32.6(i) Introduction

PIPVI can be written as a Hamiltonian system

32.6.1 qz =Hp,
pz =-Hq,

for suitable (non-autonomous) Hamiltonian functions H(q,p,z).

§32.6(ii) First Painlevé Equation

The Hamiltonian for PI is

32.6.2 HI(q,p,z)=12p2-2q3-zq,

and so

32.6.3 q=p,
32.6.4 p=6q2+z.

Then q=w satisfies PI. The function

32.6.5 σ=HI(q,p,z),

defined by (32.6.2) satisfies

32.6.6 (σ′′)2+4(σ)3+2zσ-2σ=0.

Conversely, if σ is a solution of (32.6.6), then

32.6.7 q=-σ,
32.6.8 p=-σ′′,

are solutions of (32.6.3) and (32.6.4).

§32.6(iii) Second Painlevé Equation

The Hamiltonian for PII is

32.6.9 HII(q,p,z)=12p2-(q2+12z)p-(α+12)q,

and so

32.6.10 q=p-q2-12z,
32.6.11 p=2qp+α+12.

Then q=w satisfies PII and p satisfies

32.6.12 pp′′=12(p)2+2p3-zp2-12(α+12)2.

The function σ(z)=HII(q,p,z) defined by (32.6.9) satisfies

32.6.13 (σ′′)2+4(σ)3+2σ(zσ-σ)=14(α+12)2.

Conversely, if σ(z) is a solution of (32.6.13), then

32.6.14 q=(4σ′′+2α+1)/(8σ),
32.6.15 p=-2σ,

are solutions of (32.6.10) and (32.6.11).

§32.6(iv) Third Painlevé Equation

The Hamiltonian for PIII is

32.6.16 zHIII(q,p,z)=q2p2-(κzq2+(2θ0+1)q-κ0z)p+κ(θ0+θ)zq,

and so

32.6.17 zq =2q2p-κzq2-(2θ0+1)q+κ0z,
32.6.18 zp =-2qp2+2κzqp+(2θ0+1)p-κ(θ0+θ)z.

Then q=w satisfies PIII with

32.6.19 (α,β,γ,δ)=(-2κθ,2κ0(θ0+1),κ2,-κ02).

The function

32.6.20 σ=zHIII(q,p,z)+pq+θ02-12κ0κz2

defined by (32.6.16) satisfies

32.6.21 (zσ′′-σ)2+2((σ)2-κ02κ2z2)(zσ-2σ)+8κ0κθ0θzσ=4κ02κ2(θ02+θ2)z2.

Conversely, if σ is a solution of (32.6.21), then

32.6.22 q=κ0(zσ′′-(2θ0+1)σ+2κ0κθz)κ02κ2z2-(σ)2,
32.6.23 p=(σ+κ0κz)/(2κ0),

are solutions of (32.6.17) and (32.6.18).

The Hamiltonian for PIII (§32.2(iii)) is

32.6.24 ζHIII(q,p,ζ)=q2p2-(ηq2+θ0q-η0ζ)p+12η(θ0+θ)q,

and so

32.6.25 ζq=2q2p-ηq2-θ0q+η0ζ,
32.6.26 ζp=-2qp2+2ηqp+θ0p-12η(θ0+θ1).

Then q=u satisfies PIII with

32.6.27 (α,β,γ,δ)=(-4ηθ,4η0(θ0+1),4η2,-4η02).

The function

32.6.28 σ=ζHIII(q,p,ζ)+14θ02-12η0ηζ

defined by (32.6.24) satisfies

32.6.29 ζ2(σ′′)2+(4(σ)2-η02η2)(ζσ-σ)+η0ηθ0θσ=14η02η2(θ02+θ2).

Conversely, if σ is a solution of (32.6.29), then

32.6.30 q=η0(ζσ′′-2θ0σ+η0ηθ)η02η2-4(σ)2,
32.6.31 p=(2σ+η0ηζ)/(2η0),

are solutions of (32.6.25) and (32.6.26).

The Hamiltonian for PIII with γ=0 is

32.6.32 zHIII(q,p,z)=q2p2+(θq-κ0z)p-κzq,

and so

32.6.33 zq=2q2p+θq-κ0z,
32.6.34 zp=-2qp2-θp+κz.

Then q=w satisfies PIII with

32.6.35 (α,β,γ,δ)=(2κ,κ0(θ-1),0,-κ02).

The function

32.6.36 σ=zHIII(q,p,z)+pq+14(θ+1)2

defined by (32.6.32) satisfies

32.6.37 (zσ′′-σ)2+2(σ)2(zσ-2σ)-4κ0κ(θ+1)θzσ=4κ02κ2z2.

Conversely, if σ is a solution of (32.6.37), then

32.6.38 q=κ0(zσ′′-θσ+2κ0κz)/(σ)2,
32.6.39 p=σ/(2κ0),

are solutions of (32.6.33) and (32.6.34).

§32.6(v) Other Painlevé Equations

For Hamiltonian structure for PIV see Jimbo and Miwa (1981), Okamoto (1986); also Forrester and Witte (2001).

For Hamiltonian structure for PV see Jimbo and Miwa (1981), Okamoto (1987b); also Forrester and Witte (2002).

For Hamiltonian structure for PVI see Jimbo and Miwa (1981) and Okamoto (1987a); also Forrester and Witte (2004).