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32 Painlevé TranscendentsProperties

§32.6 Hamiltonian Structure

Contents
  1. §32.6(i) Introduction
  2. §32.6(ii) First Painlevé Equation
  3. §32.6(iii) Second Painlevé Equation
  4. §32.6(iv) Third Painlevé Equation
  5. §32.6(v) Other Painlevé Equations

§32.6(i) Introduction

PIPVI can be written as a Hamiltonian system

32.6.1 dqdz =Hp,
dpdz =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)=12p22q3zq,

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+2zσ2σ=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=pq212z,
32.6.11 p=2qp+α+12.

Then q=w satisfies PII and p satisfies

32.6.12 pp′′=12(p)2+2p3zp212(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(zσσ)=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)qκ0z)p+κ(θ0+θ)zq,

and so

32.6.17 zq =2q2pκzq2(2θ0+1)q+κ0z,
32.6.18 zp =2qp2+2κzqp+(2θ0+1)pκ(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+θ0212κ0κz2

defined by (32.6.16) satisfies

32.6.21 (zσ′′σ)2+2((σ)2κ02κ2z2)(zσ2σ)+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+θ0qη0ζ)p+12η(θ0+θ)q,

and so

32.6.25 ζq=2q2pηq2θ0q+η0ζ,
32.6.26 ζp=2qp2+2ηqp+θ0p12η(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)+14θ0212η0ηζ

defined by (32.6.24) satisfies

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

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

32.6.30 q=η0(ζσ′′2θ0σ+η0ηθ)η02η24(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+(θqκ0z)pκzq,

and so

32.6.33 zq=2q2p+θqκ0z,
32.6.34 zp=2qp2θp+κz.

Then q=w satisfies PIII with

32.6.35 (α,β,γ,δ)=(2κ,κ0(θ1),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(zσ2σ)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).