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32 Painlevé TranscendentsProperties

§32.4 Isomonodromy Problems

Contents
  1. §32.4(i) Definition
  2. §32.4(ii) First Painlevé Equation
  3. §32.4(iii) Second Painlevé Equation
  4. §32.4(iv) Third Painlevé Equation
  5. §32.4(v) Other Painlevé Equations

§32.4(i) Definition

PIPVI can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose

32.4.1 𝚿λ =𝐀(z,λ)𝚿,
𝚿z =𝐁(z,λ)𝚿,

is a linear system in which 𝐀 and 𝐁 are matrices and λ is independent of z. Then the equation

32.4.2 2𝚿zλ=2𝚿λz,

is satisfied provided that

32.4.3 𝐀z𝐁λ+𝐀𝐁𝐁𝐀=0.

(32.4.3) is the compatibility condition of (32.4.1). Isomonodromy problems for Painlevé equations are not unique.

§32.4(ii) First Painlevé Equation

PI is the compatibility condition of (32.4.1) with

32.4.4 𝐀(z,λ)=(4λ4+2w2+z)[1001]i(4λ2w+2w2+z)[0ii0](2λw+12λ)[0110],
32.4.5 𝐁(z,λ)=(λ+wλ)[1001]iwλ[0ii0].

§32.4(iii) Second Painlevé Equation

PII is the compatibility condition of (32.4.1) with

32.4.6 𝐀(z,λ)=i(4λ2+2w2+z)[1001]2w[0ii0]+(4λwαλ)[0110],
32.4.7 𝐁(z,λ)=[iλwwiλ].

See Flaschka and Newell (1980).

§32.4(iv) Third Painlevé Equation

The compatibility condition of (32.4.1) with

32.4.8 𝐀(z,λ)=[14z0014z]+[12θu0u112θ]1λ+[v014zv1v0(v012z)/v114zv0]1λ2,
32.4.9 𝐁(z,λ)=[140014]λ+[0u0u10]1z[v014zv1v0(v012z)/v114zv0]1zλ,

where θ is an arbitrary constant, is

32.4.10 zu0=θu0zv0v1,
32.4.11 zu1=θu1(z(2v0z)/(2v1)),
32.4.12 zv0=2v0u1v1+v0+(u0(2v0z)/v1),
32.4.13 zv1=2u02u1v12θv1.

If w=u0/(v0v1), then

32.4.14 zw=(4v0z)w2+(2θ1)w+z,

and w satisfies PIII with

32.4.15 (α,β,γ,δ)=(2θ0,2(1θ),1,1),

where

32.4.16 θ0=4v0z(θ(1z4v0)+z2v02v0v1u0+u1v1).

Note that the right-hand side of the last equation is a first integral of the system (32.4.10)–(32.4.13).

§32.4(v) Other Painlevé Equations

For isomonodromy problems for PIV, PV, and PVI see Jimbo and Miwa (1981).