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32 Painlevé TranscendentsProperties

§32.4 Isomonodromy Problems

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§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 Az-Bλ+AB-BA=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)[100-1]-i(4λ2w+2w2+z)[0-ii0]-(2λw+12λ)[0110],
32.4.5 B(z,λ)=(λ+wλ)[100-1]-iwλ[0-ii0].

§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)[100-1]-2w[0-ii0]+(4λw-αλ)[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)=[14z00-14z]+[-12θu0u112θ]1λ+[v0-14z-v1v0(v0-12z)/v114z-v0]1λ2,
32.4.9 B(z,λ)=[1400-14]λ+[0u0u10]1z-[v0-14z-v1v0(v0-12z)/v114z-v0]1zλ,

where θ is an arbitrary constant, is

32.4.10 zu0=θu0-zv0v1,
32.4.11 zu1=-θu1-(z(2v0-z)/(2v1)),
32.4.12 zv0=2v0u1v1+v0+(u0(2v0-z)/v1),
32.4.13 zv1=2u0-2u1v12-θv1.

If w=-u0/(v0v1), then

32.4.14 zw=(4v0-z)w2+(2θ-1)w+z,

and w satisfies PIII with

32.4.15 (α,β,γ,δ)=(2θ0,2(1-θ),1,-1),

where

32.4.16 θ0=4v0z(θ(1-z4v0)+z-2v02v0v1u0+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).