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32 Painlevé TranscendentsProperties

§32.2 Differential Equations

Contents

§32.2(i) Introduction

The six Painlevé equations PIPVI are as follows:

32.2.1 2wz2=6w2+z,
32.2.2 2wz2=2w3+zw+α,
32.2.3 2wz2=1w(wz)2-1zwz+αw2+βz+γw3+δw,
32.2.4 2wz2=12w(wz)2+32w3+4zw2+2(z2-α)w+βw,
32.2.5 2wz2=(12w+1w-1)(wz)2-1zwz+(w-1)2z2(αw+βw)+γwz+δw(w+1)w-1,
32.2.6 2wz2=12(1w+1w-1+1w-z)(wz)2-(1z+1z-1+1w-z)wz+w(w-1)(w-z)z2(z-1)2(α+βzw2+γ(z-1)(w-1)2+δz(z-1)(w-z)2),

with α, β, γ, and δ arbitrary constants. The solutions of PIPVI are called the Painlevé transcendents. The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions.

Let

32.2.7 2wz2=F(z,w,wz),

be a nonlinear second-order differential equation in which F is a rational function of w and w/z, and is locally analytic in z, that is, analytic except for isolated singularities in . In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.

There are fifty equations with the Painlevé property. They are distinct modulo Möbius (bilinear) transformations

32.2.8 W(ζ) =a(z)w+b(z)c(z)w+d(z),
ζ =ϕ(z),

in which a(z), b(z), c(z), d(z), and ϕ(z) are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of PIPVI.

For arbitrary values of the parameters α, β, γ, and δ, the general solutions of PIPVI are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations PIIPVI have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF.

§32.2(ii) Renormalizations

If γδ0 in PIII, then set γ=1 and δ=-1, without loss of generality, by rescaling w and z if necessary. If γ=0 and αδ0 in PIII, then set α=1 and δ=-1, without loss of generality. Lastly, if δ=0 and βγ0, then set β=-1 and γ=1, without loss of generality.

If δ0 in PV, then set δ=-12, without loss of generality.

§32.2(iii) Alternative Forms

In PIII, if w(z)=ζ-1/2u(ζ) with ζ=z2, then

32.2.9 2uζ2=1u(uζ)2-1ζuζ+u2(α+γu)4ζ2+β4ζ+δ4u,

which is known as PIII.

In PIII, if w(z)=exp(-u(z)), β=-α, and δ=-γ, then

32.2.10 2uz2+1zuz=2αzsinu+2γsin(2u).

In PIV, if w(z)=22(u(ζ))2 with ζ=2z and α=2ν+1, then

32.2.11 2uζ2=3u5+2ζu3+(14ζ2-ν-12)u+β32u3.

When β=0 this is a nonlinear harmonic oscillator.

In PV, if w(z)=(cothu(ζ))2 with ζ=lnz, then

32.2.12 2uζ2=-αcoshu2(sinhu)3-βsinhu2(coshu)3-14γζsinh(2u)-18δ2ζsinh(4u).

See also Okamoto (1987c), McCoy et al. (1977), Bassom et al. (1992), Bassom et al. (1995), and Takasaki (2001).

§32.2(iv) Elliptic Form

PVI can be written in the form

32.2.13 z(1-z)I(wtt(t-1)(t-z))=w(w-1)(w-z)(α+βzw2+γ(z-1)(w-1)2+(δ-12)z(z-1)(w-z)2),

where

32.2.14 I=z(1-z)2z2+(1-2z)z-14.

See Fuchs (1907), Painlevé (1906), Gromak et al. (2002, §42); also Manin (1998).

§32.2(v) Symmetric Forms

Let

32.2.15 f1z+f1(f2-f3)+2μ1 =0,
f2z+f2(f3-f1)+2μ2 =0,
f3z+f3(f1-f2)+2μ3 =0,

where μ1, μ2, μ3 are constants, f1, f2, f3 are functions of z, with

32.2.16 μ1+μ2+μ3=1,
32.2.17 f1(z)+f2(z)+f3(z)+2z=0.

Then w(z)=f1(z) satisfies PIV with

32.2.18 (α,β)=(μ3-μ2,-2μ12).

See Noumi and Yamada (1998).

Next, let

32.2.19 zf1z =f1f3(f2-f4)+(12-μ3)f1+μ1f3,
zf2z =f2f4(f3-f1)+(12-μ4)f2+μ2f4,
zf3z =f3f1(f4-f2)+(12-μ1)f3+μ3f1,
zf4z =f4f2(f1-f3)+(12-μ2)f4+μ4f2,

where μ1, μ2, μ3, μ4 are constants, f1, f2, f3, f4 are functions of z, with

32.2.20 μ1+μ2+μ3+μ4=1,
32.2.21 f1(z)+f3(z)=z,
32.2.22 f2(z)+f4(z)=z.

Then w(z)=1-(z/f1(z)) satisfies PV with

32.2.23 (α,β,γ,δ)=(12μ12,-12μ32,μ4-μ2,-12).

§32.2(vi) Coalescence Cascade

PIPV are obtained from PVI by a coalescence cascade:

32.2.24 PVIPVPIVPIIIPIIPI

For example, if in PII

32.2.25 w(z;α)=ϵW(ζ)+1ϵ5,
32.2.26 z =ϵ2ζ-6ϵ10,
α =4ϵ15,

then

32.2.27 2Wζ2=6W2+ζ+ϵ6(2W3+ζW);

thus in the limit as ϵ0, W(ζ) satisfies PI with z=ζ.

If in PIII

32.2.28 w(z;α,β,γ,δ)=1+2ϵW(ζ;a),
32.2.29 z =1+ϵ2ζ,
α =-12ϵ-6,
β =12ϵ-6+2aϵ-3,
γ =-δ
=14ϵ-6,

then as ϵ0, W(ζ;a) satisfies PII with z=ζ, α=a.

If in PIV

32.2.30 w(z;α,β)=22/3ϵ-1W(ζ;a)+ϵ-3,
32.2.31 z =2-2/3ϵζ-ϵ-3,
α =-2a-12ϵ-6,
β =-12ϵ-12,

then as ϵ0, W(ζ;a) satisfies PII with z=ζ, α=a.

If in PV

32.2.32 w(z;α,β,γ,δ)=1+ϵζW(ζ;a,b,c,d),
32.2.33 z =ζ2,
α =14aϵ-1+18cϵ-2,
β =-18cϵ-2,
γ =14ϵb,
δ =18ϵ2d,

then as ϵ0, W(ζ;a,b,c,d) satisfies PIII with z=ζ, α=a, β=b, γ=c, δ=d.

If in PV

32.2.34 w(z;α,β,γ,δ)=122ϵW(ζ;a,b),
32.2.35 z =1+2ϵζ,
α =12ϵ-4,
β =14b,
γ =-ϵ-4,
δ =aϵ-2-12ϵ-4,

then as ϵ0, W(ζ;a,b) satisfies PIV with z=ζ, α=a, β=b.

Lastly, if in PVI

32.2.36 w(z;α,β,γ,δ)=W(ζ;a,b,c,d),
32.2.37 z =1+ϵζ,
γ =cϵ-1-dϵ-2,
δ =dϵ-2,

then as ϵ0, W(ζ;a,b,c,d) satisfies PV with z=ζ, α=a, β=b, γ=c, δ=d.