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32 Painlevé TranscendentsProperties

§32.7 Bäcklund Transformations


§32.7(i) Definition

With the exception of PI, a Bäcklund transformation relates a Painlevé transcendent of one type either to another of the same type but with different values of the parameters, or to another type.

§32.7(ii) Second Painlevé Equation

Let w=w(z;α) be a solution of PII. Then the transformations

32.7.1 𝒮:w(z;-α)=-w,


32.7.2 𝒯±:w(z;α±1)=-w-2α±12w2±2w+z,

furnish solutions of PII, provided that α12. PII also has the special transformation

32.7.3 W(ζ;12ε)=2-1/3εw(z;0)ddzw(z;0),

or equivalently,

32.7.4 w2(z;0)=2-1/3(W2(ζ;12ε)-εddζW(ζ;12ε)+12ζ),

with ζ=-21/3z and ε=±1, where W(ζ;12ε) satisfies PII with z=ζ, α=12ε, and w(z;0) satisfies PII with α=0.

The solutions wα=w(z;α), wα±1=w(z;α±1), satisfy the nonlinear recurrence relation

32.7.5 α+12wα+1+wα+α-12wα+wα-1+2wα2+z=0.

See Fokas et al. (1993).

§32.7(iii) Third Painlevé Equation

Let wj=w(z;αj,βj,γj,δj), j=0,1,2, be solutions of PIII with

32.7.6 (α1,β1,γ1,δ1)=(-α0,-β0,γ0,δ0),
32.7.7 (α2,β2,γ2,δ2)=(-β0,-α0,-δ0,-γ0).


32.7.8 𝒮1:w1 =-w0,
32.7.9 𝒮2:w2 =1/w0.

Next, let Wj=W(z;αj,βj,1,-1), j=0,1,2,3,4, be solutions of PIII with

32.7.10 α1 =α3=α0+2,
α2 =α4=α0-2,
β1 =β2=β0+2,
β3 =β4=β0-2.


32.7.11 𝒯1:W1 =zW0+zW02-βW0-W0+zW0(zW0+zW02+αW0+W0+z),
32.7.12 𝒯2:W2 =-zW0-zW02-βW0-W0+zW0(zW0-zW02-αW0+W0+z),
32.7.13 𝒯3:W3 =-zW0+zW02+βW0-W0-zW0(zW0+zW02+αW0+W0-z),
32.7.14 𝒯4:W4 =zW0-zW02+βW0-W0-zW0(zW0-zW02-αW0+W0-z).

See Milne et al. (1997).

If γ=0 and αδ0, then set α=1 and δ=-1, without loss of generality. Let uj=w(z;1,βj,0,-1), j=0,5,6, be solutions of PIII with

32.7.15 β5 =β0+2,
β6 =β0-2.


32.7.16 𝒯5:u5 =(zu0+z-(β0+1)u0)/u02,
32.7.17 𝒯6:u6 =-(zu0-z+(β0-1)u0)/u02.

Similar results hold for PIII with δ=0 and βγ0.


32.7.18 w(z;a,b,0,0) =W2(ζ;0,0,a,b),
z =12ζ2.

§32.7(iv) Fourth Painlevé Equation

Let w0=w(z;α0,β0) and wj±=w(z;αj±,βj±), j=1,2,3,4, be solutions of PIV with

32.7.19 α1± =14(2-2α0±3-2β0),
β1± =-12(1+α0±12-2β0)2,
α2± =-14(2+2α0±3-2β0),
β2± =-12(1-α0±12-2β0)2,
α3± =32-12α034-2β0,
β3± =-12(1-α0±12-2β0)2,
α4± =-32-12α034-2β0,
β4± =-12(-1-α0±12-2β0)2.


32.7.20 𝒯1±:w1± =w0-w02-2zw0-2β02w0,
32.7.21 𝒯2±:w2± =-w0+w02+2zw0-2β02w0,
32.7.22 𝒯3±:w3± =w0+2(1-α012-2β0)w0w0±-2β0+2zw0+w02,
32.7.23 𝒯4±:w4± =w0+2(1+α0±12-2β0)w0w0-2β0-2zw0-w02,

valid when the denominators are nonzero, and where the upper signs or the lower signs are taken throughout each transformation. See Bassom et al. (1995).

§32.7(v) Fifth Painlevé Equation

Let wj(zj)=w(zj;αj,βj,γj,δj), j=0,1,2, be solutions of PV with

32.7.24 z1 =-z0,
z2 =z0,
(α1,β1,γ1,δ1) =(α0,β0,-γ0,δ0),
(α2,β2,γ2,δ2) =(-β0,-α0,-γ0,δ0).


32.7.25 𝒮1:w1(z1) =w(z0),
32.7.26 𝒮2:w2(z2) =1/w(z0).

Let W0=W(z;α0,β0,γ0,-12) and W1=W(z;α1,β1,γ1,-12) be solutions of PV, where

32.7.27 α1 =18(γ0+ε1(1-ε3-2β0-ε22α0))2,
β1 =-18(γ0-ε1(1-ε3-2β0-ε22α0))2,
γ1 =ε1(ε3-2β0-ε22α0),

and εj=±1, j=1,2,3, independently. Also let

32.7.28 Φ=zW0-ε22α0W02+ε3-2β0+(ε22α0-ε3-2β0+ε1z)W0,

and assume Φ0. Then

32.7.29 𝒯ε1,ε2,ε3:W1=(Φ-2ε1zW0)/Φ,

provided that the numerator on the right-hand side does not vanish. Again, since εj=±1, j=1,2,3, independently, there are eight distinct transformations of type 𝒯ε1,ε2,ε3.

§32.7(vi) Relationship Between the Third and Fifth Painlevé Equations

Let w=w(z;α,β,1,-1) be a solution of PIII and

32.7.30 v=w-εw2+((1-εα)w/z),

with ε=±1. Then

32.7.31 W(ζ;α0,β0,γ0,δ0) =v-1v+1,
z =2ζ,

satisfies PV with

32.7.32 (α0,β0,γ0,δ0)=((β-εα+2)2/32,-(β+εα-2)2/32,-ε,0).

§32.7(vii) Sixth Painlevé Equation

Let wj(zj)=wj(zj;αj,βj,γj,δj), j=0,1,2,3, be solutions of PVI with

32.7.33 z1 =1/z0,
32.7.34 z2 =1-z0,
32.7.35 z3 =1/z0,
32.7.36 (α1,β1,γ1,δ1)=(α0,β0,-δ0+12,-γ0+12),
32.7.37 (α2,β2,γ2,δ2)=(α0,-γ0,-β0,δ0),
32.7.38 (α3,β3,γ3,δ3)=(-β0,-α0,γ0,δ0).


32.7.39 𝒮1:w1(z1) =w0(z0)/z0,
32.7.40 𝒮2:w2(z2) =1-w0(z0),
32.7.41 𝒮3:w3(z3) =1/w0(z0).

The transformations 𝒮j, for j=1,2,3, generate a group of order 24. See Iwasaki et al. (1991, p. 127).

Let w(z;α,β,γ,δ) and W(z;A,B,C,D) be solutions of PVI with

32.7.42 (α,β,γ,δ)=(12(θ-1)2,-12θ02,12θ12,12(1-θ22)),
32.7.43 (A,B,C,D)=(12(Θ-1)2,-12Θ02,12Θ12,12(1-Θ22)),


32.7.44 θj=Θj+12σ,

for j=0,1,2,, where

32.7.45 σ=θ0+θ1+θ2+θ-1=1-(Θ0+Θ1+Θ2+Θ).


32.7.46 σw-W=z(z-1)WW(W-1)(W-z)+Θ0W+Θ1W-1+Θ2-1W-z=z(z-1)ww(w-1)(w-z)+θ0w+θ1w-1+θ2-1w-z.

PVI also has quadratic and quartic transformations. Let w=w(z;α,β,γ,δ) be a solution of PVI. The quadratic transformation

32.7.47 u1(ζ1) =(1-w)(w-z)(1+z)2w,
ζ1 =(1-z1+z)2,

transforms PVI with α=-β and γ=12-δ to PVI with (α1,β1,γ1,δ1)=(4α,-4γ,0,12). The quartic transformation

32.7.48 u2(ζ2) =(w2-z)24w(w-1)(w-z),
ζ2 =z,

transforms PVI with α=-β=γ=12-δ to PVI with (α2,β2,γ2,δ2)=(16α,0,0,12). Also,

32.7.49 u3(ζ3)=(1-z1/41+z1/4)2(w+z1/4w-z1/4)2,
32.7.50 ζ3=(1-z1/41+z1/4)4,

transforms PVI with α=β=0 and γ=12-δ to PVI with α3=β3 and γ3=12-δ3.

§32.7(viii) Affine Weyl Groups

See Okamoto (1986, 1987a, 1987b, 1987c), Sakai (2001), Umemura (2000).