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32 Painlevé TranscendentsProperties

§32.7 Bäcklund Transformations

  1. §32.7(i) Definition
  2. §32.7(ii) Second Painlevé Equation
  3. §32.7(iii) Third Painlevé Equation
  4. §32.7(iv) Fourth Painlevé Equation
  5. §32.7(v) Fifth Painlevé Equation
  6. §32.7(vi) Relationship Between the Third and Fifth Painlevé Equations
  7. §32.7(vii) Sixth Painlevé Equation
  8. §32.7(viii) Affine Weyl Groups

§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)=w2α±12w2±2w+z,

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

32.7.3 W(ζ;12ε)=21/3εw(z;0)ddzw(z;0),

or equivalently,

32.7.4 w2(z;0)=21/3(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+α12wα+wα1+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=α02,
β1 =β2=β0+2,
β3 =β4=β02.


32.7.11 𝒯1:W1 =zW0+zW02βW0W0+zW0(zW0+zW02+αW0+W0+z),
32.7.12 𝒯2:W2 =zW0zW02βW0W0+zW0(zW0zW02αW0+W0+z),
32.7.13 𝒯3:W3 =zW0+zW02+βW0W0zW0(zW0+zW02+αW0+W0z),
32.7.14 𝒯4:W4 =zW0zW02+βW0W0zW0(zW0zW02αW0+W0z).

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 =β02.


32.7.16 𝒯5:u5 =(zu0+z(β0+1)u0)/u02,
32.7.17 𝒯6:u6 =(zu0z+(β01)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(22α0±32β0),
β1± =12(1+α0±122β0)2,
α2± =14(2+2α0±32β0),
β2± =12(1α0±122β0)2,
α3± =3212α0342β0,
β3± =12(1α0±122β0)2,
α4± =3212α0342β0,
β4± =12(1α0±122β0)2.


32.7.20 𝒯1±:w1± =w0w022zw02β02w0,
32.7.21 𝒯2±:w2± =w0+w02+2zw02β02w0,
32.7.22 𝒯3±:w3± =w0+2(1α0122β0)w0w0±2β0+2zw0+w02,
32.7.23 𝒯4±:w4± =w0+2(1+α0±122β0)w0w02β02zw0w02,

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(1ε32β0ε22α0))2,
β1 =18(γ0ε1(1ε32β0ε22α0))2,
γ1 =ε1(ε32β0ε22α0),

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

32.7.28 Φ=zW0ε22α0W02+ε32β0+(ε22α0ε32β0+ε1z)W0,

and assume Φ0. Then

32.7.29 𝒯ε1,ε2,ε3:W1=(Φ2ε1zW0)/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=wεw2+((1εα)w/z),

with ε=±1. Then

32.7.31 W(ζ;α0,β0,γ0,δ0) =v1v+1,
z =2ζ,

satisfies PV with

32.7.32 (α0,β0,γ0,δ0)=((βεα+2)2/32,(β+εα2)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 =1z0,
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) =1w0(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(θ1)2,12θ02,12θ12,12(1θ22)),
32.7.43 (A,B,C,D)=(12(Θ1)2,12Θ02,12Θ12,12(1Θ22)),


32.7.44 θj=Θj+12σ,

for j=0,1,2,, where

32.7.45 σ=θ0+θ1+θ2+θ1=1(Θ0+Θ1+Θ2+Θ).


32.7.46 σwW=z(z1)WW(W1)(Wz)+Θ0W+Θ1W1+Θ21Wz=z(z1)ww(w1)(wz)+θ0w+θ1w1+θ21wz.

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) =(1w)(wz)(1+z)2w,
ζ1 =(1z1+z)2,

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

32.7.48 u2(ζ2) =(w2z)24w(w1)(wz),
ζ2 =z,

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

32.7.49 u3(ζ3)=(1z1/41+z1/4)2(w+z1/4wz1/4)2,
32.7.50 ζ3=(1z1/41+z1/4)4,

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

§32.7(viii) Affine Weyl Groups

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