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32 Painlevé TranscendentsProperties

§32.9 Other Elementary Solutions

Contents
  1. §32.9(i) Third Painlevé Equation
  2. §32.9(ii) Fifth Painlevé Equation
  3. §32.9(iii) Sixth Painlevé Equation

§32.9(i) Third Painlevé Equation

Elementary nonrational solutions of PIII are

32.9.1 w(z;μ,0,0,μκ3)=κz1/3,
32.9.2 w(z;0,2κ,0,4κμλ2)=z(κ(lnz)2+λlnz+μ),
32.9.3 w(z;ν2λ,0,ν2(λ24κμ),0)=zν1κz2ν+λzν+μ,

with κ, λ, μ, and ν arbitrary constants.

In the case γ=0 and αδ0 we assume, as in §32.2(ii), α=1 and δ=1. Then PIII has algebraic solutions iff

32.9.4 β=2n,

with n. These are rational solutions in ζ=z1/3 of the form

32.9.5 w(z)=Pn2+1(ζ)/Qn2(ζ),

where Pn2+1(ζ) and Qn2(ζ) are polynomials of degrees n2+1 and n2, respectively, with no common zeros. For examples and plots see Clarkson (2003a) and Milne et al. (1997). Similar results hold when δ=0 and βγ0.

PIII with β=δ=0 has a first integral

32.9.6 z2(w)2+2zww=(C+2αzw+γz2w2)w2,

with C an arbitrary constant, which is solvable by quadrature. A similar result holds when α=γ=0. PIII with α=β=γ=δ=0, has the general solution w(z)=Czμ, with C and μ arbitrary constants.

§32.9(ii) Fifth Painlevé Equation

Elementary nonrational solutions of PV are

32.9.7 w(z;μ,18,μκ2,0)=1+κz1/2,
32.9.8 w(z;0,0,μ,12μ2)=κexp(μz),

with κ and μ arbitrary constants.

PV, with δ=0, has algebraic solutions if either

32.9.9 (α,β,γ)=(12μ2,18(2n1)2,1),

or

32.9.10 (α,β,γ)=(18(2n1)2,12μ2,1),

with n and μ arbitrary. These are rational solutions in ζ=z1/2 of the form

32.9.11 w(z)=Pn2n+1(ζ)/Qn2n(ζ),

where Pn2n+1(ζ) and Qn2n(ζ) are polynomials of degrees n2n+1 and n2n, respectively, with no common zeros.

PV, with γ=δ=0, has a first integral

32.9.12 z2(w)2=(w1)2(2αw2+Cw2β),

with C an arbitrary constant, which is solvable by quadrature. For examples and plots see Clarkson (2005). PV, with α=β=0 and γ2+2δ=0, has solutions w(z)=Cexp(±2δz), with C an arbitrary constant.

§32.9(iii) Sixth Painlevé Equation

An elementary algebraic solution of PVI is

32.9.13 w(z;12κ2,12κ2,12μ2,12(1μ2))=z1/2,

with κ and μ arbitrary constants.

Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of PVI with β=γ=0, δ=12. For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b).