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32 Painlevé TranscendentsProperties

§32.9 Other Elementary Solutions

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§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,0,4κμ-λ2)=z(κ(lnz)2+λlnz+μ),
32.9.3 w(z;-ν2λ,0,ν2(λ2-4κμ),0)=zν-1κz2ν+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(2n-1)2,-1),

or

32.9.10 (α,β,γ)=(18(2n-1)2,-12μ2,1),

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

32.9.11 w(z)=Pn2-n+1(ζ)/Qn2-n(ζ),

where Pn2-n+1(ζ) and Qn2-n(ζ) are polynomials of degrees n2-n+1 and n2-n, respectively, with no common zeros.

PV, with γ=δ=0, has a first integral

32.9.12 z2(w)2=(w-1)2(2αw2+Cw-2β),

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(1-μ2))=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).