32 Painlevé Transcendents Properties 32.8 Rational Solutions 32.10 Special Function Solutions

§32.9 Other Elementary Solutions

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Permalink:: http://dlmf.nist.gov/32.9
See also:: Annotations for Ch.32

§32.9(i) Third Painlevé Equation
§32.9(ii) Fifth Painlevé Equation
§32.9(iii) Sixth Painlevé Equation

§32.9(i) Third Painlevé Equation

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Notes:: See Lukaševič (1965, 1967b); also Gromak et al. (2002, §33).
Permalink:: http://dlmf.nist.gov/32.9.i
See also:: Annotations for §32.9 and Ch.32

Elementary nonrational solutions of $\mbox{P}_{\mbox{\scriptsize III}}$ are

32.9.1 $w(z;\mu,0,0,-\mu\kappa^{3})=\kappa z^{1/3},$

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Symbols:: : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(i), §32.9 and Ch.32

32.9.2 $w(z;0,-2\kappa,0,4\kappa\mu-\lambda^{2})=z(\kappa(\ln z)^{2}+\lambda\ln z+\mu),$

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real, $\kappa$ : constant, $\lambda$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(i), §32.9 and Ch.32

32.9.3 $w(z;-\nu^{2}\lambda,0,\nu^{2}(\lambda^{2}-4\kappa\mu),0)=\dfrac{z^{\nu-1}}{% \kappa z^{2\nu}+\lambda z^{\nu}+\mu},$

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Symbols:: : real, $\kappa$ : constant, $\lambda$ : constant, $\mu$ : constant and $\nu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(i), §32.9 and Ch.32

with $\kappa$ , $\lambda$ , $\mu$ , and $\nu$ arbitrary constants.

In the case $\gamma=0$ and $\alpha\delta\neq 0$ we assume, as in §32.2(ii), $\alpha=1$ and $\delta=-1$ . Then $\mbox{P}_{\mbox{\scriptsize III}}$ has algebraic solutions iff

32.9.4 $\beta=2n,$

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Symbols:: : integer and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.9.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(i), §32.9 and Ch.32

with $n\in\mathbb{Z}$ . These are rational solutions in $\zeta=z^{1/3}$ of the form

32.9.5 $w(z)=\ifrac{P_{n^{2}+1}(\zeta)}{Q_{n^{2}}(\zeta)},$

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Symbols:: : integer, : real, $P_{n^{2}+1}(\zeta)$ : polynomials and $Q_{n^{2}}(\zeta)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.9.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(i), §32.9 and Ch.32

where $P_{n^{2}+1}(\zeta)$ and $Q_{n^{2}}(\zeta)$ are polynomials of degrees $n^{2}+1$ and $n^{2}$ , respectively, with no common zeros. For examples and plots see Clarkson (2003a) and Milne et al. (1997). Similar results hold when $\delta=0$ and $\beta\gamma\neq 0$ .

$\mbox{P}_{\mbox{\scriptsize III}}$ with $\beta=\delta=0$ has a first integral

32.9.6 $z^{2}(w^{\prime})^{2}+2zww^{\prime}=(C+2\alpha zw+\gamma z^{2}w^{2})w^{2},$

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Symbols:: : real, $\alpha$ : arbitrary constant, $\gamma$ : arbitrary constant and : constant
Permalink:: http://dlmf.nist.gov/32.9.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(i), §32.9 and Ch.32

with an arbitrary constant, which is solvable by quadrature. A similar result holds when $\alpha=\gamma=0$ . $\mbox{P}_{\mbox{\scriptsize III}}$ with $\alpha=\beta=\gamma=\delta=0$ , has the general solution $w(z)=Cz^{\mu}$ , with and $\mu$ arbitrary constants.

§32.9(ii) Fifth Painlevé Equation

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Notes:: See Lukaševič (1967b), Gromak and Lukaševič (1982); also Gromak et al. (2002, §38).
Permalink:: http://dlmf.nist.gov/32.9.ii
See also:: Annotations for §32.9 and Ch.32

Elementary nonrational solutions of $\mbox{P}_{\mbox{\scriptsize V}}$ are

32.9.7 $w(z;\mu,-\tfrac{1}{8},-\mu\kappa^{2},0)=1+\kappa z^{1/2},$

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Symbols:: : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(ii), §32.9 and Ch.32

32.9.8 $w(z;0,0,\mu,-\tfrac{1}{2}\mu^{2})=\kappa\exp\left(\mu z\right),$

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Symbols:: $\exp\NVar{z}$ : exponential function, : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(ii), §32.9 and Ch.32

with $\kappa$ and $\mu$ arbitrary constants.

$\mbox{P}_{\mbox{\scriptsize V}}$ , with $\delta=0$ , has algebraic solutions if either

32.9.9 $\mspace{17.0mu}(\alpha,\beta,\gamma)=(\tfrac{1}{2}\mu^{2},-\tfrac{1}{8}(2n-1)^% {2},-1),$

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Symbols:: : integer, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(ii), §32.9 and Ch.32

32.9.10 $\mspace{5.0mu}(\alpha,\beta,\gamma)=(\tfrac{1}{8}(2n-1)^{2},-\tfrac{1}{2}\mu^{% 2},1),$

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Symbols:: : integer, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(ii), §32.9 and Ch.32

with $n\in\mathbb{Z}$ and $\mu$ arbitrary. These are rational solutions in $\zeta=z^{1/2}$ of the form

32.9.11 $w(z)=\ifrac{P_{n^{2}-n+1}(\zeta)}{Q_{n^{2}-n}(\zeta)},$

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Symbols:: : integer, : real, $P_{n^{2}+1}(\zeta)$ : polynomials and $Q_{n^{2}}(\zeta)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.9.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(ii), §32.9 and Ch.32

where $P_{n^{2}-n+1}(\zeta)$ and $Q_{n^{2}-n}(\zeta)$ are polynomials of degrees $n^{2}-n+1$ and $n^{2}-n$ , respectively, with no common zeros.

$\mbox{P}_{\mbox{\scriptsize V}}$ , with $\gamma=\delta=0$ , has a first integral

32.9.12 $z^{2}(w^{\prime})^{2}=(w-1)^{2}(2\alpha w^{2}+Cw-2\beta),$

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Symbols:: : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant and : constant
Permalink:: http://dlmf.nist.gov/32.9.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(ii), §32.9 and Ch.32

with an arbitrary constant, which is solvable by quadrature. For examples and plots see Clarkson (2005). $\mbox{P}_{\mbox{\scriptsize V}}$ , with $\alpha=\beta=0$ and $\gamma^{2}+2\delta=0$ , has solutions $w(z)=C\exp\left(\pm\sqrt{-2\delta}z\right)$ , with an arbitrary constant.

§32.9(iii) Sixth Painlevé Equation

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Keywords:: Painlevé equations, elementary solutions
Notes:: See Hitchin (1995).
Permalink:: http://dlmf.nist.gov/32.9.iii
See also:: Annotations for §32.9 and Ch.32

An elementary algebraic solution of $\mbox{P}_{\mbox{\scriptsize VI}}$ is

32.9.13 $w(z;\tfrac{1}{2}\kappa^{2},-\tfrac{1}{2}\kappa^{2},\tfrac{1}{2}\mu^{2},\tfrac{% 1}{2}(1-\mu^{2}))=z^{1/2},$

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Symbols:: : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §32.9(iii), §32.9 and Ch.32

with $\kappa$ and $\mu$ arbitrary constants.

Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\beta=\gamma=0$ , $\delta=\tfrac{1}{2}$ . 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).

32.8 Rational Solutions 32.10 Special Function Solutions

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§32.9 Other Elementary Solutions

Contents

§32.9(i) Third Painlevé Equation

§32.9(ii) Fifth Painlevé Equation

§32.9(iii) Sixth Painlevé Equation