§32.9 Other Elementary Solutions

Permalink:: http://dlmf.nist.gov/32.9

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

§32.9(i) Third Painlevé Equation

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

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

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

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E1
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : real, $\kappa$ : constant, $\lambda$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E2
Encodings:: TeX, pMML, png

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

Symbols:: $z$ : real, $\kappa$ : constant, $\lambda$ : constant, $\mu$ : constant and $\nu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E3
Encodings:: TeX, pMML, png

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 $\beta=2n,$

Symbols:: $n$ : integer and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.9.E4
Encodings:: TeX, pMML, png

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

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

Symbols:: $n$ : integer, $z$ : 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

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

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\gamma$ : arbitrary constant and $C$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E6
Encodings:: TeX, pMML, png

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

§32.9(ii) Fifth Painlevé Equation

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

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

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

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E7
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E8
Encodings:: TeX, pMML, png

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

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

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

Symbols:: $n$ : 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

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

Symbols:: $n$ : 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

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

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

Symbols:: $n$ : integer, $z$ : 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

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

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant and $C$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E12
Encodings:: TeX, pMML, png

with $C$ 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 $w(z)=C\mathop{\exp\/}\nolimits\!\left(\pm\sqrt{-2\delta}z\right)$ , with $C$ an arbitrary constant.

§32.9(iii) Sixth Painlevé Equation

Notes:: See Hitchin (1995).
Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.9.iii

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

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

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.9.E13
Encodings:: TeX, pMML, png

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).

§32.9 Other Elementary Solutions

Contents

§32.9(i) Third Painlevé Equation

§32.9(ii) Fifth Painlevé Equation

§32.9(iii) Sixth Painlevé Equation