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

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

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

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

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

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

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

32.9.4 | $$\beta =2n,$$ | ||

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

32.9.5 | $$w(z)={P}_{{n}^{2}+1}(\zeta )/{Q}_{{n}^{2}}(\zeta ),$$ | ||

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 $\beta \gamma \ne 0$.

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

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

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

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

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

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

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

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

32.9.9 | $$(\alpha ,\beta ,\gamma )=(\frac{1}{2}{\mu}^{2},-\frac{1}{8}{(2n-1)}^{2},-1),$$ | ||

or

32.9.10 | $$(\alpha ,\beta ,\gamma )=(\frac{1}{8}{(2n-1)}^{2},-\frac{1}{2}{\mu}^{2},1),$$ | ||

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

32.9.11 | $$w(z)={P}_{{n}^{2}-n+1}(\zeta )/{Q}_{{n}^{2}-n}(\zeta ),$$ | ||

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.

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

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

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

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

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

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

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