§32.2 Differential Equations

The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows:

with $\alpha$, $\beta$, $\gamma$, and $\delta$ arbitrary constants. The solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are called the Painlevé transcendents. The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions.

Let

be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.

There are fifty equations with the Painlevé property. They are distinct modulo Möbius (bilinear) transformations

in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$.

For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF.

If $\gamma \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\gamma =1$ and $\delta =-1$, without loss of generality, by rescaling $w$ and $z$ if necessary. If $\gamma =0$ and $\alpha \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\alpha =1$ and $\delta =-1$, without loss of generality. Lastly, if $\delta =0$ and $\beta \gamma \ne 0$, then set $\beta =-1$ and $\gamma =1$, without loss of generality.

If $\delta \ne 0$ in ${\text{P}}_{\text{V}}$, then set $\delta =-\frac{1}{2}$, without loss of generality.

In ${\text{P}}_{\text{III}}$, if $w(z)={\zeta }^{-1/2}u(\zeta )$ with $\zeta =z^2$, then

which is known as ${\text{P}}_{\text{III}}^{\prime }$.

In ${\text{P}}_{\text{III}}$, if $w(z)=\exp \left(-\mathrm{i}u(z)\right)$, $\beta =-\alpha$, and $\delta =-\gamma$, then

In ${\text{P}}_{\text{IV}}$, if $w(z)=2\sqrt{2}{(u(\zeta ))}^2$ with $\zeta =\sqrt{2}z$ and $\alpha =2\nu +1$, then

When $\beta =0$ this is a nonlinear harmonic oscillator.

In ${\text{P}}_{\text{V}}$, if $w(z)={(\coth u(\zeta ))}^2$ with $\zeta =\ln z$, then

See also Okamoto (1987c), McCoy et al. (1977), Bassom et al. (1992), Bassom et al. (1995), and Takasaki (2001).

${\text{P}}_{\text{VI}}$ can be written in the form

where

See Fuchs (1907), Painlevé (1906), Gromak et al. (2002, §42); also Manin (1998).

Let

where ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ are constants, $f_1$, $f_2$, $f_3$ are functions of $z$, with

Then $w(z)=f_1(z)$ satisfies ${\text{P}}_{\text{IV}}$ with

See Noumi and Yamada (1998).

Next, let

where ${\mu }_1$, ${\mu }_2$, ${\mu }_3$, ${\mu }_4$ are constants, $f_1$, $f_2$, $f_3$, $f_4$ are functions of $z$, with

Then $w(z)=1-(\sqrt{z}/f_1(z))$ satisfies ${\text{P}}_{\text{V}}$ with

${\text{P}}_{\text{I}}$–${\text{P}}_{\text{V}}$ are obtained from ${\text{P}}_{\text{VI}}$ by a coalescence cascade:

For example, if in ${\text{P}}_{\text{II}}$

then

thus in the limit as $ϵ\to 0$, $W(\zeta )$ satisfies ${\text{P}}_{\text{I}}$ with $z=\zeta$.

If in ${\text{P}}_{\text{III}}$

then as $ϵ\to 0$, $W(\zeta ;a)$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =a$.

If in ${\text{P}}_{\text{IV}}$

then as $ϵ\to 0$, $W(\zeta ;a)$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =a$.

If in ${\text{P}}_{\text{V}}$

then as $ϵ\to 0$, $W(\zeta ;a,b,c,d)$ satisfies ${\text{P}}_{\text{III}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =d$.

If in ${\text{P}}_{\text{V}}$

then as $ϵ\to 0$, $W(\zeta ;a,b)$ satisfies ${\text{P}}_{\text{IV}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$.

Lastly, if in ${\text{P}}_{\text{VI}}$

then as $ϵ\to 0$, $W(\zeta ;a,b,c,d)$ satisfies ${\text{P}}_{\text{V}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =d$.