# §32.2 Differential Equations

## §32.2(i) Introduction

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

 32.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=6w^{2}+z,$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $z$: real Referenced by: §32.11(i), §32.11(i) Permalink: http://dlmf.nist.gov/32.2.E1 Encodings: TeX, pMML, png See also: Annotations for 32.2(i), 32.2 and 32
 32.2.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=2w^{3}+zw+\alpha,$
 32.2.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{1}{w}\left(\frac{\mathrm{d}w% }{\mathrm{d}z}\right)^{2}-\frac{1}{z}\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha w^{2}+\beta}{z}+\gamma w^{3}+\frac{\delta}{w},$
 32.2.4 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{1}{2w}\left(\frac{\mathrm{d}% w}{\mathrm{d}z}\right)^{2}+\frac{3}{2}w^{3}+4zw^{2}+2(z^{2}-\alpha)w+\frac{% \beta}{w},$
 32.2.5 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(\frac{1}{2w}+\frac{1}{w-1}% \right)\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-\frac{1}{z}\frac{% \mathrm{d}w}{\mathrm{d}z}+\frac{(w-1)^{2}}{z^{2}}\left(\alpha w+\frac{\beta}{w% }\right)+\frac{\gamma w}{z}+\frac{\delta w(w+1)}{w-1},$
 32.2.6 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{1}{2}\left(\frac{1}{w}+\frac% {1}{w-1}+\frac{1}{w-z}\right)\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-% \left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}\right)\frac{\mathrm{d}w}{\mathrm% {d}z}+\frac{w(w-1)(w-z)}{z^{2}(z-1)^{2}}\left(\alpha+\frac{\beta z}{w^{2}}+% \frac{\gamma(z-1)}{(w-1)^{2}}+\frac{\delta z(z-1)}{(w-z)^{2}}\right),$

with $\alpha$, $\beta$, $\gamma$, and $\delta$ arbitrary constants. The solutions of $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize 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

 32.2.7 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=F\left(z,w,\frac{\mathrm{d}w}{% \mathrm{d}z}\right),$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $z$: real Permalink: http://dlmf.nist.gov/32.2.E7 Encodings: TeX, pMML, png See also: Annotations for 32.2(i), 32.2 and 32

be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $\mathbb{C}$. 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

 32.2.8 $\displaystyle W(\zeta)$ $\displaystyle=\frac{a(z)w+b(z)}{c(z)w+d(z)},$ $\displaystyle\zeta$ $\displaystyle=\phi(z),$

in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\phi(z)$ 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 $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$.

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

## §32.2(ii) Renormalizations

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

If $\delta\neq 0$ in $\mbox{P}_{\mbox{\scriptsize V}}$, then set $\delta=-\tfrac{1}{2}$, without loss of generality.

## §32.2(iii) Alternative Forms

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

 32.2.9 $\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=\frac{1}{u}\left(\frac{\mathrm% {d}u}{\mathrm{d}\zeta}\right)^{2}-\frac{1}{\zeta}\frac{\mathrm{d}u}{\mathrm{d}% \zeta}+\frac{u^{2}(\alpha+\gamma u)}{4\zeta^{2}}+\frac{\beta}{4\zeta}+\frac{% \delta}{4u},$

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

In $\mbox{P}_{\mbox{\scriptsize III}}$, if $w(z)=\exp\left(-iu(z)\right)$, $\beta=-\alpha$, and $\delta=-\gamma$, then

 32.2.10 $\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}u}{% \mathrm{d}z}=\frac{2\alpha}{z}\sin u+2\gamma\sin\left(2u\right).$

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

 32.2.11 $\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=3u^{5}+2\zeta u^{3}+\left(% \tfrac{1}{4}\zeta^{2}-\nu-\tfrac{1}{2}\right)u+\frac{\beta}{32u^{3}}.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $\nu$ and $\beta$: arbitrary constant Referenced by: §32.3(iii) Permalink: http://dlmf.nist.gov/32.2.E11 Encodings: TeX, pMML, png See also: Annotations for 32.2(iii), 32.2 and 32

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

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

 32.2.12 $\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=-\frac{\alpha\cosh u}{2(\sinh u% )^{3}}-\frac{\beta\sinh u}{2(\cosh u)^{3}}-\tfrac{1}{4}\gamma e^{\zeta}\sinh% \left(2u\right)-\tfrac{1}{8}\delta e^{2\zeta}\sinh\left(4u\right).$

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

## §32.2(iv) Elliptic Form

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

 32.2.13 $z(1-z)I\left(\int_{\infty}^{w}\frac{\mathrm{d}t}{\sqrt{t(t-1)(t-z)}}\right)=% \sqrt{w(w-1)(w-z)}\*\left(\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1% )^{2}}+(\delta-\tfrac{1}{2})\frac{z(z-1)}{(w-z)^{2}}\right),$

where

 32.2.14 $I=z(1-z)\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}+(1-2z)\frac{\mathrm{d}}{% \mathrm{d}z}-\frac{1}{4}.$

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

## §32.2(v) Symmetric Forms

Let

 32.2.15 $\displaystyle\frac{\mathrm{d}f_{1}}{\mathrm{d}z}+f_{1}(f_{2}-f_{3})+2\mu_{1}$ $\displaystyle=0,$ $\displaystyle\frac{\mathrm{d}f_{2}}{\mathrm{d}z}+f_{2}(f_{3}-f_{1})+2\mu_{2}$ $\displaystyle=0,$ $\displaystyle\frac{\mathrm{d}f_{3}}{\mathrm{d}z}+f_{3}(f_{1}-f_{2})+2\mu_{3}$ $\displaystyle=0,$

where $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ are constants, $f_{1}$, $f_{2}$, $f_{3}$ are functions of $z$, with

 32.2.16 $\mu_{1}+\mu_{2}+\mu_{3}=1,$ ⓘ Symbols: $\mu_{j}$: constants Permalink: http://dlmf.nist.gov/32.2.E16 Encodings: TeX, pMML, png See also: Annotations for 32.2(v), 32.2 and 32
 32.2.17 $f_{1}(z)+f_{2}(z)+f_{3}(z)+2z=0.$ ⓘ Symbols: $z$: real and $f_{j}(z)$: solutions Permalink: http://dlmf.nist.gov/32.2.E17 Encodings: TeX, pMML, png See also: Annotations for 32.2(v), 32.2 and 32

Then $w(z)=f_{1}(z)$ satisfies $\mbox{P}_{\mbox{\scriptsize IV}}$ with

 32.2.18 $(\alpha,\beta)=(\mu_{3}-\mu_{2},-2\mu_{1}^{2}).$ ⓘ Symbols: $(\NVar{a},\NVar{b})$: open interval, $\alpha$: arbitrary constant, $\mu_{j}$: constants and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E18 Encodings: TeX, pMML, png See also: Annotations for 32.2(v), 32.2 and 32

Next, let

 32.2.19 $\displaystyle z\frac{\mathrm{d}f_{1}}{\mathrm{d}z}$ $\displaystyle=f_{1}f_{3}(f_{2}-f_{4})+(\tfrac{1}{2}-\mu_{3})f_{1}+\mu_{1}f_{3},$ $\displaystyle z\frac{\mathrm{d}f_{2}}{\mathrm{d}z}$ $\displaystyle=f_{2}f_{4}(f_{3}-f_{1})+(\tfrac{1}{2}-\mu_{4})f_{2}+\mu_{2}f_{4},$ $\displaystyle z\frac{\mathrm{d}f_{3}}{\mathrm{d}z}$ $\displaystyle=f_{3}f_{1}(f_{4}-f_{2})+(\tfrac{1}{2}-\mu_{1})f_{3}+\mu_{3}f_{1},$ $\displaystyle z\frac{\mathrm{d}f_{4}}{\mathrm{d}z}$ $\displaystyle=f_{4}f_{2}(f_{1}-f_{3})+(\tfrac{1}{2}-\mu_{2})f_{4}+\mu_{4}f_{2},$

where $\mu_{1}$, $\mu_{2}$, $\mu_{3}$, $\mu_{4}$ are constants, $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ are functions of $z$, with

 32.2.20 $\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}=1,$ ⓘ Symbols: $\mu_{j}$: constants Permalink: http://dlmf.nist.gov/32.2.E20 Encodings: TeX, pMML, png See also: Annotations for 32.2(v), 32.2 and 32
 32.2.21 $f_{1}(z)+f_{3}(z)=\sqrt{z},$ ⓘ Symbols: $z$: real and $f_{j}(z)$: solutions Permalink: http://dlmf.nist.gov/32.2.E21 Encodings: TeX, pMML, png See also: Annotations for 32.2(v), 32.2 and 32
 32.2.22 $f_{2}(z)+f_{4}(z)=\sqrt{z}.$ ⓘ Symbols: $z$: real and $f_{j}(z)$: solutions Permalink: http://dlmf.nist.gov/32.2.E22 Encodings: TeX, pMML, png See also: Annotations for 32.2(v), 32.2 and 32

Then $w(z)=1-(\sqrt{z}/f_{1}(z))$ satisfies $\mbox{P}_{\mbox{\scriptsize V}}$ with

 32.2.23 $(\alpha,\beta,\gamma,\delta)=(\tfrac{1}{2}\mu_{1}^{2},-\tfrac{1}{2}\mu_{3}^{2}% ,\mu_{4}-\mu_{2},-\tfrac{1}{2}).$

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

 32.2.24 $\begin{array}[]{ccccccc}\mbox{\mbox{P}_{\mbox{\scriptsize VI}}}&% \longrightarrow&\mbox{\mbox{P}_{\mbox{\scriptsize V}}}&\longrightarrow&\mbox% {\mbox{P}_{\mbox{\scriptsize IV}}}\\ &&\big{\downarrow}&&\big{\downarrow}\\ &&\mbox{\mbox{P}_{\mbox{\scriptsize III}}}&\longrightarrow&\mbox{\mbox{P}_{% \mbox{\scriptsize II}}}&\longrightarrow&\mbox{\mbox{P}_{\mbox{\scriptsize I}% }}\end{array}$ ⓘ Permalink: http://dlmf.nist.gov/32.2.E24 Encodings: TeX, pMML, png See also: Annotations for 32.2(vi), 32.2 and 32

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

 32.2.25 $w(z;\alpha)=\epsilon W(\zeta)+\frac{1}{\epsilon^{5}},$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant and $W(\zeta)$: bilinear transformation Permalink: http://dlmf.nist.gov/32.2.E25 Encodings: TeX, pMML, png See also: Annotations for 32.2(vi), 32.2 and 32
 32.2.26 $\displaystyle z$ $\displaystyle=\epsilon^{2}\zeta-\frac{6}{\epsilon^{10}},$ $\displaystyle\alpha$ $\displaystyle=\frac{4}{\epsilon^{15}},$ ⓘ Symbols: $z$: real and $\alpha$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E26 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 32.2(vi), 32.2 and 32

then

 32.2.27 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\zeta}^{2}}=6W^{2}+\zeta+\epsilon^{6}(2W^{% 3}+\zeta W);$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $W(\zeta)$: bilinear transformation Permalink: http://dlmf.nist.gov/32.2.E27 Encodings: TeX, pMML, png See also: Annotations for 32.2(vi), 32.2 and 32

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

If in $\mbox{P}_{\mbox{\scriptsize III}}$

 32.2.28 $w(z;\alpha,\beta,\gamma,\delta)=1+2\epsilon W(\zeta;a),$
 32.2.29 $\displaystyle z$ $\displaystyle=1+\epsilon^{2}\zeta,$ $\displaystyle\alpha$ $\displaystyle=-\tfrac{1}{2}\epsilon^{-6},$ $\displaystyle\beta$ $\displaystyle=\tfrac{1}{2}\epsilon^{-6}+2a\epsilon^{-3},$ $\displaystyle\gamma$ $\displaystyle=-\delta=\tfrac{1}{4}\epsilon^{-6},$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E29 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 32.2(vi), 32.2 and 32

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

If in $\mbox{P}_{\mbox{\scriptsize IV}}$

 32.2.30 $w(z;\alpha,\beta)=2^{2/3}\epsilon^{-1}W(\zeta;a)+\epsilon^{-3},$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant and $W(\zeta)$: bilinear transformation Permalink: http://dlmf.nist.gov/32.2.E30 Encodings: TeX, pMML, png See also: Annotations for 32.2(vi), 32.2 and 32
 32.2.31 $\displaystyle z$ $\displaystyle=2^{-2/3}\epsilon\zeta-\epsilon^{-3},$ $\displaystyle\alpha$ $\displaystyle=-2a-\tfrac{1}{2}\epsilon^{-6},$ $\displaystyle\beta$ $\displaystyle=-\tfrac{1}{2}\epsilon^{-12},$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E31 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 32.2(vi), 32.2 and 32

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

If in $\mbox{P}_{\mbox{\scriptsize V}}$

 32.2.32 $w(z;\alpha,\beta,\gamma,\delta)=1+\epsilon\zeta W(\zeta;a,b,c,d),$
 32.2.33 $\displaystyle z$ $\displaystyle=\zeta^{2},$ $\displaystyle\alpha$ $\displaystyle=\tfrac{1}{4}a\epsilon^{-1}+\tfrac{1}{8}c\epsilon^{-2},$ $\displaystyle\beta$ $\displaystyle=-\tfrac{1}{8}c\epsilon^{-2},$ $\displaystyle\gamma$ $\displaystyle=\tfrac{1}{4}\epsilon b,$ $\displaystyle\delta$ $\displaystyle=\tfrac{1}{8}\epsilon^{2}d,$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E33 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for 32.2(vi), 32.2 and 32

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

If in $\mbox{P}_{\mbox{\scriptsize V}}$

 32.2.34 $w(z;\alpha,\beta,\gamma,\delta)=\tfrac{1}{2}\sqrt{2}\epsilon W(\zeta;a,b),$
 32.2.35 $\displaystyle z$ $\displaystyle=1+\sqrt{2}\epsilon\zeta,$ $\displaystyle\alpha$ $\displaystyle=\tfrac{1}{2}\epsilon^{-4},$ $\displaystyle\beta$ $\displaystyle=\tfrac{1}{4}b,$ $\displaystyle\gamma$ $\displaystyle=-\epsilon^{-4},$ $\displaystyle\delta$ $\displaystyle=a\epsilon^{-2}-\tfrac{1}{2}\epsilon^{-4},$ ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E35 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for 32.2(vi), 32.2 and 32

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

Lastly, if in $\mbox{P}_{\mbox{\scriptsize VI}}$

 32.2.36 $w(z;\alpha,\beta,\gamma,\delta)=W(\zeta;a,b,c,d),$
 32.2.37 $\displaystyle z$ $\displaystyle=1+\epsilon\zeta,$ $\displaystyle\gamma$ $\displaystyle=c\epsilon^{-1}-d\epsilon^{-2},$ $\displaystyle\delta$ $\displaystyle=d\epsilon^{-2},$ ⓘ Symbols: $z$: real, $\gamma$: arbitrary constant and $\delta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.2.E37 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 32.2(vi), 32.2 and 32

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