§32.8 Rational Solutions

Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.8

§32.8(i) Introduction
§32.8(ii) Second Painlevé Equation
§32.8(iii) Third Painlevé Equation
§32.8(iv) Fourth Painlevé Equation
§32.8(v) Fifth Painlevé Equation
§32.8(vi) Sixth Painlevé Equation

§32.8(i) Introduction

Permalink:: http://dlmf.nist.gov/32.8.i

$\mbox{P}_{{\mbox{\scriptsize II}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. See Airault (1979).

§32.8(ii) Second Painlevé Equation

Notes:: See Vorob’ev (1965), Yablonskiĭ (1959), Flaschka and Newell (1980), Kajiwara and Ohta (1996); also Gromak et al. (2002, §20).
Permalink:: http://dlmf.nist.gov/32.8.ii

Rational solutions of $\mbox{P}_{{\mbox{\scriptsize II}}}$ exist for $\alpha=n(\in\Integer)$ and are generated using the seed solution $w(z;0)=0$ and the Bäcklund transformations (32.7.1) and (32.7.2). The first four are

32.8.1 $w(z;1)=-\ifrac{1}{z},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E1
Encodings:: TeX, pMML, png

32.8.2 $w(z;2)=\frac{1}{z}-\frac{3z^{2}}{z^{3}+4},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E2
Encodings:: TeX, pMML, png

32.8.3 $w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: TeX, pMML, png

32.8.4 $w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{3}+40)}{z^{9}+60z^{6}+11200}.$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: TeX, pMML, png

More generally,

32.8.5 $w(z;n)=\frac{d}{dz}\left(\mathop{\ln\/}\nolimits\!\left(\frac{Q_{{n-1}}(z)}{Q_{n}(z)}\right)\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : integer, $z$ : real and $Q_{n}(z)$ : polynomial
Permalink:: http://dlmf.nist.gov/32.8.E5
Encodings:: TeX, pMML, png

where the $Q_{n}(z)$ are monic polynomials (coefficient of highest power of $z$ is 1) satisfying

32.8.6 $Q_{{n+1}}(z)Q_{{n-1}}(z)={zQ_{n}^{2}(z)+4\left(Q_{n}^{{\prime}}(z)\right)^{2}-4Q_{n}(z)Q_{n}^{{\prime\prime}}(z)},$

Symbols:: $n$ : integer, $z$ : real and $Q_{n}(z)$ : polynomial
Permalink:: http://dlmf.nist.gov/32.8.E6
Encodings:: TeX, pMML, png

with $Q_{0}(z)=1$ , $Q_{1}(z)=z$ . Thus

32.8.7

$Q_{2}(z)=z^{3}+4,$

$Q_{3}(z)=z^{6}+20z^{3}-80,$

$Q_{4}(z)=z^{{10}}+60z^{7}+11200z,$

$Q_{5}(z)=z^{{15}}+140z^{{12}}+2800z^{9}+78400z^{6}-3\; 13600z^{3}-62\; 72000,$

$Q_{6}(z)=z^{{21}}+280z^{{18}}+18480z^{{15}}+6\; 27200z^{{12}}-172\; 48000z^{9}+14488\; 32000z^{6}+1\; 93177\; 60000z^{3}-3\; 86355\; 20000.$

Symbols:: $z$ : real and $Q_{n}(z)$ : polynomial
Permalink:: http://dlmf.nist.gov/32.8.E7
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

Next, let $p_{m}(z)$ be the polynomials defined by $p_{m}(z)=0$ for $m<0$ , and

32.8.8 $\sum _{{m=0}}^{\infty}p_{m}(z)\lambda^{m}=\mathop{\exp\/}\nolimits\!\left(z\lambda-\tfrac{4}{3}\lambda^{3}\right).$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $m$ : integer, $z$ : real and $p_{m}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E8
Encodings:: TeX, pMML, png

Then for $n\geq 2$

32.8.9 $w(z;n)=\frac{d}{dz}\left(\mathop{\ln\/}\nolimits\!\left(\frac{\tau _{{n-1}}(z)}{\tau _{n}(z)}\right)\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : integer, $z$ : real and $\tau _{n}(z)$ : determinant
Permalink:: http://dlmf.nist.gov/32.8.E9
Encodings:: TeX, pMML, png

where $\tau _{n}(z)$ is the $n\times n$ determinant

32.8.10 $\tau _{n}(z)=\begin{vmatrix}p_{1}(z)&p_{3}(z)&\cdots&p_{{2n-1}}(z)\\ p_{1}^{{\prime}}(z)&p_{3}^{{\prime}}(z)&\cdots&p_{{2n-1}}^{{\prime}}(z)\\ \vdots&\vdots&\ddots&\vdots\\ p_{1}^{{(n-1)}}(z)&p_{3}^{{(n-1)}}(z)&\cdots&p_{{2n-1}}^{{(n-1)}}(z)\end{vmatrix}.$

Symbols:: $n$ : integer, $z$ : real, $p_{m}(z)$ : polynomials and $\tau _{n}(z)$ : determinant
Permalink:: http://dlmf.nist.gov/32.8.E10
Encodings:: TeX, pMML, png

For plots of the zeros of $Q_{n}(z)$ see Clarkson and Mansfield (2003).

§32.8(iii) Third Painlevé Equation

Notes:: See Gromak and Lukaševič (1982), Lukaševič (1967b), Murata (1995); also Gromak et al. (2002, §35).
Permalink:: http://dlmf.nist.gov/32.8.iii

Special rational solutions of $\mbox{P}_{{\mbox{\scriptsize III}}}$ are

32.8.11 $w(z;\mu,-\mu\kappa^{2},\lambda,-\lambda\kappa^{4})=\kappa,$

Symbols:: $z$ : real, $\kappa$ : contstant, $\lambda$ : contstant and $\mu$ : contstant
Permalink:: http://dlmf.nist.gov/32.8.E11
Encodings:: TeX, pMML, png

32.8.12 $w(z;0,-\mu,0,\mu\kappa)=\kappa z,$

Symbols:: $z$ : real, $\kappa$ : contstant and $\mu$ : contstant
Permalink:: http://dlmf.nist.gov/32.8.E12
Encodings:: TeX, pMML, png

32.8.13 $w(z;2\kappa+3,-2\kappa+1,1,-1)=\dfrac{z+\kappa}{z+\kappa+1},$

Symbols:: $z$ : real and $\kappa$ : contstant
Permalink:: http://dlmf.nist.gov/32.8.E13
Encodings:: TeX, pMML, png

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

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

32.8.14 $\alpha\pm\beta=4n,$

Symbols:: $n$ : integer, $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.8.E14
Encodings:: TeX, pMML, png

with $n\in\Integer$ . These solutions have the form

32.8.15 $w(z)=\ifrac{P_{m}(z)}{Q_{m}(z)},$

Symbols:: $m$ : integer, $z$ : real, $P_{m}(z)$ : polynomials and $Q_{m}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E15
Encodings:: TeX, pMML, png

where $P_{m}(z)$ and $Q_{m}(z)$ are polynomials of degree $m$ , with no common zeros.

For examples and plots see Milne et al. (1997); also Clarkson (2003a). For determinantal representations see Kajiwara and Masuda (1999).

§32.8(iv) Fourth Painlevé Equation

Notes:: See Lukaševič (1967a), Gromak (1987), Murata (1985); also Gromak et al. (2002, §26).
Permalink:: http://dlmf.nist.gov/32.8.iv

Special rational solutions of $\mbox{P}_{{\mbox{\scriptsize IV}}}$ are

32.8.16 $w_{1}(z;\pm 2,-2)=\pm\ifrac{1}{z},$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E16
Encodings:: TeX, pMML, png

32.8.17 $w_{2}(z;0,-2)=-2z,$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E17
Encodings:: TeX, pMML, png

32.8.18 $w_{3}(z;0,-\tfrac{2}{9})=-\tfrac{2}{3}z.$

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E18
Encodings:: TeX, pMML, png

There are also three families of solutions of $\mbox{P}_{{\mbox{\scriptsize IV}}}$ of the form

32.8.19 $w_{1}(z;\alpha _{1},\beta _{1})=\ifrac{P_{{1,n-1}}(z)}{Q_{{1,n}}(z)},$

Symbols:: $n$ : integer, $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $Q_{{j,n}}(z)$ : polynomials and $P_{{j,n}}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E19
Encodings:: TeX, pMML, png

32.8.20 $w_{2}(z;\alpha _{2},\beta _{2})=-2z+(\ifrac{P_{{2,n-1}}(z)}{Q_{{2,n}}(z)}),$

Symbols:: $n$ : integer, $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $Q_{{j,n}}(z)$ : polynomials and $P_{{j,n}}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E20
Encodings:: TeX, pMML, png

32.8.21 $w_{3}(z;\alpha _{3},\beta _{3})=-\tfrac{2}{3}z+(\ifrac{P_{{3,n-1}}(z)}{Q_{{3,n}}(z)}),$

Symbols:: $n$ : integer, $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $Q_{{j,n}}(z)$ : polynomials and $P_{{j,n}}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E21
Encodings:: TeX, pMML, png

where $P_{{j,n-1}}(z)$ and $Q_{{j,n}}(z)$ are polynomials of degrees $n-1$ and $n$ , respectively, with no common zeros.

In general, $\mbox{P}_{{\mbox{\scriptsize IV}}}$ has rational solutions iff either

32.8.22

$\alpha=m,$

$\beta=-2(1+2n-m)^{2},$

Symbols:: $m$ : integer, $n$ : integer, $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Referenced by:: §32.8(iv)
Permalink:: http://dlmf.nist.gov/32.8.E22
Encodings:: TeX, TeX, pMML, pMML, png, png

32.8.23

$\mspace{12.0mu}\alpha=m,$

$\beta=-2(\tfrac{1}{3}+2n-m)^{2},$

Symbols:: $m$ : integer, $n$ : integer, $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.8.E23
Encodings:: TeX, TeX, pMML, pMML, png, png

with $m,n\in\Integer$ . The rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv).

For examples and plots see Bassom et al. (1995); also Clarkson (2003b). For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999).

§32.8(v) Fifth Painlevé Equation

Notes:: See Gromak and Lukaševič (1982), Kitaev et al. (1994); also Gromak et al. (2002, §40).
Permalink:: http://dlmf.nist.gov/32.8.v

Special rational solutions of $\mbox{P}_{{\mbox{\scriptsize V}}}$ are

32.8.24 $w(z;\tfrac{1}{2},-\tfrac{1}{2}\mu^{2},\kappa(2-\mu),-\tfrac{1}{2}\kappa^{2})=\kappa z+\mu,$

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E24
Encodings:: TeX, pMML, png

32.8.25 $w(z;\tfrac{1}{2},\kappa^{2}\mu,2\kappa\mu,\mu)=\kappa/(z+\kappa),$

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E25
Encodings:: TeX, pMML, png

32.8.26 $w(z;\tfrac{1}{8},-\tfrac{1}{8},-\kappa\mu,\mu)=(\kappa+z)/(\kappa-z),$

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E26
Encodings:: TeX, pMML, png

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

In the general case assume $\delta\neq 0$ , so that as in §32.2(ii) we may set $\delta=-\tfrac{1}{2}$ . Then $\mbox{P}_{{\mbox{\scriptsize V}}}$ has a rational solution iff one of the following holds with $m,n\in\Integer$ and $\varepsilon=\pm 1$ :

$\alpha=\tfrac{1}{2}(m+\varepsilon\gamma)^{2}$ and $\beta=-\tfrac{1}{2}n^{2}$ , where $n>0$ , $m+n$ is odd, and $\alpha\neq 0$ when $|m|<n$ .
$\alpha=\tfrac{1}{2}n^{2}$ and $\beta=-\tfrac{1}{2}(m+\varepsilon\gamma)^{2}$ , where $n>0$ , $m+n$ is odd, and $\beta\neq 0$ when $|m|<n$ .
$\alpha=\tfrac{1}{2}a^{2}$ , $\beta=-\tfrac{1}{2}(a+n)^{2}$ , and $\gamma=m$ , with $m+n$ even.
$\alpha=\tfrac{1}{2}(b+n)^{2}$ , $\beta=-\tfrac{1}{2}b^{2}$ , and $\gamma=m$ , with $m+n$ even.
$\alpha=\tfrac{1}{8}(2m+1)^{2}$ , $\beta=-\tfrac{1}{8}(2n+1)^{2}$ , and $\gamma\notin\Integer$ .

These rational solutions have the form

32.8.27 $w(z)=\lambda z+\mu+(\ifrac{P_{{n-1}}(z)}{Q_{{n}}(z)}),$

Symbols:: $n$ : integer, $z$ : real, $\mu$ : constant, $P_{n}(z)$ : polynomials and $Q_{n}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E27
Encodings:: TeX, pMML, png

where $\lambda$ , $\mu$ are constants, and $P_{{n-1}}(z)$ , $Q_{{n}}(z)$ are polynomials of degrees $n-1$ and $n$ , respectively, with no common zeros. Cases (a) and (b) are special cases of §32.10(v).

For examples and plots see Clarkson (2005). For determinantal representations see Masuda et al. (2002). For the case $\delta=0$ see Airault (1979) and Lukaševič (1968).

§32.8(vi) Sixth Painlevé Equation

Notes:: See Mazzocco (2001a).
Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.8.vi

Special rational solutions of $\mbox{P}_{{\mbox{\scriptsize VI}}}$ are

32.8.28 $w(z;\mu,-\mu\kappa^{2},\tfrac{1}{2},\tfrac{1}{2}-\mu(\kappa-1)^{2})=\kappa z,$

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E28
Encodings:: TeX, pMML, png

32.8.29 $w(z;0,0,2,0)=\kappa z^{2},$

Symbols:: $z$ : real and $\kappa$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E29
Encodings:: TeX, pMML, png

32.8.30 $w(z;0,0,\tfrac{1}{2},-\tfrac{3}{2})=\ifrac{\kappa}{z},$

Symbols:: $z$ : real and $\kappa$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E30
Encodings:: TeX, pMML, png

32.8.31 $w(z;0,0,2,-4)=\ifrac{\kappa}{z^{2}},$

Symbols:: $z$ : real and $\kappa$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E31
Encodings:: TeX, pMML, png

32.8.32 $w(z;\tfrac{1}{2}(\kappa+\mu)^{2},-\tfrac{1}{2},\tfrac{1}{2}(\mu-1)^{2},\tfrac{1}{2}\kappa(2-\kappa))=\dfrac{z}{\kappa+\mu z},$

Symbols:: $z$ : real, $\kappa$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/32.8.E32
Encodings:: TeX, pMML, png

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

In the general case, $\mbox{P}_{{\mbox{\scriptsize VI}}}$ has rational solutions if

32.8.33 $a+b+c+d=2n+1,$

Symbols:: $n$ : integer
Permalink:: http://dlmf.nist.gov/32.8.E33
Encodings:: TeX, pMML, png

where $n\in\Integer$ , $a=\varepsilon _{1}\sqrt{2\alpha}$ , $b=\varepsilon _{2}\sqrt{-2\beta}$ , $c=\varepsilon _{3}\sqrt{2\gamma}$ , and $d=\varepsilon _{4}\sqrt{1-2\delta}$ , with $\varepsilon _{j}=\pm 1$ , $j=1,2,3,4$ , independently, and at least one of $a$ , $b$ , $c$ or $d$ is an integer. These are special cases of §32.10(vi).