# §32.8 Rational Solutions

## §32.8(i) Introduction

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

Rational solutions of  exist for and are generated using the seed solution and the Bäcklund transformations (32.7.1) and (32.7.2). The first four are

32.8.1
32.8.2
32.8.3
32.8.4

More generally,

where the are monic polynomials (coefficient of highest power of is 1) satisfying

32.8.6

with , . Thus

32.8.7

Next, let be the polynomials defined by for , and

Then for

where is the determinant

32.8.10

For plots of the zeros of see Clarkson and Mansfield (2003).

## §32.8(iii) Third Painlevé Equation

Special rational solutions of  are

32.8.11
32.8.12
32.8.13

with , , and arbitrary constants.

In the general case assume , so that as in §32.2(ii) we may set and . Then  has rational solutions iff

with . These solutions have the form

32.8.15

where and are polynomials of degree , 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

Special rational solutions of  are

32.8.16
32.8.17
32.8.18

There are also three families of solutions of  of the form

where and are polynomials of degrees and , respectively, with no common zeros.

In general,  has rational solutions iff either

32.8.22

or

with . 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

Special rational solutions of  are

32.8.24
32.8.25
32.8.26

with and arbitrary constants.

In the general case assume , so that as in §32.2(ii) we may set . Then  has a rational solution iff one of the following holds with and :

1. (a)

and , where , is odd, and when .

2. (b)

and , where , is odd, and when .

3. (c)

, , and , with even.

4. (d)

, , and , with even.

5. (e)

, , and .

These rational solutions have the form

where , are constants, and , are polynomials of degrees and , 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 see Airault (1979) and Lukaševič (1968).

## §32.8(vi) Sixth Painlevé Equation

Special rational solutions of  are

32.8.28
32.8.29
32.8.30
32.8.31
32.8.32

with and arbitrary constants.

In the general case,  has rational solutions if

32.8.33

where , , , , and , with , , independently, and at least one of , , or is an integer. These are special cases of §32.10(vi).