§32.8 Rational Solutions

${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ 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).

Rational solutions of ${\text{P}}_{\text{II}}$ exist for $\alpha =n(\in \mathbb{Z})$ 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

More generally,

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

with $Q_0(z)=1$, $Q_1(z)=z$. Thus

Next, let $p_m(z)$ be the polynomials defined by $p_m(z)=0$ for , and

Then for $n\ge 2$

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

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

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

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

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

with $n\in \mathbb{Z}$. These solutions have the form

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).

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

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

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, ${\text{P}}_{\text{IV}}$ has rational solutions iff either

or

with $m,n\in \mathbb{Z}$. 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).

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

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

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

1. (a)

   $\alpha =\frac{1}{2}{(m+\epsilon \gamma )}^2$ and $\beta =-\frac{1}{2}{n}^2$, where $n>0$, $m+n$ is odd, and $\alpha \ne 0$ when .

2. (b)

   $\alpha =\frac{1}{2}{n}^2$ and $\beta =-\frac{1}{2}{(m+\epsilon \gamma )}^2$, where $n>0$, $m+n$ is odd, and $\beta \ne 0$ when .

3. (c)

   $\alpha =\frac{1}{2}{a}^2$, $\beta =-\frac{1}{2}{(a+n)}^2$, and $\gamma =m$, with $m+n$ even.

4. (d)

   $\alpha =\frac{1}{2}{(b+n)}^2$, $\beta =-\frac{1}{2}{b}^2$, and $\gamma =m$, with $m+n$ even.

5. (e)

   $\alpha =\frac{1}{8}{(2m+1)}^2$, $\beta =-\frac{1}{8}{(2n+1)}^2$, and $\gamma \notin \mathbb{Z}$.

These rational solutions have the form

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).

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

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

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

where $n\in \mathbb{Z}$, $a={\epsilon }_1\sqrt{2\alpha }$, $b={\epsilon }_2\sqrt{-2\beta }$, $c={\epsilon }_3\sqrt{2\gamma }$, and $d={\epsilon }_4\sqrt{1-2\delta }$, with ${\epsilon }_j=\pm 1$, $j=1,2,3,4$, independently, and at least one of $a$, $b$, $c$ or $d$ is an integer. These are special cases of §32.10(vi).