– 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 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,
32.8.5 | |||
where the are monic polynomials (coefficient of highest power of is ) satisfying
32.8.6 | |||
with , . Thus
32.8.7 | ||||
Next, let be the polynomials defined by for , and
32.8.8 | |||
Then for
32.8.9 | |||
where is the determinant
32.8.10 | |||
For plots of the zeros of see Clarkson and Mansfield (2003).
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
32.8.14 | |||
with . These solutions have the form
32.8.15 | |||
where and are polynomials of degree , with no common zeros.
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
32.8.19 | |||
32.8.20 | |||
32.8.21 | |||
where and are polynomials of degrees and , respectively, with no common zeros.
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 :
and , where , is odd, and when .
and , where , is odd, and when .
, , and , with even.
, , and , with even.
, , and .
These rational solutions have the form
32.8.27 | |||
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).
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).