…
►
–
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.
…
►More
generally,
…
►In
general,
has rational solutions iff either
…
►In the
general case assume
, so that as in §
32.2(ii) we may set
.
…
►In the
general case,
has rational solutions if
…
…
►From this graph we estimate an initial
value
.
…
►No explicit
general formulas exist when
.
…
►Table
3.8.3 gives the successive
values of
and
.
…
►Consider
and
.
We have
and
.
…
…
►Wrench (1968) gives exact
values of
up to
.
Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D
values of
for
.
…
►uniformly for bounded real
values of
.
…
►In terms of
generalized Bernoulli polynomials
(§
24.16(i)), we have for
,
►
5.11.17
…
…
►They
generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§
36.4).
…
►The first sheet corresponds to
and is
generated as a solution of Equations (
36.5.6)–(
36.5.9).
The second sheet corresponds to
and it intersects the bifurcation set (§
36.4) smoothly along the line
generated by
,
.
For
the second sheet is
generated by a second solution of (
36.5.6)–(
36.5.9), and for
it is
generated by the roots of the polynomial equation
…
►the intersection lines with the bifurcation set are
generated by
,
.
…