Natural Science Forum / Physics / General Physics / April 2007

Katsunori Iwasak and Takato Uehara (Kyushu U., Japan), "Periodic
solutions to Painleve VI and dynamical system on cubic surface," math/
0512583 v3 (math.A.G.) 6 Jan 2006, 26 pages, show us some of the cubic
and second order operator generalizations/modifications/relationships
of the Riccati Differential Equation, although their interest happens
to be more in the Painleve VI equations because they are so difficult
to solve and analyze.

The Painleve Equations belong to 6 major classes which are
generalization-modifications of the Riccati Differential Equation:

1) dy/dt = A(t) + B(t)y + C(t)y^2

With the Painleve Equations, time (t) is not necessarily involved, and
it can be replaced by any other continuous variable v such that dy/dv
is defined.

For example, P_I, the first type of Painleve Equation, is in the
notation of Peter A. Clarkson of U. Kent, U.K., "The Painleve
equations - nonlinear special functions,"
www.ima.umn.edu/talks/workshops/7-22-8-2-2002/clarkson/SFO2L.pdf, 87
pages:

2) Dzz(w) = 6w^2 + z

where Dzz is the second derivative with respect to z.

The second Painleve Equation is:

3) Dzz(w) = 2w^3 + zw + a

I'll try to continue this shortly.

Osher Doctorow