Symmetry, the Chazy equation and Chazy hierarchies
Peter Clarkson
Abstract:
There are three different actions of the unimodular Lie group SL(2) on a
two-dimensional space. In every case, it is shown how an ordinary differential
equation admitting SL(2) as a symmetry group can be reduced in order by three,
and the solution recovered from that of the reduced equation via a pair of
quadratures and the solution to a linear second order equation. A particular
example is the Chazy equation, a nonlinear third order ordinary differential
equation whose general solution can be expressed as a ratio of two solutions of
a hypergeometric equation. The reduction method leads to an alternative formula
in terms of solutions to the Lamé equation, resulting in a surprising
transformation between the Lamé and hypergeometric equations.
This theory is a generalisation of Lie's group-theoretic method for reducing
the order of ODEs which opens up new possibilities for Painlevé
classification, especially at fifth order and above, which will be illustrated.
It will also be shown that the theory yields non-Painlevé equations with
very complicated singularity structure which can nevertheless be solved in
closed form or mapped into Painlevé equations.