Symmetry and the Chazy equation
It begins with a discussion on the Chazy o.d.e., (1) \(y_{xxx} = 2yy_{xx} - 3y^2_x\), arising from the study of third order equations having the ``Painlevé property'', namely solutions having poles for movable singularities. Equation (1) also is the simplest o.d.e. having a movable natural boundary which means a closed curve in \(\mathbb{C}\) plane beyond which the solution cannot be analytically continued. Chazy relates (1) to the linear hypergeometric equation \[ t(1 - t) {dx^2 \over dt^2} + \left( {1 \over 2} - {7 \over 6} t \right) {dx \over dt} + {1 \over 144} x = 0. \tag{2} \] The author then shows how solutions to (1) can be constructed from that of the Lamé equations, which can be further related to the hypogeometric equation (2) via an elliptic change of variables. The main focus of the paper is how a unimodular Lie group \(SL (2, \mathbb{C})\) works on a two-dimensional space. In each case a careful analysis is presented on how an o.d.e. admitting \(SL (2)\) as a symmetry group can be reduced in order by 3 and the solution recovered from that of the reduced equation implementing a pair of quadratures and the solution to a linear second order equation. This general result is then focused on the Chazy equation whose general solution can be expressed as a ratio of two solutions of (2). A more general form of (1) is studied by adding the term, \(\alpha (6y_x - y^2)^2\), and when \(\alpha = 0\) or \(\alpha = {4 \over 36 - k^2}\), \(0 < k \in \mathbb{N}\) then the nontrivial solution to the generalized Chazy equation has a movable circular natural boundary. The reduction method leads to an alternative formula in terms of the solutions to Lamé equations resulting in a remarkable transformation between Lamé and hypergeometric equations. The paper concludes with a Painlevé analysis on the singular solutions to the generalized Chazy equation.
scientific article; zbMATH DE number 838716
scientific article