Article

Nonlocal representation of the sl(2, R) algebra for the Chazy equation

Published in: Quaestiones Mathematicae
Volume 42 , issue 1 , 2019 , pages: 125–133

DOI: 10.2989/16073606.2018.1441199

Author(s): Sameerah Jamal School of Mathematics and Centre for Differential Equations, Continuum Mechanics and Applications, South Africa , P.G.L. Leach Department of Mathematics and Institute of Systems Science, Research and Postgraduate Support, South Africa , Andronikos Paliathanasis Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Chile

Keywords: 34M25 , 34M55 , 70G65 , 34M25 , 34M55 , 70G65

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Abstract

A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.

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