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arXiv 1908.045632019-08-13nlin.SImath.CA

A Systematic Analysis of the Properties of the Generalised Painlevé--Ince Equation

Andronikos Paliathanasis, P. G. L. Leach

备注：: 5 pages, to appear in Quaestiones Mathematicae

英文摘要：

We consider the generalized Painlevé--Ince equation, \begin{equation*} \ddot{x}+αx\dot{x}+βx^{3}=0 \end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $β=α^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlevé test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlevé--Ince equation fails at the Painlevé test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlevé--Ince equation is integrable is terms of Lie symmetries and of the Painlevé test.

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