A New Technique in Nonlinear Singularity Analysis

Abstract

The test for singularity structure proposed by Ablowitz, Ramani and Segur (1980) has been widely used to obtain necessary conditions for non-linear ordinary differential equations (ODEs) to have the Painlevé property. In this paper, we provide an example of an ODE for which this standard test breaks down in a new and surprising way. We show how to overcome the problems presented by this equation by developing a new technique based on regular singular points of bilinear equations.