Movable algebraic singularities of second-order ordinary differential equations

TL;DR

This paper investigates the nature of movable algebraic singularities in second-order nonlinear ODEs, showing that under certain conditions, solutions only have algebraic singularities reachable by finite paths, extending previous results.

Contribution

It generalizes earlier work by characterizing the only movable singularities in a broad class of second-order nonlinear equations, excluding more complex accumulation points.

Findings

Movable algebraic singularities are the only finite-curve reachable singularities.

Solutions in the considered class do not exhibit accumulation point singularities.

The results extend and generalize previous findings by Shimomura.

Abstract

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here.