Movable algebraic singularities of second-order ordinary differential equations

Abstract

Any nonlinear equation of the form y″=∑n=0Nan(z)yn has a solution with leading behavior proportional to (z−z0)−2/(N−1) about a point z0, where the coefficients an are analytic at z0 and aN(z0)≠0. Equations are considered for which each possible leading term of this form extends to a Laurent series solution in fractional powers of z−z0. For these 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 generalizes results of Shimomura [“On second order nonlinear differential equations with the quasi-Painlevé property II,” RIMS Kokyuroku 1424, 177 (2005)]. The possibility that these algebraic singularities could accumulate along infinitely long paths ending at a finite point is considered. Smith [“On the singularities in the complex plane of the solutions of y″+y′f(y)+g(y)=P(x),” Proc. Lond. Math. Soc. 3, 498 (1953)] showed that such singularities do occur in solutions of a simple equation outside this class.