Rod Halburd (UCL)
Movable singularities of nonlinear differential equations
Abstract: If the location of a singularity of an ODE varies as the initial
conditions change, then the singularity is said to be movable. This is on
contrast to fixed singularities, which only occur at points where the
equation is singular in some sense. Movable singularities in the complex
domain play a special role in the theory of integrable systems. In this
talk it will be shown that for several classes of second-order ordinary
differential equations, the only movable singularities that can be reached
by analytic continuation along finite-length curves are poles or algebraic
branch points. Although these solutions have simple local singularity
structure, globally they can be very complicated. We use Nevanlinna
theory to classify some equations that admit solutions that are algebraic
over the field of meromorphic functions.