Painleve equations and applications

The theme running throughout this thesis is the Painlevé equations, in their differential, discrete and ultra-discrete versions. The differential Painlevé equations have the Painlevé property. If all solutions of a differential equation are meromorphic functions then it necessarily has the Painlevé property. Any ODE with the Painlevé property is necessarily a reduction of an integrable PDE. Nevanlinna theory studies the value distribution and characterizes the growth of meromorphic functions, by using certain averaged properties on a disc of variable radius. We shall be interested in its well-known use as a tool for detecting integrability in difference equations—a difference equation may be integrable if it has sufficiently many finite-order solutions in the sense of Nevanlinna theory. This does not provide a sufficient test for integrability; additionally it must satisfy the well-known singularity confinement test. [Continues.]