Painlevé equations and applications

TLDR: In this article, the authors use averaged properties on a disc of variable radius to detect the Painleve property of a difference equation, which is used as a tool for detecting the integrability of difference equations.

Abstract: The theme running throughout this thesis is the Painleve equations, in their differential, discrete and ultra-discrete versions The differential Painleve equations have the Painleve property If all solutions of a differential equation are meromorphic functions then it necessarily has the Painleve property Any ODE with the Painleve 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]