Integrability tests for discrete and differential equations

Differential and difference equations arise in all areas of the mathematical sciences from population dynamics to relativity to the most abstract areas of pure mathematics. When one is considers a particular equation it is natural to ask whether this equation can be solved explicitly or at least whether we can characterise the solutions in a nice way. In other words, is the equation integrable? This question lies at the heart of much of this project.A large part of the project is directed towards the use of well-defined properties that are correlated with integrability and the development of powerful tests for these properties. For difference and discrete equations, three such properties will be used. One property involves studying difference equations in the complex plane. Another property involves studying how complicated rational functions (ratios of polynomials) get when iterated by a discrete equation. The final property involves looking at rational numbers that are generated by discrete equations and considering how quickly their numerators and denominates grow under iteration. This project is not only aimed at providing useful tests for applied mathematicians and scientists to apply to real world equations, but it is also aimed at providing a rigorous basis for part of the theory.For ordinary differential equations (ODEs) it has long been known that equations whose solutions have a a very simple singularity structure are integrable. Another main aim of this research is to understand the types of (movable) singularities that solutions of classes of ODEs can develop. This is aimed at a better understanding of what makes certain integrable equations special and also towards developing tests to determine when an equation has a global singularity structure that is neither trivial nor very bad so the equation might be integrable. This work has also led to methods for detecting special classes of solutions in otherwise non-integrable equations.