Integrability tests for discrete and differential equations

微分方程式和差分方程式出现在数学科学的所有领域，从人口动力学到相对论，再到最抽象的纯数学领域。当一个人考虑一个特定的方程时，很自然地会问这个方程是否可以显式地解，或者至少我们是否可以用一种好的方式来刻画解的特征。换句话说，这个方程是可积的吗？这个问题是这个项目的核心。这个项目的很大一部分是针对与可积性相关的定义良好的性质的使用，以及对这些性质的强大测试的开发。对于差分方程组和离散方程，将使用三个这样的性质。其中一个性质涉及研究复平面中的差分方程组。另一个性质涉及研究复杂的有理函数(多项式的比率)在离散方程迭代时如何得到。最后一个性质涉及查看由离散方程生成的有理数，并考虑它们的分子和分母在迭代下增长得有多快。这个项目不仅旨在为应用数学家和科学家应用于现实世界的方程提供有用的测试，而且还旨在为部分理论提供严格的基础。对于常微分方程组(ODE)，人们很早就知道，解具有非常简单奇点结构的方程是可积的。这项研究的另一个主要目的是了解一类常微分方程组的解可以发展出的(可移动的)奇点的类型。这是为了更好地理解某些可积方程的特殊之处，也是为了开发测试来确定一个方程何时具有既不是微不足道也不是非常糟糕的全局奇点结构，因此该方程可能是可积的。这项工作还导致了检测不可积方程中特殊类型的解的方法。

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.