权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Integrability tests for discrete and differential equations

离散方程和微分方程的可积性检验

基本信息

批准号：
EP/C54319X/2
负责人：
Rodney Halburd
金额：
--
依托单位：
University College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FC54319X%2F2
关键词：
Integrability tests discrete differential equations

项目摘要

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.

微分方程和差分方程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。当人们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求解，或者至少我们是否可以用一种很好的方式来求解。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使用与可积性相关的定义良好的属性，以及为这些属性开发强大的测试。对于差分和离散方程，将使用三个这样的性质。一个属性涉及研究复平面中的差分方程。另一个性质涉及研究当通过离散方程迭代时，有理函数（多项式的比率）有多复杂。最后一个性质涉及到研究由离散方程生成的有理数，并考虑它们的分子和命名在迭代下增长的速度。该项目不仅旨在为应用数学家和科学家提供有用的测试，以应用于真实的世界方程，而且旨在为部分理论提供严格的基础。对于常微分方程（ODE），人们早就知道解具有非常简单的奇异结构的方程是可积的。本研究的另一个主要目的是了解的类型（可移动）奇点的解决方案的常微分方程类可以开发。这是为了更好地理解是什么使某些可积方程的特殊性，也是为了开发测试，以确定当一个方程具有全局奇异结构，既不是平凡的，也不是非常糟糕，所以方程可能是可积的。这项工作也导致了方法检测特殊类的解决方案，否则不可积方程。