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Applications of Nevanlinna theory to differential and difference equations

Nevanlinna 理论在微分方程和差分方程中的应用

基本信息

批准号：
EP/K041266/1
负责人：
Rodney Halburd
金额：
$ 3.04万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK041266%2F1
关键词：
Applications Nevanlinna theory differential difference

项目摘要

Strong hints as to whether certain types of equations are integrable (in some sense solvable) can be obtained by looking at the behaviour of solutions in the complex domain, even when we are only concerned with finding real solutions. For differential equations the most widely used property of this type is the Painlevé property. An ordinary differential equation has the Painlevé property if all solutions are single-valued about all movable singularities. Although proving that a given equation actually has the Painlevé property is sometimes difficult, there are very powerful but simple methods that can be applied to very general equations that will show that they do not have the Painlevé property. These classifications are based on purely local methods such as series expansions. Unfortunately the Painlevé property is destroyed by almost any rational change of dependent variable. However, although the transformed equation will have movable branch points, the global branching structure will remain simple. In particular, if the solution has no fixed singularities then it is algebroid (algebriac over the meromorphic functions) so its Riemann surface has only a finite number of sheets over any point. The first part of this project is to develop methods to determine when an equation possesses algebroid solutions. This should be a more sensitive test than the usual Painlevé analysis. Local series expansions will not be enough here and more global methods are required. Here Nevanlinna theory, the theory of the value distribution of meromorphic functions, will play a central role.The second part of the project again uses Nevanlinna theory as a way of detecting integrable equations, but this time we will consider differential-delay equations, e.g., equations of the form F(z,y(z+1),y(z-1),y(z),y'(z))=0. Based on earlier work on difference equations, differential-delay equations that admit meromorphic solutions that are finite-order (i.e., not too complicated) in the sense of Nevanlinna theory of a particular type will be classified. Very little work on the integrability of differential-delay equations has been carried out. Differential-delay equations appear in many models, especially in mathematical biology.

关于某些类型的方程是否可积(在某种意义上是可解的)的强烈提示，可以通过观察复数域中解的行为来获得，即使我们只关心寻找实数解。对于微分方程，最广泛使用的这类性质是Painlevé性质。一个常微分方程具Painlevé性质，如果所有解都是关于所有可移动奇点的单值。虽然证明一个给定的方程确实具有Painlevé性质有时很困难，但有一些非常强大但简单的方法可以应用于非常一般的方程，这些方法将表明它们不具有Painlevé性质。这些分类是基于纯粹的局部方法，例如级数展开。不幸的是，Painlevé性质几乎被因变量的任何合理变化所破坏。然而，尽管变换后的方程将具有可移动的分支点，但全局分支结构将保持简单。特别地，如果解没有固定奇点，则它是代数体(亚纯函数上的代数体)，因此它的Riemann曲面在任何点上只有有限数量的薄片。这个项目的第一部分是开发确定方程何时有代数体解的方法。这应该是一个比通常的Painlevé分析更敏感的测试。局部级数展开在这里是不够的，需要更多的全局方法。在这里，关于亚纯函数的值分布理论，将发挥核心作用。项目的第二部分再次使用了Nevanlinna理论作为一种检测可积方程的方法，但这一次我们将考虑微分延迟方程，例如形式为F(z，y(z+1)，y(z-1)，y(Z)，y‘(Z))=0的方程。在前人关于差分方程组的工作的基础上，我们将对具有有限阶(即不太复杂)的亚纯解的微分时滞方程进行分类。关于时滞微分方程可积性的研究工作很少。微分时滞方程出现在许多模型中，特别是在数学生物学中。