Geometric Aspects of Complex Differential Equations

复微分方程的几何方面

• 批准号: EP/W012251/1
• 负责人: Thomas Kecker
• 金额: $ 25.66万
• 依托单位: University of Portsmouth
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW012251%2F1
• 关键词: Geometric; Aspects; Complex; Differential; Equations

Differential equations play a key role in describing a wide range of dynamical processes, from mechanical systems, fluid dynamics, optical phenomena and quantum mechanics amongst many others, where they relate the state of a physical system to the rate of change that the system is momentarily undergoing. It is important in applications to know when a system has solutions that are, in a certain mathematical sense, well-behaved, or whether the solutions exhibit chaotic behaviour. Although in applications one is usually interested in the time-evolution of a system, time being a real, 1-dimensional parameter, it has long been understood that the nature of the solutions of an equation is best explained when time is considered as member of an extended number system, mathematically realised as points in a 2-dimenional plane, known as the complex numbers.The solutions of a differential equation, considered in this 2-dimensional complex plane, can have singularities, i.e. points where a solution becomes infinite or is otherwise ill-defined. The nature of these singularities, in turn, has an impact on the behaviour of the solutions in the 1-dimensional time parameter space. It allows one to determine whether an equation is integrable, i.e. exactly solvable in a certain mathematical sense. This project is about investigating the nature of the singularities of certain wide classes of equations that are motivated from physical applications. To investigate the singularities, we will consider the solutions of the equations in spaces of the dependent variables (those variables describing the state of a system and its rate of change) to which certain points have been added to include points where these become infinite. In this way, one can investigate what happens when the solutions develop a singularity. It turns out, however, that even in these augmented spaces there are points at which the rate of change of a system is indeterminate, which happens when both the numerator and the denominator of the fractions expressing the rate of change approach zero at the same time. Such 'base points' can be removed by a well-defined mathematical procedure known as a blow-up. The blow-up of a point is a geometrical notion which adds further points to the solution space by introducing an 'exceptional line', each point of which corresponds to a direction emerging from the point in question. In this way, the solutions of an equation are separated out over the exceptional line, making it possible to investigate them further. The 'exceptional line' introduced by the blow-up, when considered as an object in complex geometry (where the coordinates take complex numbers as values) turns out to be equivalent to a sphere. In this way, the original point at which the rate of change of the solution was ill-defined, becomes inflated to a sphere, hence the name blow-up for this process. It turns out that for most equations, a single blow-up of the equation at one point is not sufficient to render the equation free of indeterminacies, i.e., even after one blow-up has been performed, further ones may be necessary. For the Painlevé equations, an important class of equations in mathematical physics, it turns out that a total of 9 blow-ups is required to bring these equations into a form where no points with indeterminate behaviour remain. The solution space thus constructed, first obtained by the Japanese mathematician K. Okamoto, is called the 'space of initial values' of the equation. For the Painlevé equations, using this space one can explain the nature of the singularities, which in this case are poles: points at which the solutions tend to infinity in a controlled way.In the proposed project, we will apply the method of constructing the space of initial values to much wider classes of equations to investigate how this method can be utilised to determine the nature of more complicated singularities that certain differential equations can exhibit.

微分方程在描述广泛的动力学过程中起着关键作用，从机械系统，流体动力学，光学现象和量子力学等等，它们将物理系统的状态与系统瞬间经历的变化率联系起来。在应用程序中，重要的是要知道一个系统的解决方案，在一定的数学意义上，良好的行为，或解决方案是否表现出混沌行为。虽然在应用中人们通常对系统的时间演化感兴趣，时间是真实的一维参数，但长期以来人们已经理解，当时间被认为是扩展的数系统的成员时，方程的解的性质得到了最好的解释，数学上实现为二维平面中的点，称为复数。在该二维复平面中考虑的，可以具有奇点，即解变为无穷大或以其它方式不明确的点。这些奇点的性质，反过来，有影响的行为的解决方案在1维时间参数空间。它允许人们确定一个方程是否是可积的，即在某种数学意义上精确可解。这个项目是关于调查的性质的奇异性的某些广泛的类方程的动机从物理应用。为了研究奇点，我们将考虑因变量（描述系统状态及其变化率的变量）空间中方程的解，其中某些点已被添加到包括这些点变为无穷大的点。通过这种方式，人们可以研究当解发展为奇点时会发生什么。然而，事实证明，即使在这些增广空间中，也存在系统的变化率不确定的点，当表示变化率的分数的分子和分母同时接近零时，就会发生这种情况。这样的“基点”可以通过一个定义明确的数学过程被称为爆破来移除。一个点的爆破是一个几何概念，它通过引入一条“例外线”将更多的点添加到解空间中，其中每个点对应于从所讨论的点出现的方向。通过这种方式，方程的解在例外线上被分离出来，使得进一步研究它们成为可能。由爆破引入的“例外线”，当被认为是复杂几何中的对象时（其中坐标以复数作为值），结果等价于球体。这样，原来的点，在那里的变化率的解决方案是不明确的，成为膨胀到一个领域，因此名称爆破为这个过程。事实证明，对于大多数方程，方程在一个点的单次爆破不足以使方程摆脱不确定性，即，即使在已经执行了一次吹胀之后，也可能需要进一步的吹胀。对于Painlevé方程，数学物理中的一类重要方程，事实证明，总共需要9次爆破才能使这些方程成为一种形式，其中没有不确定行为的点。这样构造的解空间，首先由日本数学家K。冈本，被称为“空间的初始值”的方程。对于Painlevé方程，使用这个空间可以解释奇点的性质，在这种情况下，奇点是极点：点的解决方案趋于无穷大的控制方式。在拟议的项目，我们将把构造初值空间的方法应用于更广泛的方程类，以研究如何利用这种方法来确定更复杂的奇点的性质，微分方程可以展示。