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.

微分方程在描述机械系统、流体动力学、光学现象和量子力学等广泛的动力学过程中发挥着关键作用，它们将物理系统的状态与系统瞬时变化率联系起来。尽管在应用中人们通常对系统的时间演化感兴趣，但在应用中了解系统何时具有在某种数学意义上表现良好的解或解是否表现出混沌行为非常重要。 ，时间是一个真实的、一维的参数，人们很早就认识到，当时间被视为扩展数系的成员时，方程解的本质得到了最好的解释，在数学上将其实现为二维平面上的点，称为复数。在这个二维复平面中考虑的微分方程可能具有奇点，即解变得无穷大或定义不明确的点。这些奇点的性质反过来又会影响解的行为。在一维时间参数空间。它允许人们确定方程是否可积，即在某种数学意义上完全可解。该项目旨在研究由物理应用激发的某些广泛方程组的奇点性质。 ，我们将考虑因变量（描述系统状态及其变化率的变量）空间中的方程的解，其中添加了某些点以包含这些点变得无穷大，这样，一个。可以调查发生了什么然而，当解出现奇点时，事实证明，即使在这些增广空间中，系统的变化率也存在不确定的点，当分数的分子和分母都表示变化率时，就会发生这种情况。同时改变接近零，这样的“基点”可以通过定义明确的数学过程（称为放大）来删除。点的放大是一个几何概念，它通过以下方式向解空间添加更多点。引入“特殊”线”，其中的每个点对应于从该点出现的一个方向，通过这种方式，方程的解在异常线上被分离出来，使得进一步研究它们成为可能。当将爆炸视为复杂几何中的对象（其中坐标以复数为值）时，结果相当于一个球体，这样，解的变化率就变成了病态的原始点。 -定义，膨胀成球体，因此得名事实证明，对于大多数方程，在某一点对方程进行一次放大不足以使方程摆脱不确定性，即，即使在执行了一次放大之后，仍会进一步消除不确定性。对于 Painlevé 方程（数学物理学中的一类重要方程）来说，事实证明总共需要 9 次爆炸才能将这些方程转化为不存在不确定行为的点的解空间。因此由日本数学家 K. Okamoto 首先构建的，称为方程的“初始值空间”。对于 Painlevé 方程，使用该空间可以解释奇点的性质，在本例中奇点是极点：点。在该项目中，我们将把构造初始值空间的方法应用于更广泛的方程组，以研究如何利用该方法来确定更复杂的奇点某些微分方程可以表现出来。