Stability of coherent structures in evolutionary partial differential equations: a geometric approach

演化偏微分方程中相干结构的稳定性：几何方法

• 批准号: RGPIN-2017-04259
• 负责人: Cox, Graham
• 金额: $ 1.53万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=648235
• 关键词: Stability; coherent; structures; evolutionary; partial

Most continuously evolving physical processes can be described by differential equations. Examples include water waves, biological populations, interacting atoms and molecules, and the constituents of a chemical reaction. In any such system a distinguished role is played by steady states—solutions that do not change in time because all external forces are in perfect equilibrium. It is important to know whether such states are stable, in the sense that small perturbations to the initial condition or external forces will eventually fade away, or unstable, meaning the perturbations will be amplified exponentially and lead to radically different behaviour in the long run. For this reason it is typically only the stable states that can be observed in nature, or physically realized in a laboratory setting, so it is important to identify them and understand what properties lead to their stability.******The ultimate goal is to predict a state's stability from its general shape and structure, and to determine what properties are indicative of instability. A classical problem describes a signal propagating in one dimension (such as light traveling along an optical fibre, or an electrical impulse in a neuron). In this case it is known that a pulse solution (which looks like a small bump moving along the fibre) is unstable, whereas a front (which is shaped like a cliff or a step) is stable. The difference between the two is that the pulse has a local maximum while the front does not, and this is enough to distinguish stability from instability.******When the problem involves multiple spatial dimension (as all real physical systems do), it is much more difficult, and results from the one-dimensional case no longer apply. The proposed research addresses this shortcoming by simultaneously developing two different tools for higher-dimensional problems: 1) the Maslov index; and 2) the Evans function. Both methods are well understood in the one-dimensional context, but have only recently begun to receive attention in a more general setting. Thus the proposed research is likely to have a strong impact on both the mathematical and physical sciences, with theoretical advancements allowing for new applications to problems in fluid dynamics, materials science and nonlinear optics, to name just a few examples.******These new theoretical tools will be advanced through the consideration of a large, robust family of physical applications. Student researchers will have the opportunity to communicate with scientists in related disciplines to determine the most important applications of these methods, and guide their efforts accordingly. As a result, the work outlined in the proposal will effectively train these researchers not just as mathematicians, but as active, productive members of the general scientific community, and as such will promote the development of innovative new scientific methods in Canada.

大多数连续演化的物理过程可以用微分方程来描述。例子包括水波、生物种群、相互作用的原子和分子以及化学反应的成分。在任何这样的系统中，稳态都扮演着重要的角色——由于所有外力都处于完美平衡，因此不会随时间变化的解决方案。重要的是要知道这些状态是否稳定，即对初始条件或外力的微小扰动最终会消失，或者不稳定，这意味着扰动将呈指数级放大，并从长远来看导致完全不同的行为。因此，通常只有稳定状态才能在自然界中观察到，或在实验室环境中物理实现，因此识别它们并了解哪些属性导致其稳定性非常重要。******最终目标是从状态的一般形状和结构预测其稳定性，并确定哪些属性表明不稳定。一个经典问题描述在一维中传播的信号（例如沿着光纤传播的光，或神经元中的电脉冲）。在这种情况下，我们知道脉冲解（看起来像沿着光纤移动的小凸起）是不稳定的，而前沿（形状像悬崖或台阶）是稳定的。两者之间的区别在于，脉冲具有局部最大值，而前沿没有，这足以区分稳定性和不稳定。 *****当问题涉及多个空间维度时（就像所有真实的物理系统一样），它就困难得多，并且一维情况的结果不再适用。拟议的研究通过同时开发两种不同的工具来解决高维问题来解决这一缺点：1）马斯洛夫指数； 2）埃文斯函数。这两种方法在一维环境中都得到了很好的理解，但最近才开始在更一般的环境中受到关注。因此，拟议的研究可能会对数学和物理科学产生重大影响，理论进步允许在流体动力学、材料科学和非线性光学等问题上有新的应用，仅举几个例子。******这些新的理论工具将通过考虑大量、强大的物理应用而得到推进。学生研究人员将有机会与相关学科的科学家进行交流，以确定这些方法最重要的应用，并相应地指导他们的工作。因此，提案中概述的工作将有效地培训这些研究人员，使其不仅成为数学家，而且成为整个科学界积极、富有成效的成员，从而促进加拿大创新科学方法的发展。