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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758938
• 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 statessolutions 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.

大多数不断发展的物理过程可以通过微分方程来描述。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，稳定的稳定效果都会扮演着杰出的角色，这些角色不会随时间变化而变化，因为所有外力都处于完美的均衡状态。重要的是要知道这种状态是否稳定，从某种意义上说，对初始状况或外部力量的小扰动最终会消失或不稳定，这意味着扰动将被指数置，并从长远来看会导致根本不同的行为。因此，通常只有稳定状态可以在本质上观察到稳定状态，或者在实验室环境中物理实现，因此重要的是要识别它们并了解什么特性会导致其稳定性。最终目标是从其一般形状和结构中预测国家的稳定性，并确定哪些特性表明了不稳定的性能。经典问题描述了一个在一个维度中传播的信号（例如沿光纤传播的光或神经元中的电脉冲）。在这种情况下，众所周知，脉冲溶液（看起来像是沿纤维移动的小凹凸）是不稳定的，而前部（形状像悬崖或台阶形状）是稳定的。两者之间的区别在于，脉冲具有局部最大值，而正面则没有，这足以区分稳定性与不稳定性。当问题涉及多个空间维度（如所有真实物理系统所做的那样）时，这要困难得多，并且不再应用一维情况。拟议的研究通过同时开发两个不同的工具来解决高维问题：1）Maslov指数来解决这一缺点； 2）Evans功能。两种方法在一维环境中都得到了充分理解，但直到最近才开始在更一般的环境中受到关注。因此，拟议的研究可能对数学科学和物理科学产生很大的影响，理论上的进步允许将新的应用于流体动力学，材料科学和非线性光学的问题，仅列举了几个示例。这些新的理论工具将通过考虑大型，可靠的物理应用家庭来提出。学生研究人员将有机会与相关学科的科学家进行沟通，以确定这些方法的最重要应用，并相应地指导他们的努力。结果，该提案中概述的工作将不仅有效地培训这些研究人员，而且还将作为一般科学界的积极，有效的成员，因此将促进加拿大创新的新科学方法的发展。