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

大多数不断演化的物理过程都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，稳态都扮演着一个特殊的角色--因为所有外力都处于完美平衡中，所以它们的解不会随时间变化。重要的是要知道这些状态是稳定的，也就是说，对初始条件或外力的微小扰动最终会消失，或者是不稳定的，这意味着扰动将以指数级放大，并从长远来看导致截然不同的行为。由于这个原因，通常只有在自然界中可以观察到的稳定状态，或者在实验室环境中物理实现的稳定状态，所以重要的是识别它们并了解什么属性导致它们的稳定性。*最终目标是根据状态的一般形状和结构来预测其稳定性，并确定哪些属性表示不稳定。经典问题描述的是在一维空间中传播的信号(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。两者的不同之处在于，脉冲存在局部极大值，而锋面没有，这足以区分稳定性和不稳定性。*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)，它要困难得多，一维情况的结果不再适用。这项研究通过同时开发两种不同的工具来解决这一缺陷：1)Maslov指数；2)Evans函数。这两种方法在一维背景下都很好地理解，但直到最近才开始在更一般的背景下受到关注。因此，拟议的研究可能会对数学和物理科学产生强烈的影响，理论的进步允许对流体力学、材料科学和非线性光学中的问题进行新的应用，仅举几个例子。*这些新的理论工具将通过考虑一大类强大的物理应用来推进。学生研究人员将有机会与相关学科的科学家交流，以确定这些方法的最重要应用，并相应地指导他们的努力。因此，提案中概述的工作将有效地将这些研究人员培训为不仅是数学家，而且是一般科学界积极、富有成效的成员，并因此将促进加拿大创新新科学方法的发展。

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.