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

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，稳定状态都扮演着一个突出的角色，因为所有的外力都处于完美的平衡状态，所以稳定状态的解不会随时间而改变。重要的是要知道这样的状态是稳定的，从某种意义上说，对初始条件或外力的小扰动最终会消失，还是不稳定的，这意味着扰动将呈指数级放大，并导致从长远来看完全不同的行为。由于这个原因，通常只有稳定状态才能在自然界中观察到，或者在实验室环境中物理实现，所以重要的是要识别它们并了解什么属性导致它们的稳定性。最终目标是从一个状态的一般形状和结构来预测它的稳定性，并确定哪些性质是不稳定的指标。一个经典问题描述了一个在一维中传播的信号（例如光沿着沿着光纤传播，或者神经元中的电脉冲）。在这种情况下，已知脉冲解（其看起来像沿着光纤沿着移动的小凸起）是不稳定的，而前沿（其形状像悬崖或台阶）是稳定的。两者之间的区别在于脉冲有局部最大值，而波前没有，这足以区分稳定性和不稳定性。当这个问题涉及到多维空间时（就像所有真实的物理系统一样），它就困难得多，一维情况下的结果不再适用。拟议的研究通过同时开发两种不同的工具来解决这一缺点：1）Maslov指数; 2）Evans函数。这两种方法在一维背景下都很好理解，但最近才开始在更一般的环境中受到关注。因此，拟议中的研究可能会对数学和物理科学产生强烈影响，理论上的进步允许在流体动力学，材料科学和非线性光学等问题上的新应用，仅举几例。这些新的理论工具将通过考虑一个大的，强大的物理应用程序的家庭。学生研究人员将有机会与相关学科的科学家进行交流，以确定这些方法的最重要应用，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，不仅是数学家，而且是一般科学界的积极、富有成效的成员，因此将促进加拿大创新科学方法的发展。

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.