Eigenvalues, geometry and instability in conservative models in applied mathematics.

大多数为模拟物理现象而导出的方程都支持某些形式的特殊解，如驻波或行波，或其他相干结构。这些特殊解通常对应于这些模型方程所要表示的系统中真实的、可观察到的物理现象。这种建模的一个重要方面是这些特殊溶液的稳定性，因为溶液的稳定性决定了这些溶液是否可能被物理观察到。本项目旨在了解这些相干结构的稳定性和不稳定性。这通常相当于计算对这些相干结构之一进行线性化的算子的索引，即正实部特征值的个数。这计算了解的不稳定流形的维度，给出了附近解的动力学的有价值的信息。虽然这个项目的重点是保守系统，但我们同时考虑了保守系统和耗散系统。我们使用解析、渐近和几何技术来识别不稳定特征值并计算这些特征值的数量。有许多数学模型对科学和工程很重要，它们支持行波形式的解决方案。一个例子是一个被称为Korteweg-DeVries方程的方程，这个方程控制了水在狭窄的浅体（如运河）中的行为。Korteweg-DeVries方程的解被称为孤波，它的行为和经验所暗示的一样——它们对应于一定量的水在不改变形状的情况下传播。类似的公式也适用于光波的传播，以及通过电网等网络的扰动传播。了解这些解决方案在多大程度上准确地模拟底层系统的行为是很重要的。这就是稳定性问题，它衡量了这些特殊解的健壮性。如果特解像行波一样是稳定的，就意味着它的邻近解也有类似的行为。这意味着这样的解是健壮的，并且很可能被观察到：如果条件不完全是产生行波的必要条件，但很接近，我们期望看到一个接近特殊解的解。另一方面，不稳定的解决方案不是健壮的。为了观察这些不稳定的溶液，必须制造出完全合适的条件，这在实践中是很难做到的。这意味着这样的解在数学上的意义大于在物理上的重要性。该项目主要关注发展数学技术，以理解许多模型中行波的稳定性，包括Kuramoto模型（电网行为模型）和几个非线性色散方程（控制非线性介质中的光波，水波，等离子体中的波和许多其他现象）。

Most of the equations derived to model physical phenomena support some formof special solutions such as standing or traveling waves, or other coherentstructures. These special solutions often correspond to real, observablephysical phenomena in the systems that these model equations are meant torepresent. One important aspect of this modeling is the stability of thesespecial solutions, as the stability of solutions determines if thesesolutions are likely to be physically observed. This project is aimed atunderstanding the stability and instability these coherent structures.This usually amounts to computing the index of the operator found bylinearizing about one of these coherent structures - that is the number ofeigenvalues of positive real part. This counts the dimension of theunstable manifold to the solution, giving valuable information on thedynamics of nearby solutions. While the emphasis of this project is onconservative systems we consider both conservative and dissipative systems.We use analytical, asymptotic and geometric techniques to identify unstableeigenvalues and to count the number of such eigenvalues.There are many mathematical models that are important for science andengineering that support solutions in the form of traveling waves. Oneexample is an equation known as the Korteweg-DeVries equation, an equationwhich governs the behavior of water in a narrow shallow body such as acanal. The Korteweg-DeVries equation has solutions called solitary waveswhich behave just as experience would suggest - they correspond to aquantity of water which propagates along without changing shape. Similarequations govern the propagation of light waves, propagation of adisturbance through a network such as the power grid, etc. It isimportant to understand the extent to which these solutions accuratelymodel the behavior of the underlying system. This is the question ofstability, which measures how robust these special solutions are. If aspecial solution like a traveling wave is stable it means that nearbysolutions behave in a similar way. This means such solutions are robust,and are likely to be observed: if the conditions are not exactly thosenecessary to produce a traveling wave but are close we expect to see asolution which is close to the special one. An unstable solution, on the other hand, is not robust. In order to observe these unstable solutionone must produce exactly the right conditions, which is very difficultto achieve in practice. This means that such solutions are more of mathematical interest than of physical importance. This project isprimarily concerned with developing mathematical techniques to understand the stability of traveling waves in a number of models including the Kuramoto model (a model for the behavior of power networks) and several nonlinear dispersive equations (which govern light waves in nonlinear media, water waves, waves in plasmas and many other phenomena).