Mathematical Sciences: Geometric Approaches to the Stability of Travelling Waves & Other Problems

行波是沿固定方向传输信息的偏微分方程解。物理上可观测到的波必须对其剖面中的扰动是稳定的。这项赠款下的工作将旨在开发区分稳定和不稳定波的技术。在扩散问题中，设想了一种由波的结构决定任何不稳定性的理论。对这类系统的分析必然涉及简化为一个更简单的系统，例如通过奇异摄动。Kopell正在发展一种新的几何方法来处理奇异摄动，它被设计用来解决诸如Hodgkin-Huxley神经脉冲传播方程的稳定性问题。其他要解决的稳定性问题包括流体和光学问题中出现的微妙影响。在这些情况下，波的指数衰减不会发生，需要开发新的技术。动力系统中的横截性理论也将被发展并应用于椭圆问题中的自旋-轨道共振和唯一性问题。可以分析的物理系统的复杂性受到可用的数学技术的限制。具有一些简单特征的系统，例如那些线性的系统或具有单一未知函数的方程，都会产生成熟的分析方法，这些方法现在是工程师工具箱的一部分。然而，对于最好的应用数学家来说，更复杂的系统仍然是一个挑战，许多重要的现象被简化为易于分析的系统而遗漏。未来的科学进步很可能是由用于分析这种复杂系统的技术的数学发展推动的。这种系统需要数学解的一个方面是基本结构在外界扰动影响下的稳定性问题。这笔赠款的工作将主要用于开发评估此类结构稳定性的技术。人们希望，在未来的某个时候，这些技术将成为工程师的工具箱。

Travelling waves are the solutions of partial differential equations that transport information in a fixed direction. A physically observable wave must be stable to perturbations in its profile. The work under this grant will be aimed at the development of techniques for discriminating betwen stable and unstable waves. In diffusive problems, there is envisioned a theory in which the structure of the wave determines any instabilities. The analysis of such systems necessarily involves a reduction to a simpler system, such as through a singular perturbation. A new geometric approach to singular perturbations is being developed with Kopell which is designed to solve stability problems in such equations as those of Hodgkin-Huxley for nerve impulse propagation. Other stability problems to be attacked involve the subtle effects ocurring in fluid and optical problems. Exponential decay to waves does not happen in these cases and new techniques need to be developed. The theory of transversality in dynamical systems will also be developed and aplied to problems in spin-orbit resonance and uniqueness in elliptic problems. The complexity of physical systems that can be analyzed is confined by the mathematical techniques available. Systems with some simple character, such as those which are linear or an equation with a single unknown function, yield to well-developed methods of analysis that are now part of the engineer's toolbox. However, more complicated systems remain a challenge to the best applied mathematicians and many important phenomena are missed by the simplification to systems that can be easily analyzed. Future advances in science may well be driven by the mathematical development of techniques for the analysis of such complicated systems. One aspect of such a system that demands a mathematical solution is the question of the stability of basic structures, under the perturbing influences suplied by the outside world. The work in this grant will be largely devoted to the development of techniques for assessing the stability of such structures. It is hoped that at some point in the future these techniques will be in the engineer's toolbox.