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

行波是偏微分方程的解 以固定方向传输信息的方程。 一 物理上可观察到的波必须对它的扰动是稳定的。 profile. 这项赠款下的工作将针对 发展技术，区分稳定和 不稳定波 在扩散问题中，设想了 波的结构决定任何 不稳定性 对这些系统的分析必然涉及到 简化为一个更简单的系统，例如通过一个单一的 扰动 奇异摄动的一种新的几何方法 正在与Kopell一起开发，旨在解决 Hodgkin-Huxley等方程的稳定性问题 神经脉冲的传播。 其他稳定性问题 攻击涉及流体和光学发生的微妙影响， 问题 波的指数衰减不会发生在 需要开发新的案例和技术。 理论 动力系统中的横截性也将得到发展， 应用于自旋-轨道共振和唯一性问题， 椭圆问题 可以分析的物理系统的复杂性是 受限于可用的数学技术。 系统与 一些简单的字符，例如线性字符或 方程与一个单一的未知函数，产量良好的发展 分析方法，现在是工程师工具箱的一部分。 然而，更复杂的系统仍然是对最好的挑战 应用数学家和许多重要的现象都错过了 简化到可以容易分析的系统。 未来科学的进步很可能是由数学驱动的。 发展分析这种复杂的 系统. 这样一个系统的一个方面，需要一个数学 解决方案是基本结构的稳定性问题， 在外部世界的干扰下。 这笔赠款的工作将主要用于开发 评估这种结构稳定性的技术。 它 希望在未来的某个时候，这些技术将 在工程师的工具箱里

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.