Nonlinear Waves and Dynamical Systems

这个项目涉及(1)各种物理系统中非线性波的稳定性和非线性波之间的相互作用，以及(2)计算机辅助几何设计(CAGD)数学工具在动力系统问题中的应用。第一个项目解决了对物理系统进行建模的方程中的对称破坏分叉问题，该系统可以“捕获”光脉冲。它还将考察所谓的孤立波相互作用的集体坐标模型--这一方法的数学有效性(仍然是一个主要的未解领域)以及在这些模型中出现的一些数学结构。这包括将新的和强大的拓扑方法应用于由PI导出的简单迭代映射，作为集体坐标动力学的近似。第二个项目将使用最近发展的样条工具作为计算迭代映射和矢量场的不变流形的新算法的基础，与现有方法相比，这将大大提高计算的速度、精度和简单性。该项目涉及(1)各种物理系统中非线性波的稳定性和非线性波之间的相互作用，以及(2)计算机辅助几何设计(CAGD)数学工具在动力系统问题中的应用。第一个项目涉及在许多物理和工程领域的系统建模中出现的非线性波，这里的重点是从光学派生的模型。它将研究这些波如何以及在什么条件下可能变得不稳定，例如它们可能如何分裂。它还将研究当这种波碰撞时会发生什么，将强大的数学(拓扑)技术应用于简化模型，以获得更深层次的理解，以及研究这种简化模型是否有效。第二个项目将研究计算不变流形的新的、更有效的方法，不变流形是理解混沌的基本对象。它将把计算机图形学领域的研究整合到一个对数学物理很重要的新背景中。其目标是将开发的曲线和曲面类型用于设计汽车和飞机的软件，以及用于电影和电视的计算机动画，用于意想不到的科学目的。

This project is concerned with (1) the stability of nonlinear waves and the interaction between nonlinear waves in a variety of physical systems, and (2) the application of mathematical tools from computer-aided geometric design (CAGD) to problems in dynamical systems. The first project addresses symmetry-breaking bifurcations in equations modeling a physical system that can "trap" pulses of light. It will also look at so-called collective-coordinate models of solitary wave interactions - both the mathematical validity of this approach (still a major unsolved area) and some of the mathematical structures that arise in these models. This includes the application of new and powerful topological methods to simple iterated maps derived by the PI as approximations to the collective-coordinate dynamics. The second project will use recently developed spline tools as the basis for new algorithms for computing invariant manifolds of iterated maps and vector fields, which should greatly improve the speed, accuracy, and simplicity of such computations in comparison with current methods.This project is concerned with (1) the stability of nonlinear waves and the interaction between nonlinear waves in a variety of physical systems, and (2) the application of mathematical tools from computer-aided geometric design (CAGD) to problems in dynamical systems. The first project deals with nonlinear waves that arise in the modeling of systems from many areas of physics and engineering, with an emphasis here on models derived from optics. It will study how and under what conditions such waves may become unstable, for example how they might break apart. It will also study what happens when such waves collide, applying powerful mathematical (topological) techniques to simplified models in order to gain a deeper understanding, as well as studying whether such reduced models are valid. The second project will study new, and more efficient, ways to compute invariant manifolds, which are fundamental objects in understanding chaos. It will integrate research from the field of computer graphics into a new context important for mathematical physics. The goal is to use types of curves and surfaces developed for application to software for designing automobiles and aircraft, and for computer animation used in movies and television, for an unexpected scientific purpose.