Dynamical Systems Approaches to Partial Differential Equations

韦恩教授将研究偏微分方程的长期行为，如流体力学和光学研究中出现的纳维斯托克斯方程、欧拉方程和麦克斯韦方程。他将使用动力系统理论的方法和技术，对这些方程的解的行为做出定性和定量的预测，并将重点放在五个主要领域：(I)流体表面波动模型方程的推导、证明和实验验证；(Ii)Navier-Stokes方程解的长期行为，特别是涡旋解；(Iii)光学介质中极短脉冲的模拟；(Iv)耦合光学系统中脉冲解的存在性和稳定性；以及(V)薄弹性介质运动的近似。特别是，他将尝试将从理解常微分方程相空间中的不变几何对象获得的见解扩展到偏微分方程的无限维设置，特别是在无界空间域上的问题的情况下，其中连续谱的存在可能导致有限维系统和无限维系统的行为之间的定性差异。韦恩教授将研究的系统类型出现在许多应用中，包括非线性光学通信、流体力学和弹性材料的行为。例如，通过光纤获得越来越快的信息传输的尝试通常使用极短的光脉冲。虽然有一种众所周知的和大量研究的方法来近似通过这种光纤的长脉冲传输，但光学技术现在达到了这种近似被打破的地步，需要新的模型。上述项目清单中的第(3)点旨在制定这种近似值。这个问题，像这个项目中将研究的其他问题一样，其特点是虽然不可能精确地求解控制系统的方程，但应用程序至少需要对解的行为有一个定性的了解。韦恩教授的研究有三个目标：第一，可以用来近似真实物理系统的行为的模型方程的推导--无论它发生在光学、流体还是其他地方。第二，精确估计的计算，以控制真实系统的行为可以偏离模型系统的多大程度，以及最后，几何洞察力的发展，它可以提供一种定性的手段来理解和预测这种类型的复杂物理系统的行为，即使实际的解无法计算。

Professor Wayne will study the long-time behavior of partialdifferential equations like the Navier-Stokes, Euler, and Maxwell'sequations which arise in the study of fluid dynamics and optics. Hewill use methods and techniques from dynamical systems theory to makequalitative and quantitative predictions about the behavior of thesolutions of these equations and will focus on five main areas: (i)The derivation, justification and experimental validation of modelequations for waves on fluid surfaces; (ii) The long-time behavior ofsolutions of the Navier-Stokes equations; particularly vortexsolutions; (iii) The modeling of very short pulses in optical media;(iv) The existence and stability of pulse solutions in coupled opticalsystems; and (v) The approximation of the motion of thin elasticmedia. In particular, he will attempt to extend the insights gainedfrom understanding the invariant geometrical objects in the phasespace of ordinary differential equations to the infinite dimensionalsetting of partial differential equations, focusing particularly onthe case of problems on unbounded spatial domains where the presenceof continuous spectrum may cause qualitative differences between thebehavior of the finite and infinite dimensional systems.The types of systems that Professor Wayne will study arise in manyapplications, including nonlinear optical communication, fluidmechanics and the behavior of elastic materials. For instance,attempts to obtain faster and faster transmission of informationthrough optical fibers often utilize extremely short optical pulses.While there is a well understood and much studied method forapproximating the transmission of long pulses through such fibersoptical technology is now reaching the point where this approximationbreaks down and new models are needed. Point (iii) in the list ofprojects above aims to develop such approximations. This problem,like the others that will be studied in this project, is characterizedby the fact that while it is impossible to solve the equationsgoverning the system exactly, applications require at least aqualitative understanding of the behavior of solutions. ProfessorWayne's research has three goals: First, the derivation of modelequations which can be used to approximate the behavior of the truephysical system -- whether it arises in optics, fluids, or elsewhere.Secondly, the computation of accurate estimates to control how muchthe behavior of the true system can deviate from that of the modelsystem, and finally the development of geometrical insights which canprovide a qualitative means of understanding and predicting thebehavior of this type of complicated physical system even if theactual solution can not be computed.