Topics in Global Behavior of Solutions to nonlinear dispersive PDE

这个项目追求发展偏微分方程理论的几个研究途径。主要研究者将研究非线性色散方程解的全局行为。这些方程描述了波的演化，特别是在不同组的波以不同的速度传播的情况下，波随着时间的推移而“分散”。典型的方程是非线性薛定谔方程和波动方程，因为它们在数学上代表了最简单的非线性色散模型，在物理上描述了波在非线性光学、流体力学、等离子体物理、量子力学和玻色-爱因斯坦凝聚等应用中的传播。色散方程中的非线性项产生了复杂但迷人的动力学，如移动和相互作用的孤子，收缩或固定奇点，崩溃等。表现出这些特性的解决方案的研究将是本提案的重点。坍缩现象可以说是演化方程中“最”可能发生的非线性现象，所以一般来说，它很难研究。最近，在色散方程奇异解的理解方面取得了重大的分析进展，因此，现在是非常及时地将研究工作集中在这个方向上。因此，本项目的重点将是提供奇点的形成和动力学的描述，并研究在非线性色散方程的解中区分不同类型的全局行为的各种阈值。本项目研究描述在各种物理背景下出现的波的色散的微分方程：激光光学、流体和空气动力学以及量子和等离子体物理学，仅举几例。这里开发的方法不仅可以应用于数学和物理学中的问题，还可以应用于统计学，工程学，生物学和医学（例如，研究海啸或心律失常的形成等广泛现象）。了解奇点何时以及如何发生对于数值模拟和实时工程应用也很重要。首席研究员将通过一个（可免费访问的）机构网站和各种免费预印本服务器在线传播该项目的结果，她将在国家和国际会议上介绍最新进展。此外，该提案还包括最高水平的科学和数学能力的辅导和培训活动。主要研究者将组织研讨会，座谈会，本科生讲座，以及专门设计的类似活动，以指导学生对项目的主题，以及对偏微分方程和分析的一般领域。她参加了由她的家乡机构和附近大学组织的夏季“本科生研究经验”项目，该项目将加强。特别是，她将扩大和发展她的迷你课程和客座讲座在“夏季计划的妇女在数学”在她的家乡机构，以鼓励妇女追求职业生涯在数学和科学。

This project pursues several avenues of research in the theory of evolution partial differential equations. The principal investigator will study global behavior of solutions to nonlinear dispersive equations. These equations describe the evolution of waves, in particular in situations where different groups of waves propagate at different velocities so that waves "disperse" over time. Typical equations are the nonlinear Schrodinger and wave equations, since mathematically they represent the simplest nonlinear dispersive models and physically they describe wave propagation in applications such as nonlinear optics, hydrodynamics, plasma physics, quantum mechanics, and Bose-Einstein condensates. The nonlinear terms in dispersive equations create complicated but fascinating dynamics such as moving and interacting solitons, contracting or fixed singularities, collapses, etc. The study of solutions exhibiting such properties will be the focus of this proposal. The phenomenon of collapse is arguably "the most" nonlinear occurrence possible in an evolution equation, so it is, in general, very difficult to study. Recently there have been significant analytical strides made in the understanding of singular solutions to dispersive equations, hence it is quite timely to focus a research effort in this direction. Thus, the thrust of the project will be to provide descriptions of both the formation and the dynamics of singularities and to investigate the various thresholds that separate different types of global behavior in solutions to nonlinear dispersive equations.This project studies differential equations that describe dispersion of waves appearing in various physical contexts: laser optics, fluid and air dynamics, and quantum and plasma physics, to name just a few. The methods developed here can be applied to problems not only in mathematics and physics, but also in statistics, engineering, biology, and medicine (for example,to study phenomena as wide-ranging as the formation of tsunamis or errant cardiac rhythms). Understanding when and how singularities occur is important for numerical simulations and real-time engineering applications as well. The principal investigator will disseminate the results of this project online, through a (freely accessible) institutional website and various free preprint servers, and she will present the latest advances at national and international conferences. Furthermore, this proposal includes mentoring and training activities at the highest levels of scientific and mathematical competency. The principal investigator will organize seminars, colloquia, undergraduate talks, and similar activities specifically designed to instruct students on the topics of the project, as well as on the general areas of partial differential equations and analysis. Her participation in the summer "Research Experiences for Undergraduates" programs organized by her home institution and nearby universities will be enhanced by this project. In particular, she will extend and develop her mini-courses and guest lectures at the "Summer Program for Women in Math" at her home institution in order to encourage women to pursue careers in mathematics and science.