Topics in Global Behavior of Solutions to nonlinear dispersive PDE

本项目致力于发展偏微分方程组理论的几个方面的研究。主要研究人员将研究非线性色散方程解的整体行为。这些方程描述了波的演化，特别是在不同的波组以不同的速度传播，从而使波随着时间“分散”的情况下。典型的方程是非线性薛定谔方程和波动方程，因为它们在数学上表示最简单的非线性色散模型，在物理上描述波在诸如非线性光学、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引人入胜的动力学，如运动和相互作用的孤子、收缩或固定的奇点、崩塌等。对具有这些性质的解的研究将是本提案的重点。崩塌现象可以说是演化方程中可能出现的“最”非线性现象，因此，总的来说，研究它是非常困难的。最近，对色散方程奇异解的理解在分析方面取得了重大进展，因此，现在就将研究集中在这个方向上是非常及时的。因此，该项目的主旨将是提供奇点的形成和动力学的描述，并调查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。该项目研究描述出现在各种物理环境中的波的色散的微分方程式：激光光学、流体和空气动力学、量子和等离子体物理，仅举几例。这里开发的方法不仅可以应用于数学和物理问题，也可以应用于统计学、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)。了解奇点发生的时间和方式对于数值模拟和实时工程应用也很重要。首席调查员将通过一个(可免费访问的)机构网站和各种免费预印本服务器在网上传播这一项目的成果，她将在国家和国际会议上介绍最新进展。此外，这项建议还包括在最高水平的科学和数学能力方面开展指导和培训活动。首席研究员将组织研讨会、座谈会、本科生讲座和类似的活动，专门为指导学生有关项目主题以及偏微分方程式和分析的一般领域而设计。她参加了由她的母校和附近大学组织的暑期“本科生研究体验”项目，这一项目将促进她的参与。特别是，她将在她所在的机构扩大和发展她的迷你课程和“数学女性暑期计划”的客座讲座，以鼓励女性在数学和科学领域追求职业。

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.