Infinite-Dimensional Dynamical Systems - Stability and Long-Time Behavior

无限维动力系统 - 稳定性和长期行为

基本信息

批准号：
2210867
负责人：
Milena Stanislavova
金额：
$ 19.7万
依托单位：
University of Alabama at Birmingham
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-10-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2210867&HistoricalAwards=false
关键词：
Infinite Dimensional Dynamical Systems Stability

项目摘要

A variety of phenomena arising in nature and applications, such as ocean waves (rogue waves) and wave breaking, optical transmission lines and optical communications, novel materials (graphene) among others, involve single-wave structures (solitons) that travel without change of shape. This project is aimed at the study of the nonlinear partial differential equations, which are instrumental for analysis and modeling of such soliton-like structures. Many important phenomena most readily manifest themselves through the behavior of the special solutions such as traveling or standing waves, which not only serve as basis for the behavior of the system, but also determine the related nearby dynamics. These structures and their stability are of great importance and are essential in practical applications. Stable states of the system attract all nearby configurations, while the loss of stability or being unable to control the dynamics is of practical importance as well. The investigator will involve undergraduate and graduate students in various stages of the project and will aim to recruit and retain them to continue working in the field of applied mathematics. Exposing students to parts of the project that require broad interaction with other sciences such as optics, water waves and liquid crystals, will be particularly beneficial for the students' training.Throughout the project, the point of view in working with these partial differential equations will be one of infinite-dimensional dynamical systems, which allows us to take advantage of the classical tools by adapting them to the infinite-dimensional setting. The project focuses on Hamiltonian models with sign indefinite energy functionals such as various Dirac systems. The goal is to investigate the linear and spectral stability for certain solitary waves, but also to prove results on uniform bounds for the spectrally stable solutions. Investigating the dynamics near solitary waves for some exotic NLS models and for water wave models such as the Benney-Luke equations is another focus of this proposal. The study of the long-term dynamics and asymptotic profiles in the Landau - de Gennes models of liquid crystals rounds up this research program. All these directions will require new techniques and tools from diverse areas such as functional analysis, dynamical systems, and harmonic analysis as well as numerical simulations. The local and long-time behavior of solutions, as well as their stability is of great practical importance as they describe systems in optics, liquid crystals, and water waves, among others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在自然界和应用中产生的各种现象，例如海浪（流氓波）和波浪破裂，光传输线和光学通信，新型材料（石墨烯）等，涉及而没有变化的单波结构（孤子）。该项目旨在研究非线性偏微分方程，该方程对这种类似孤子的结构的分析和建模起了重要作用。许多重要的现象最容易通过特殊解决方案（例如旅行或驻波）的行为来表现出来，这不仅是系统行为的基础，而且还确定了附近的相关动力学。这些结构及其稳定性非常重要，在实际应用中至关重要。系统的稳定状态吸引了附近的所有配置，而稳定性或无法控制动态的损失也很重要。调查人员将在项目的各个阶段参与本科生和研究生，并旨在招募和保留他们，以继续在应用数学领域工作。将学生暴露于需要与其他科学（例如光波，水波和液晶）进行广泛互动的项目中，将对学生的培训特别有益。进行该项目，使用这些部分微分方程的观点将是无限少数动力学的系统之一，这使我们可以通过适应性的工具来启用无限型工具。该项目着重于具有符号无限能量功能（例如各种狄拉克系统）的哈密顿模型。目的是研究某些孤立波的线性和光谱稳定性，但也证明了光谱稳定溶液的均匀界限。研究某些外来NLS模型的孤立波附近的动力学以及Benney-Luke方程等水浪模型是该提案的另一个重点。液晶液晶模型中对长期动力学和渐近谱的研究使该研究计划结束。所有这些方向都需要来自不同领域的新技术和工具，例如功能分析，动力学系统和谐波分析以及数值模拟。解决方案的本地和长期行为以及它们的稳定性非常重要，因为它们描述了光学，液晶和水浪的系统等。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的审查标准通过评估来评估的。