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

无限维动力系统 - 稳定性和长期行为

• 批准号: 2108285
• 负责人: Milena Stanislavova
• 金额: $ 19.7万
• 依托单位: University of Kansas Center for Research Inc
• 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: 2021-07-15 至 2022-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108285&HistoricalAwards=false
• 关键词: Infinite; Dimensional; Dynamical; Systems; Stability

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.

自然界和应用中出现的各种现象，例如海浪（流氓波）和波破碎、光传输线和光通信、新型材料（石墨烯）等，都涉及在不改变形状的情况下传播的单波结构（孤子）。该项目旨在研究非线性偏微分方程，这有助于此类孤子结构的分析和建模。许多重要的现象最容易通过特殊解的行为表现出来，例如行波或驻波，它们不仅作为系统行为的基础，而且还决定了相关的附近动力学。这些结构及其稳定性非常重要，在实际应用中至关重要。系统的稳定状态会吸引所有附近的配置，而稳定性的丧失或无法控制动态也具有实际重要性。研究者将让本科生和研究生参与该项目的各个阶段，旨在招募和留住他们继续在应用数学领域工作。让学生接触到项目中需要与光学、水波和液晶等其他科学进行广泛交互的部分，将特别有利于学生的培训。在整个项目中，处理这些偏微分方程的观点将是无限维动力系统之一，这使我们能够通过将经典工具适应无限维设置来利用它们。该项目重点关注具有符号不定能量泛函的哈密顿模型，例如各种狄拉克系统。目标是研究某些孤立波的线性和光谱稳定性，同时也证明光谱稳定解的统一界限的结果。研究一些奇异的 NLS 模型和水波模型（例如 Benney-Luke 方程）的孤立波附近的动力学是该提案的另一个重点。对液晶朗道-德热尼斯模型中的长期动力学和渐近曲线的研究使该研究项目更加圆满。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、动力系统、调和分析以及数值模拟。解决方案的局部和长期行为及其稳定性具有非常重要的实际意义，因为它们描述了光学、液晶和水波等系统。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。