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

自然界和应用中出现的各种现象，如海浪（异常波）和破波、光传输线和光通信、新型材料（石墨烯）等，都涉及不改变形状的单波结构（孤子）。本项目旨在研究非线性偏微分方程，这将有助于分析和建模这类孤子结构。许多重要的现象最容易通过行波或驻波等特殊解的行为表现出来，它们不仅是系统行为的基础，而且还决定了相关的附近动力学。这些结构及其稳定性在实际应用中具有重要意义。系统的稳定状态会吸引附近的所有构型，而失去稳定性或无法控制动力学也具有重要的实际意义。研究者将在项目的各个阶段包括本科生和研究生，并将致力于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其他科学（如光学、水波和液晶）广泛互动的部分，将对学生的训练特别有益。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，这使我们能够利用经典工具，使其适应无限维环境。该项目侧重于具有符号不定能量泛函的哈密顿模型，如各种狄拉克系统。目的是研究某些孤立波的线性稳定性和谱稳定性，并证明谱稳定解在均匀界上的结果。研究一些奇异的NLS模型和水波模型（如Benney-Luke方程）在孤立波附近的动力学是本提案的另一个重点。对朗道-德-热讷液晶模型的长期动力学和渐近分布的研究使本研究计划更加完善。所有这些方向都需要来自不同领域的新技术和工具，如功能分析、动力系统、谐波分析以及数值模拟。溶液的局部和长期行为及其稳定性在描述光学、液晶和水波等系统时具有重要的实际意义。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.