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

在自然界和应用中出现的各种现象，例如海浪（流氓波）和波浪破碎，光传输线和光通信，新材料（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）。本计画的主要目的是研究非线性偏微分方程，此方程可用来分析与模拟此类孤子结构。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是非常重要和必不可少的。系统的稳定状态吸引所有附近的配置，而失去稳定性或无法控制的动态也具有实际意义。研究人员将在项目的各个阶段涉及本科生和研究生，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广泛互动的部分，如光学，水波和液晶，将对学生的训练特别有益。在整个项目中，使用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，使它们适应无限维的设置。该项目的重点是具有符号不定能量泛函的哈密顿模型，如各种狄拉克系统。我们的目标是研究某些孤立波的线性和谱稳定性，但也证明结果的一致边界的谱稳定的解决方案。研究一些奇异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本提案的另一个重点。液晶的朗道-德让尼斯模型的长期动力学和渐近分布的研究是本研究计划的补充。所有这些方向将需要新的技术和工具，从不同的领域，如功能分析，动力系统，谐波分析以及数值模拟。解决方案的局部和长期行为，以及它们的稳定性是非常实际的重要性，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，并已被认为是值得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的支持。

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.