Collaborative Research: Nonlinear Dynamics and Spectral Analysis in Dispersive Partial Differential Equations

合作研究：色散偏微分方程中的非线性动力学和谱分析

• 批准号: 2055130
• 负责人: Svetlana Roudenko
• 金额: $ 32.16万
• 依托单位: Florida International University
• 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 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055130&HistoricalAwards=false
• 关键词: Collaborative; Research; Nonlinear; Dynamics; Spectral

Internal deep-water waves, ion-acoustic waves in a plasma, laser propagation in highly refractive materials, and collective particle behavior in very low temperature gases are all physical systems whose behavior in time is modeled by a nonlinear dispersive wave equation. The term "nonlinear" refers to a property of size-dependent response and the term "dispersive" refers to how the fluctuations of the wave influence the speed and direction of motion. At the mathematical level, one studies the behavior of general solutions and special types of solutions to these equations and seeks to provide quantitative descriptions of phenomena observed in the physical setting. In this project, the investigators explore how coherent waves travel - whether they retain their shape despite encountering obstacles, break apart and dissolve, or collapse into a singularity. Each of these possibilities hinges on both the nonlinear and the dispersive character of the equations; over the past several decades, mathematical techniques have been developed to model and measure these properties. The project aims to improve existing methods and apply the methods in new directions. The project contains educational efforts at various levels of mathematical learning. These include advising undergraduate and graduate students as well as postdoctoral scholars, with special emphasis on attracting underrepresented minorities. The main objective of the research will be to provide analytical descriptions of the behavior of solutions to certain classes of nonlinear dispersive equations. The two main categories of equations considered are the Korteweg-de Vries (KdV) family and the nonlinear Schrödinger (NLS) family. Both classes of equations satisfy powerful dispersive estimates called local virial estimates, and the KdV family in addition satisfies a monotonicity property that controls the movement of mass. The multiple-scale method, spectral analysis, application of dispersive estimates, and monotonicity bounds are core methods that will be utilized and extended. Several focus problems are identified in which some classical feature, like scale-invariance or a property of localized influence, have been removed or weakened, providing the stimulus to develop new techniques, while at the same time, provide new descriptions of real physical phenomena. The phenomena of primary interest are the dynamics of coherent structures like solitary waves and line solitons as they interact with each other and their environment, and the description of how singular collapsing solutions arise and their asymptotic description.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.

内部深水波，血浆中的离子声波，高度屈光材料中的激光传播以及非常低的温度气体中的集体颗粒行为都是物理系统，其时间的行为是通过非线性色散波等效性建模的。术语“非线性”是指尺寸依赖性响应的属性和术语“分散”是指波浪的波动如何影响运动的速度和方向。在数学层面，人们研究了这些方程式的一般解决方案和特殊类型的解决方案的行为，并试图提供对物理环境中观察到的现象的定量描述。在这个项目中，调查人员探讨了连贯的波浪行进 - 他们是否保留了遇到障碍，崩溃并溶解或崩溃成奇异的目的地的形状目的地。这些可能性中的每一个都取决于方程的非线性和分散性。在过去的几十年中，已经开发了数学技术来建模和衡量这些特性。该项目旨在改善现有方法并将方法应用于新的方向。该项目在数学学习的各个层面上包含教育工作。其中包括咨询本科生和研究生以及博士后学者，特别强调吸引代表性不足的少数民族。该研究的主要目的是为某些类别的非线性分散方程提供解决方案行为的分析描述。所考虑的方程式的两个主要类别是Korteweg-de Vries（KDV）家族和非线性Schrödinger（NLS）家族。两类方程式都满足称为局部病毒估计值的强大分散估计，而KDV家族还满足控制质量运动的单调性能。多尺度方法，光谱分析，分散估计的应用和单调性界限是将被利用和扩展的核心方法。确定了一些重点问题，其中某些经典特征（例如尺度变革或具有局部影响的特性）已被删除或削弱，从而提供了开发新技术的刺激，同时，提供了对真实物理现象的新描述。主要感兴趣的现象是连贯的结构的动力学，例如固体和固体彼此相互作用和环境，以及描述奇异崩溃解决方案以及其不对称描述的描述。这项奖项反映了NSF的法定任务，并通过使用基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是宝贵的支持。