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Nonlinear Phenomena in Stochastic and Deterministic Dispersive Partial Differential Equations

随机和确定性色散偏微分方程中的非线性现象

基本信息

批准号：
1927258
负责人：
Svetlana Roudenko
金额：
$ 25.79万
依托单位：
Florida International University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1927258&HistoricalAwards=false
关键词：
Nonlinear Phenomena Stochastic Deterministic Dispersive

项目摘要

Wave motion is common for many real-life phenomena, from water waves to electro-magnetic or gravitational waves. While investigating wave-like processes, it is important to understand how the waves form, how they travel, whether they form a stable or an unstable structure, and if certain initial conditions lead to formation of singularities (like tsunamis, rogue waves, or rough turbulence). Understanding the evolution in time of such processes via various mathematical models is a goal of this project, and it will include studying various wave phenomena with or without stochastic elements. These wave-type phenomena are fundamental in nature, yet we are still at the dawn of their full analytical and numerical descriptions. This project is aimed at advancing the frontiers of the science of wave phenomena with applications to the real-life manifestations in nature. Students at all levels, from undergraduate to post-doctoral, will be involved in this research and receive training for their careers. The project is aimed at understanding the global behavior of solutions to nonlinear stochastic and deterministic dispersive partial differential equations, where the nonlinearities cause a significant difference in global behavior compared with the linear time evolution. One such nonlinear structure is a solitary wave, or soliton, which has a specific shape, exists (in the mathematical sense) for all time and in some equations travels in particular directions, while in other equations it oscillates periodically. Whether such a structure is stable or unstable and how it might be influenced by the external random perturbations is important in applications, and if unstable, it is important to investigate what it leads to. Typically, instability (the case when solitons are unstable) means that a singularity will form: for example, a freak wave in the ocean or a self-focusing burn in laser optics. Thus, it is very important to investigate the question of soliton stability, which is to be studied in this project in both stochastic and deterministic settings and via analytical and numerical approaches. A special thrust will be given to the study of formation of singularities, for example, collapses and concentrations. The research program starts from advancing the soliton stability theory as well as understanding the singularity formation process for the stochastic generalized Korteweg-de Vries equation, then progressing to one of its higher dimensional generalization (the little explored even in pure deterministic case but important in quantum mechanics and fluid dynamics) Zakharov-Kuznetsov equation, where questions of asymptotic stability, existence and the description of singularities is proposed for investigation. Using the analytical and numerical techniques the nonlinear Hartree equation with the non-local potential will be investigated, with the special emphasis on understanding the spectral structure. This type of equation arises in general relativity as well as in quantum mechanics, and so is important to study in that context. Finally, questions of description of global behavior of solutions in a wave-type equation, such as the nonlinear Klein-Gordon equation, will be investigated.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.

波动在现实生活中的许多现象中都很常见，从水波到电磁波或引力波。在研究波状过程时，重要的是要了解波是如何形成的，它们是如何传播的，它们是形成稳定的结构还是不稳定的结构，以及某些初始条件是否会导致奇点的形成(如海啸、无赖波或粗糙的湍流)。通过各种数学模型了解这些过程在时间上的演化是这个项目的一个目标，它将包括研究有或没有随机元素的各种波动现象。这些波浪型现象本质上是基本的，但我们仍处于它们的全面分析和数值描述的黎明阶段。该项目旨在推动波动现象科学的前沿，并将其应用于自然界中的现实表现。所有级别的学生，从本科生到博士后，都将参与这项研究，并接受职业培训。这个项目的目的是了解非线性随机和确定性色散偏微分方程解的整体行为，其中非线性导致整体行为与线性时间演化相比有显著不同。一种这样的非线性结构是孤波或孤子，它具有特定的形状，一直存在(在数学意义上)，在某些方程中沿特定方向传播，而在其他方程中，它周期性地振荡。这种结构是稳定的还是不稳定的，以及它可能如何受到外部随机扰动的影响，在应用中是很重要的，如果是不稳定的，研究它会导致什么是重要的。通常，不稳定性(孤子不稳定时的情况)意味着将形成奇点：例如，海洋中的反常波或激光光学中的自聚焦燃烧。因此，研究孤子稳定性问题是非常重要的，本项目将在随机和确定性环境下，通过解析和数值方法来研究孤子稳定性。将特别强调奇点形成的研究，例如，崩塌和集中。研究程序从提出孤子稳定性理论和了解随机广义Korteweg-de Vries方程的奇点形成过程开始，然后发展到它的高维推广(即使在纯确定性情况下也很少被探索，但在量子力学和流体力学中很重要)Zakharov-Kuznetsov方程，其中提出了渐近稳定性、存在性和奇点的描述等问题。利用解析和数值方法，我们将研究具有非局域势的非线性Hartree方程，特别强调对谱结构的理解。这种类型的方程出现在广义相对论和量子力学中，因此在这一背景下进行研究是重要的。最后，将研究波型方程中解的整体行为的描述问题，例如非线性Klein-Gordon方程。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。