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.

波动是许多现实现象的常见，从水波到电磁波或重力波。在研究类似波浪的过程的同时，重要的是要了解波浪的形成，它们的行进方式，无论它们是稳定还是不稳定的结构，以及某些初始条件是否会导致奇异性形成（例如海啸，流氓波或粗糙的湍流）。通过各种数学模型了解此类过程的演变是该项目的目标，它将包括研究有或没有随机元素的各种波浪现象。这些波型现象本质上是基本的，但是我们仍处于它们完整的分析和数值描述的曙光。该项目旨在通过在自然界的现实生活中应用波浪现象科学的前沿。从本科到博士后的各个级别的学生将参与这项研究，并接受其职业培训。该项目的目的是了解解决非线性随机和确定性分散偏微分方程的全球行为，在这种方程中，非线性与线性时间演化相比会导致全球行为有显着差异。一种这样的非线性结构是一个孤立的波浪，或者是有特定形状的孤岛（在数学意义上）一直存在，并且在某些方程式中在特定方向上传播，而在其他方程式中，它会定期振荡。这种结构是稳定还是不稳定，以及如何受到外部随机扰动的影响在应用中很重要，如果不稳定，研究它导致什么很重要。通常，不稳定性（孤子不稳定的情况）意味着奇异性将形成：例如，海洋中的怪胎波或激光光学中的自我焦点燃烧。因此，研究孤子稳定性问题非常重要，在随机和确定性的环境以及通过分析和数值方法中，可以在该项目中进行研究。例如，奇异性形成的研究将有特殊的推力，例如崩溃和浓度。 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提议进行调查。使用分析和数值技术，将研究具有非本地电位的非线性Hartree方程，特别强调理解频谱结构。这种方程式在一般相对论和量子力学中都产生，因此在这种情况下研究很重要。最后，将研究波浪型方程中解决方案全球行为的描述问题，例如非线性klein-gordon方程。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估的。