Nonlinear Phenomena in Stochastic and Deterministic Dispersive Partial Differential Equations

波动是许多现实生活中常见的现象，从水波到电磁波或引力波。在研究波的过程中，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，流氓波或粗糙湍流）。通过各种数学模型了解这些过程的时间演变是该项目的目标，它将包括研究各种波浪现象，无论是否有随机元素。这些波型现象在自然界中是基本的，但我们仍然处于对其进行全面分析和数值描述的黎明。该项目旨在推进波动现象科学的前沿，并将其应用于自然界的现实表现。各级学生，从本科到博士后，将参与这项研究，并接受职业培训。该项目旨在了解非线性随机和确定性色散偏微分方程解的全局行为，与线性时间演化相比，非线性导致全局行为的显着差异。一种这样的非线性结构是孤波或孤立子，它具有特定的形状，存在（在数学意义上）所有时间，在某些方程中沿特定方向传播，而在其他方程中它周期性振荡。这种结构是稳定的还是不稳定的，以及它如何受到外部随机扰动的影响，在应用中是很重要的，如果不稳定，研究它会导致什么是很重要的。通常，不稳定性（孤子不稳定的情况）意味着奇点将形成：例如，海洋中的反常波或激光光学中的自聚焦燃烧。因此，它是非常重要的调查问题的孤子的稳定性，这是要研究在这个项目中，在随机和确定性的设置，并通过分析和数值方法。一个特殊的推力将被给予研究的形成奇点，例如，崩溃和浓度。本文的研究计划是从孤子稳定性理论的提出以及对随机广义Korteweg-de弗里斯方程奇异性形成过程的理解开始，然后发展到它的一个高维推广（即使在纯确定性的情况下也很少探索，但在量子力学和流体动力学中很重要）Zakharov-Kuznetsov方程，其中渐近稳定性问题，的存在性和描述的奇异性提出的调查。使用分析和数值技术的非线性Hartree方程与非本地潜在的调查，特别强调理解的频谱结构。这种类型的方程出现在广义相对论和量子力学中，因此在这方面的研究很重要。最后，将研究非线性Klein-Gordon方程等波动型方程解的全局行为描述问题。该奖项反映了NSF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。

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.