On the Long Time Behavior of Nonlinear Dispersive Equations

波在自然界中无处不在，用所谓的色散方程来描述，从光纤中的光传输到DNA分子中碱基对之间的电荷传输，再到原子内部的粒子相互作用。这些模型在物理和生物现象中普遍存在，但它们的长期行为还远远没有得到很好的理解。在这里，长时间行为是一个通用术语，用于包含全局适定性、散射效应、解的唯一延拓和波湍流等概念。了解色散方程的演化行为在数值模拟、光学、凝聚态物质、流体力学和生物学等领域有着广泛的应用。该项目的目的是了解分散系统的长期动态，同时带来新的见解，将在应用中建立新的方法和模型。该项目包括本科生和研究生的研究培训机会。这个项目包含三个部分。第一部分涉及独特的延拓原理，从数值的角度来看，它被认为起着重要的作用。在数值模拟中，计算机通常对域或数据或两者的有限近似进行计算，这可能导致非唯一性或不稳定性问题。一个完善的独特延拓理论将为计算结果提供更好的稳定性和可靠性。第二部分考虑波导上的色散模型，这是一个重要的数据传输模型。研究者还将引入一个具有波导行为的模型，该模型将首次将众所周知的概率Ornstein-Uhlenbeck算子与色散方程联系起来，并有可能在色散偏微分方程和概率之间建立一座坚固的桥梁。最后一部分研究了能量传递现象，这与波浪湍流理论有关。波浪湍流理论具有普遍预测相互作用波系波作用谱密度演变的特点，有助于预测海洋表面重力波等。该奖项反映了美国国家科学基金会的法定使命，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。

Waves are ubiquitous in nature and described by so-called dispersive equations, from the light transmission in fiber optics to the charge transport between base pairs in the DNA molecule to the particle interaction inside atoms. These models are pervasive in physical and biological phenomena, but their long-time behavior is far from well-understood. Here long-time behavior is a general term used to encompass the notions of global well-posedness, scattering effects, unique continuation of the solutions, and wave turbulence. Understanding the evolutionary behavior of dispersive equations has a broad range of applications in fields, such as, numerical simulations, optics, condensed matter, fluid mechanics, and biology. The aim of this project is to understand the long-time dynamics of dispersive systems while bringing new insights that will establish novel approaches and models in applications. The project includes research training opportunities for undergraduate and graduate students.This project contains three components. The first component concerns the unique continuation principle, which is believed to play an important part from a numerical point of view. In numerical simulations, a computer often preforms computations on finite approximations of the domain or the data or both, which may result in non-uniqueness or instability issues. A well-developed unique continuation theory will provide better stability and reliability of the computational results. The second considers dispersive models on waveguides, an important model for data transmission. The investigator will also introduce a model with waveguide behaviors that for the first time will connect a probabilistic well-known Ornstein-Uhlenbeck operator to dispersive equations and has the potential to build a strong bridge between dispersive partial differential equations and probability. The last component investigates the energy transfer phenomenon, which relates to the wave turbulence theory. Wave turbulence theory has the feature of universally predicting the evolution of the wave action spectral density of interacting wave systems, which will help with forecasting surface gravity waves in the oceans, 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.