Applied Analysis for Emergent Nonlinear Wave Phenomena

波的形成可以在各种介质中观察到，从大气层和水圈到光纤和分子链。尽管这些介质存在物理差异，但波往往会产生具有共同特征的可识别模式和结构。这种形成的例子包括表现出快速振荡的波列，长时间持续传播的凹凸形波束，以及所谓的流氓波，这是背景状态的大扰动，不知从哪里出现，突然消失。这些和其他的波动现象可以用某些非线性微分方程来模拟，对这些方程进行详细的分析是可行的。该项目将开发，扩展和应用数学技术来研究不同物理环境中的波的形成和传播，以提取有关波的动力学的准确信息，并应用于流体力学，大气科学和光学电信等领域。例如，这种分析的结果将进一步了解异常波周围的波型。该项目的一部分将作为培训研究生和博士后研究人员的工具，该项目旨在开发和应用可积系统理论的分析和计算方法来研究非线性波传播。这些方法联合收割机工具从复杂的，渐近的，和数值分析。该项目的一部分将调查新的渐近制度，其中普遍的现象，在非线性波的形成可能会发生。研究活动包括描述和分类由形成大振幅流氓波的新型机制产生的波型，适用于各种模型，包括非线性薛定谔（NLS）和sine-Gordon方程。其他议题包括初始数据的时间演化，引起光谱奇异性的NLS方程和色散冲击波的形成和他们的相互作用与其他结构的Korteweg-de弗里斯方程。该奖项反映了NSF的法定使命，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 formations are observed in a variety of media ranging from the atmosphere and the hydrosphere to optical fibers and chains of molecules. Despite the physical differences in these media, waves tend to generate identifiable patterns and structures that share common characteristics. Examples of such formations include wave trains exhibiting rapid oscillations, bump-shaped beams propagating persistently over long periods of time, and so-called rogue waves, which are large disturbances of a background state that appear out of nowhere and disappear just as suddenly. These and other wave phenomena can be modeled by certain nonlinear differential equations for which detailed analysis is feasible. This project will develop, extend, and apply mathematical techniques to study wave formation and propagation in different physical contexts to extract accurate information about the dynamics of waves, with applications to fields including hydrodynamics, atmospheric sciences, and optical telecommunications. Outcomes of such analysis will, for example, further understanding of the wave patterns surrounding rogue waves. Parts of the project will serve as vehicles for the training of graduate students and postdoctoral researchers.This project aims to develop and apply analytical and computational methods from the theory of integrable systems to study nonlinear wave propagation. These methods combine tools from complex, asymptotic, and numerical analysis. Part of the project will investigate new asymptotic regimes in which universal phenomena in nonlinear wave formation may occur. The research activity includes describing and classifying the wave patterns generated by a new type of mechanism that forms large-amplitude rogue waves, applicable to various models, including the nonlinear Schrödinger (NLS) and the sine-Gordon equations. Other topics addressed include the time-evolution of initial data that gives rise to spectral singularities in the NLS equation and the formation of dispersive shockwaves and their interactions with other structures in the Korteweg-de Vries equation. The research will contribute to the fields of integrable systems, nonlinear waves, and special functions.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.