Applied Analysis for Emergent Nonlinear Wave Phenomena

在从大气和水圈到光纤和分子链的各种介质中都可以观察到波的形成。尽管这些介质的物理特性不同，但波往往会产生具有共同特征的可识别的图案和结构。这种形成的例子包括表现出快速振荡的波列，在长时间内持续传播的凹凸状光束，以及所谓的无赖波，即突然出现并同样突然消失的背景状态的大扰动。这些波动现象和其他波动现象可以用某些非线性微分方程组来模拟，对其进行详细分析是可行的。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，以提取关于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围绕着无赖波的波动模式。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中的分析和计算方法来研究非线性波的传播。这些方法结合了复杂分析、渐近分析和数值分析的工具。该项目的一部分将研究新的渐近制度，在这些制度中，可能会出现非线性波形成的普遍现象。研究活动包括描述和分类由一种新的形成大幅度流浪波的机制产生的波型，该机制适用于各种模型，包括非线性薛定谔(NLS)和Sine-Gordon方程。讨论的其他主题包括引起NLS方程中光谱奇异性的初始数据的时间演变，以及色散激波的形成及其与Korteweg-de Vries方程中其他结构的相互作用。这项研究将在可积系统、非线性波和特殊函数领域做出贡献。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。

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.