Collaborative RUI. Nonlinear Schroedinger Models in Fluid Dynamics: Rogue Waves and Vortex Filaments

该项目结合了两个主要研究者（PI）的互补专业知识，研究由可积聚焦非线性薛定谔（NLS）方程和高阶概括建模的流体动力学的两个领域：深水中的流氓波生成和涡丝动力学，精确的数值方法将用于指导理论工作，并研究更多的物理相关模型。Floquet理论，动力系统方法，摄动理论，数值谱诊断，多辛和保形积分，将被用来解决问题的产生和持续的流氓波。物理实验室水槽实验将测试和验证接近不稳定性和同宿数据之间的相关性分析，以及流氓波的可能性;非线性阻尼是否通过降档消除流氓波。 涡旋丝的稳定性和动力学将从理论和数值上进行研究。流氓波是海洋中极其罕见的破坏性波浪，据信是自发产生的（与海啸相反）。它们是瞬时大振幅波，其高度显著大于背景海，25米和高达35米。对异常波浪的研究旨在改进这些极端波浪事件的预报。涡旋fi涡量是局部涡量结构的重要模型，在各种物理现象中，这些涡量结构常常表现为独特的特征;例如，&涡量可用于描述&超&流体&和湍流流体的流动。在地球物理系统（如龙卷风）和磁流体动力学（如细长的等离子体-#64257;在太阳日冕）中的丝状结构也可以模拟涡#64257;感叹号。为流氓波和涡丝模型开发的分析和数值工具将对其他应用流体力学问题具有价值。 这项研究将涉及每个机构的一名本科生和一名研究生。PI将继续指导来自代表性不足群体的学生，并通过将学生带到连接数学家和实验家的会议来扩大他们的培训。

This project combines the complementary expertise of the two principal investigators (PIs) to study two areas of fluid dynamics modeled by the integrable focusing Nonlinear Schroedinger (NLS) equation and higher order generalizations: rogue wave generation in deep water, and vortex filament dynamics, for which accurate numerical methods will be used to guide theoretical work, and to study more physically relevant models. Floquet theory, dynamical systems methods, perturbation theory, numerical spectral diagnostics, and multi-symplectic and conformal integrators, will be used to address the questions of generation and persistence of rogue waves. Physical lab-tank experiments will test and validate the analysis of the correlation between proximity to instabilities and homoclinic data, and the likelihood of rogue waves; whether nonlinear damping eliminates rogue waves through downshifting. Stability and dynamics of vortex filaments will be investigated theoretically and numerically.Rogue waves are extremely rare and destructive waves in the ocean that are believed to arise spontaneously (in contrast with tsunamis). They are transient large amplitude waves whose heights are significantly larger than the background sea, 25 meters and up to 35 meters. The research on rogue waves is aimed at improving predictors of these extreme wave events. Vortex filaments are important models of localized vorticity structures that often emerge as distinctive features in various physical phenomena; for example, vortex filaments can be used to describe the flow of superfluids and of turbulent fluids. Filamentary structures in geophysical systems (e.g. tornados) and in magneto-hydrodynamics (e.g. slender plasma-filled tubes in the solar corona) can also be modeled by vortex filaments. The analytical and numerical tools developed for both rogue wave and vortex filament models will be of value for other applied fluid mechanical problems. This research will involve one undergraduate and one graduate student at each of the institutions. The PIs will continue mentoring students from underrepresented groups and broadening their training by bringing students to meetings that connect mathematicians and experimentalists.