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

该项目结合了两个主要研究人员(PI)的互补专业知识，研究了由可积聚焦非线性薛定谔(NLS)方程和高阶推广建立的两个流体动力学领域：深水中的无赖波产生和涡丝动力学，将使用精确的数值方法指导理论工作，并研究更多物理相关的模型。我们将使用Floquet理论、动力系统方法、摄动理论、数值谱诊断以及多辛和共形积分器来解决无赖波的产生和持续问题。物理实验室水箱实验将测试和验证对接近不稳定和同宿数据之间的相关性的分析，以及流氓波的可能性；非线性阻尼是否通过减速来消除流氓波。涡旋细丝的稳定性和动力学将从理论和数值上进行研究。无赖波是海洋中极其罕见的破坏性波，被认为是自发产生的(与海啸相反)。它们是瞬变的大幅度波，其高度明显大于背景海，25米到35米。对异常海浪的研究旨在提高对这些极端海浪事件的预报能力。旋涡哀叹是局部化涡度结构的重要模型，它经常作为各种物理现象的显著特征出现；例如，旋涡哀叹可以用来描述超级UID和湍流UID的旋涡结构。地球物理系统(如龙卷风)和磁流体动力学(如日冕中的细长等离子体管)中的丝状结构也可以用涡旋哀叹来模拟。为流浪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一定的参考价值。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，并通过让学生参加连接数学家和实验学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.