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.
