Motion of Complex Singularities and Integrability in Surface Dynamics

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This research project is devoted to a study of surface dynamics that arise either at the interface between different moving fluids or at the fluid's surface. Surface dynamics include breaking of water waves and whitecapping, which are the primary mechanisms for the exchange of energy between the ocean and atmosphere making them a crucial ingredient of the global climate dynamics. These are strongly nonlinear phenomena which require to solve fully nonlinear hydrodynamics equations. Rogue waves are another example of strongly nonlinear large surface waves, which occur spontaneously in the ocean. Relative motion of fluids (wind over the water) induces instability of their common interface such as Kelvin-Helmholtz instability (KHI). This instability that recently became a focus of attention of experimental scientists in the context of the interface between different components superfluid Helium, will also be addressed in this project. This research will focus on development a new type of conformal map and new tools for the efficient description of the strongly nonlinear surface dynamics both for free surface and interfaces. It was Stokes who in the 19th century first used conformal mapping as a tool for studying of the steady flow of the fluid with a free surface. In this approach domains occupied by fluids are conformally mapped into simpler domains such as a lower complex half plane. The dynamics of the surface is then reduced to the dynamics of the conformal map. It gives an enormous advantage for both the analysis and high precision simulations of surface dynamics by allowing to recover the fluid dynamics through the motion of the complex branch cuts and poles in the complex domain. This project is aimed towards advancing the fields of surface dynamics and integrability as well as developing practical tools to identify the reduced models for dissipation of surface gravity waves, affecting global climate dynamics. The research will include an analysis of rogue and breaking waves dynamics through the motion of branch cuts as well as an exploration of integrability of interface dynamics of superfluids in different experimental situations. To address statistics of high amplitude water waves, analytical methods will be employed as well as development of high-performance computing tools.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.