Nonlinear Waves in Fluids

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Fluids, such as air or water, support many different kinds of waves. For example, sound waves propagate through compressible fluids, water waves propagate on the surface of an ocean, and large-scale atmospheric waves affect the earth's weather and climate. At sufficiently low intensities, waves are modeled by linear systems of equations, which are relatively well-understood; but at higher intensities, the equations become nonlinear. Nonlinearity introduces fundamental mathematical difficulties and leads to new physical phenomena, such as the shock waves generated by supersonic aircraft, the solitons (particle-like waves) that propagate on shallow water, and nonlinear wave interactions that generate additional waves. The aim of this project is to study the dynamics of waves in fluids and uncover new nonlinear phenomena in the context of specific physical applications. This award will also provide support for the involvement of one graduate student in the project. This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of fluid systems. It focuses on the asymptotic analysis of nonlinear, nondispersive and dispersive waves, including waves that propagate on boundaries, free surfaces, or interfaces. A unifying framework is provided by Hamiltonian dynamics, which is an essential structure for the waves considered in the proposed research. Physical applications include waves on vorticity discontinuities in incompressible fluids, temperature fronts in surface quasi-geostrophic flows, current-vortex sheets in plasmas, shock wave reflection, and the resonant interaction of sound and entropy waves. A number of these problems lead to common mathematical themes, such as degenerate dispersive shocks, normal form transformations for quasilinear wave equations, and the formation and propagation of singularities.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.