Work to be done on this project continues mathematical research on nonlinear elliptic problems arising in perfect-fluid hydrodynamics. Emphasis will be placed on the analytical study of the propagation of waves in stratified media. Techniques from nonlinear analysis and partial differential equations form the basis for these studies. The primary goals are to understand better the nature of internal waves and the presence of vortex rings. The internal waves arise from density stratification due to changes in salinity or temperature, for example. Experimental work shows striking examples of internal wave phenomena, and oceanic observations reveal internal waves up to 100 meters in amplitude. This work will concentrate on the analysis of waves in the two-fluid system where two fluids of constant density meet along a smooth density profile. In this case, center-manifold theory can be applied to study the ordinary differential equations which replace the nonlinear partial differential equations for the wave motion. Recent applications of global analysis to study vortex rings which have a Heaviside function distribution have proved effective. Work will continue in an effort to understand the properties of the level set of the solutions to the equation describing the vortex displacement function. The equation has jump discontinuities, for which existence can sometimes be established, but the nature of the level sets remains a mystery.
