Abstract:

For a mathematician, a water wave can be defined as a solution to the free boundary incompressible Euler equations.  The vorticity is then the curl of the velocity field.  Over the past decade, there has been a great deal of research into the existence and qualitative properties of traveling water waves with non-trivial vorticity.  One of the most interesting sub-species of rotational waves are those for which the vorticity is localized in space. Imagine, for example, a large eddy in the interior of the fluid, or a wake of vortices created by a submerged body.

The intention of this talk is to offer a fairly broad introduction to these waves, with an emphasis on recent advances and areas of current research.  In particular, we will discuss: the existence and stability/instability of traveling waves with a point vortex or dipole; stationary waves with an exponentially localized vortex ``spike''; and time-periodic rotating vortex patches.  Mathematically, this body of work draws on ideas ranging from singularly perturbed elliptic PDE theory, nonlinear dispersive equations, infinite-dimensional Hamiltonian systems, and Riemann--Hilbert theory.