Mathematical Colloquium 
The instabilities of surface water waves 

By  Bernard Deconinck 
From  Department of Applied Mathematics University of Washington, Seattle 
When  May 3, 2010 4:00 pm 
Where  Weber 202 
Abstract  Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this talk, I discuss the stability of periodic traveling wave solutions to the full set of nonlinear equations via a nonlocal formulation of the water wave problem for a onedimensional surface. Transforming the nonlocal formulation into a traveling coordinate frame, we obtain a new equation for the stationary solutions in the traveling reference frame as a single equation for the surface in physical coordinates. We develop a numerical scheme to determine nontrivial traveling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically determine the spectral stability of the periodic traveling wave solutions by extending FourierFloquet analysis to apply to the associated linear nonlocal problem. In addition to presenting the full spectrum of this linear stability problem, we recover past wellknown results such as the BenjaminFeir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation. 
Further Information 
Yongcheng Zou 
There will be Refreshments in Weber 117 at 3.30pm
The Colloquium counts as Seminar Credit for Mathematics Students.