The stability analysis of surface water waves

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• Bernard Deconinck (University of Washington)
• Thursday 18 February 2010, 15:00-16:00
• MR14, CMS.

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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 non-local formulation introduced at Cambridge of the waterwave problem for a one-dimensional surface. Transforming the non-local 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 non-trivial 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 Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wave-length of the perturbation.

This talk is part of the Applied and Computational Analysis series.

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