"Spectral Stability of Traveling Water Waves"

David P. Nicholls
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
Chicago, IL  60607

Abstract
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The motion of the free surface of an ideal fluid under the effects of
gravity and capillarity arises in a number of problems of practical
interest, consequently, the reliable and accurate numerical simulation
of these "water waves" is of central importance.  Recently, an
efficient, stable, and high-order Boundary Perturbation scheme for
simulating these traveling water waves (due to the author and
F. Reitich) has been extended to address the equally important topic
of their dynamic (spectral) stability.  In this talk we will discuss
this algorithm and present new results on the "motion" of the spectrum
of the linearized water wave equations as the traveling waveform is
varied.  In particular, we will focus upon the radius of convergence
of a Taylor series expansion of the spectral data and its possible
connection to spectral instability of traveling water waves.