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SUMMARY:The stability analysis of surface water waves - Bernard Deconinck 
 (University of Washington)
DTSTART:20100218T150000Z
DTEND:20100218T160000Z
UID:TALK23265@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:Euler's equations describe the dynamics of gravity waves on th
 e 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 non
 linear equations via a non-local formulation introduced at Cambridge of th
 e waterwave problem for a one-dimensional surface. Transforming the non-lo
 cal formulation into a traveling coordinate frame\, we obtain a new equati
 on for the stationary solutions in the traveling reference frame as a sing
 le equation for the surface in physical coordinates. We develop a numerica
 l scheme to determine non-trivial traveling wave solutions by exploiting t
 he bifurcation structure of this new equation. Specifically\, we use the c
 ontinuous dependence of the amplitude of the solutions on their propagatio
 n 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 presen
 ting the full spectrum of this linear stability problem\, we recover past 
 well-known results such as the Benjamin–Feir instability for waves in de
 ep water. In shallow water\, we find different instabilities. These shallo
 w water instabilities are critically related to the wave-length of the per
 turbation.\n
LOCATION:MR14\, CMS
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