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Event Title : "Water-waves as a spatial reversible dynamical system, infinite depth case   (influence of an essential spectrum)"
Speaker : Prof.Gerard Iooss
Affiliation : IUF, INLN UMR CNRS-UNSA 6618, France
Subject Area : Mathematics
Date : January 20, 2005
Time : 4.00Pm
Venue : Lecture Hall I, Depat of Mathematics
Abstract
The mathematical study of traveling waves, in the context of two
dimensional potential flows in one or several layers of perfect fluid(s),
in the presence of free surface and interfaces can be set as an ill-posed
evolution problem, where the horizontal space variable plays the role if a
"time".
A case of great physical interest is the infinite depth limit. In such a
case, the classical reduction technique to a small-dimensional center
manifold fails because the linearized operator possesses an essential
spectrum filling the whole real axis, and new adapted tools are necessary.
We give a method and the results for different types of systems. An
example is with two superposed layers, the bottom one being infinitely
deep, with no surface tension at the interface and surface tension at the
free surface. In case of a strong enough surface tension at the free
surface the dominant part of the bifurcating solutions is provided when a
pair of imaginary eigenvalues merge at 0, which is part of the essential
spectrum, and disappear when a parameter is varying. In case of week
surface tension at the free surface, there is in addition an oscillating
mode. In both cases the bifurcating solutions are ruled by the
Benjamin-Ono nonlocal differential equation, coupled, in the latter case
with an oscillatory mode leading to nonzero periodic waves at infinity
(which might be of exponentially small size).
In this lecture we give quite general assumptions in infinite-dimensional
reversible systems for this types of bifurcations in presence of essential
spectrum.