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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.


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