Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room) or 216 McBryde, TBA
Speaker: Gérard Iooss of Nice
Title: The Water-wave problem as a reversible spatial dynamical system
In the finite depth case, the spectrum of the linearized operator is only composed with isolated eigenvalues of finite multiplicities, with a finite number near the imaginary axis. The study of near equilibria waves reduces to a low dimensional center manifold, leading to a reversible ordinary differential equation. In most cases, the dynamics on the center manifold is the one of a perturbed integrable system, where all types of solutions are known. We review typical results.
The infinite depth limit is indeed a case of physical interest. In such a case, the above reduction technique fails because the linearized operator possesses an essential spectrum filling the whole real axis, and new adapted tools are necessary. We give the method when the dominant part of the bifurcating solutions results from the merging of a pair of imaginary eigenvalues in the essential spectrum in 0. An example is with two superposed layers, the bottom one being infinitely deep, with no surface tension at the free surface and interface. In this later case we obtain a solitary wave of Benjamin-Ono type with small periodic oscillations at infinity whose size maybe exponentially small, in terms of the bifurcation parameter.
Bibliography
F.Dias, G.Iooss. Water-waves as a spatial dynamical system. Handbook
of math. uid dynamics. S.Friedlander, D.Serre eds (62p., to appear).
G.Iooss, E.Lombardi, S.M.Sun. Gravity travelling waves for two superposed uid layers, one being of in_nite depth: a new type of bifurcation. Phil. Trans. Roy. Soc. London A (2002) 360, 2245-2336.
G.Iooss, Institut Universitaire de France INLN, UMR 6618 CNRS-UNSA 1361 route des Lucioles, Sophia-Antipolis, F-06560 VALBONNE, France
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