Title: Travelling Water-Waves, as a Paradigm for Bifurcations in Reversible Infinite Dimensional ``Dynamical'' Systems

We first show a typical bifurcation study for a finite dimensional reversible system, near a symmetric equilibrium taken at $0$. We state the results on known small bounded solutions: periodic, quasi-periodic, homoclinic to $0$, and homoclinics to periodic solutions. The main tool for such a study is center manifold reduction and normal form theory, in presence of reversibility. This allows to prove persistence of large class of reversible (symmetric) solutions under higher order terms, not considered in the normal form. We then present water-wave problems, where we look for 2D travelling waves in a potential flow. In case of finite depth layers, the problem of finding small bounded solutions, is shown to be reducible to a finite dimensional center manifold, on which the system reduces to a reversible ODE. Bounded solutions of this ODE lead to various kinds of travelling waves which are discussed. If the bottom layer has infinite depth, which appears to be the most physically realistic case, concerning the validity of results in the parameter set, the mathematical problem is more difficult. We don't know how to reduce it to a finite dimensional one, due to the occurence of a continuous spectrum (of the linearized operator) crossing the imaginary axis. We give some hints, on how to attack this difficulty, specially for periodic and homoclinic solutions which have now a \emph{polynomial decay} at infinity .

1991 Mathematics Subject Classification: 58F39, 58F14, 76B15, 34A47, 76B25

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