On the standing wave problem in deep water Gerard Iooss Institut Universitaire de France INLN UMR CNRS-UNSA 6618 1361 route des Lucioles, F-06560 Valbonne e-mail: Abstract We present a new formulation of the classical two-dimensional stand- ing wave problem which makes transparent the (seemingly mysterious) elimination of the quadratic terms made in [6]. Despite the presence of infinitely many resonances, corresponding to an infinite dimensional kernel of the linearized operator, we solve the infinite dimensional bi- furcation equation by uncoupling the critical modes up to cubic order, via a Lyapunov-Schmidt like process. This is done without using a nor- malization of the cubic order terms as in [6], where the computation contains a mistake, although the conclusion was in the end correct. Then we give all possible bifurcating formal solutions, as powers series of the amplitude (as in [6]), with an arbitrary number, possibly infinite, of dominant modes. 1 Introduction The two-dimensional standing wave problem for a potential flow with a free surface has attracted lot of interest since Stokes, and specially very recently. In particular the existence question in the cae of finite depth has a solution thanks to the work of Plotnikov and Toland [7]. They use, in an essential way, the fact that for most of the values of the depth, the kernel of the linearized operator is one dimensional.
- give infinitely
- lyapunov-schmidt like
- many terms
- dimensional stand- ing
- cos px
- infinitely many
- qt cos