On the standing wave problem in deep water

Abstract : We present a new formulation of the classical two-dimensional standing 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 normalization 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.
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  • HAL Id : hal-01271053, version 1

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Gérard Iooss. On the standing wave problem in deep water. Journal of Mathematical Fluid Mechanics, Springer Verlag, 2002, 4, pp.31. 〈hal-01271053〉

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