On the standing wave problem in deep water
type: Item
instance of: scholarly article
title: On the standing wave problem in deep water
zbMATH Open document ID: 1078.76014
author: Gérard Iooss
published in: Journal of Mathematical Fluid Mechanics
publication date: 2002-09-10T00:00:00Z
full work available at URL: standWaves3.pdf
review text: 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 the author's paper [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1989, 3513--3526 (1999; Zbl 0949.76014)]. Despite the presence of infinitely many resonances, corresponding to an infinite dimensional kernel of the linearized operator, we solve the infinite dimensional bifurcation 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 the paper cited above, where the computation contains a mistake, although the conclusion was in the end correct. Then we give all possible bifurcating formal solutions, as power series of the amplitude, with an arbitrary number, possibly infinite, of dominant modes .
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76B07
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47J15
zbMATH DE Number: 1798860
zbMATH Keywords: two-dimensional standing wave problem
zbMATH Keywords: infinite dimensional bifurcation equation
zbMATH Keywords: bifurcating formal solutions
MaRDI profile type: Publication
OpenAlex ID: W2076253063
DOI: 10.1007/S00021-002-8541-Z
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description: scientific article; zbMATH DE number 1798860