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.
Publié le : 2002-07-04
Classification:
standing waves,
Nonlinear water waves,
normal forms,
bifurcation theory,
35B34, 74J30, 76B07, 76B15, 37G05,
[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph],
[NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS],
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
@article{hal-01271053,
author = {Iooss, G\'erard},
title = {On the standing wave problem in deep water},
journal = {HAL},
volume = {2002},
number = {0},
year = {2002},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-01271053}
}
Iooss, Gérard. On the standing wave problem in deep water. HAL, Tome 2002 (2002) no. 0, . http://gdmltest.u-ga.fr/item/hal-01271053/