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Normal form transformations for modulated deep-water gravity waves
Philippe Guyenne, Adilbek Kairzhan, and Catherine Sulem
Volume 64, no. 2
(2023),
pp. 281–308
https://doi.org/10.33044/revuma.2918
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Abstract
Modulation theory is a well-known tool to describe the long-time evolution
and stability of small-amplitude, oscillating solutions to dispersive
nonlinear partial differential equations. There have been a number of
approaches to deriving envelope equations for weakly nonlinear waves. Here
we review a systematic method based on Hamiltonian transformation theory
and averaging Hamiltonians. In the context of the modulation of two- or
three-dimensional deep-water surface waves, this approach leads to a Dysthe
equation that preserves the Hamiltonian character of the water wave
problem. An explicit calculation of the third-order Birkhoff normal form
that eliminates all non-resonant cubic terms yields a non-perturbative
procedure for the reconstruction of the free surface. We also present new
numerical simulations of this weakly nonlinear approximation using the
version with exact linear dispersion. We compare them against computations
from the full water wave system and find very good agreement.
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