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Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues

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Title: Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues
Author(s): Akers, Benjamin; Nicholls, David P.
Subject(s): spectrum waves resonance
Abstract: The spectral stability problem for periodic traveling waves on a two-dimensional fluid of infinite depth is investigated via a perturbative approach, computing the spectrum as a function of the wave amplitude beginning with a flat surface. We generalize our previous results by considering the crucially important situation of eigenvalues with multiplicity greater than one (focusing on the generic case of multiplicity two) in the flat water configuration. We use this extended method of transformed field expansions (which now accounts for the resonant spectrum) to numerically simulate the evolution of the eigenvalues as the wave amplitude is increased. We observe that there are no instabilities that are analytically connected to the flat state: The spectrum loses its analyticity at the Benjamin–Feir threshold. We complement the numerical results with an explicit calculation of the first nonzero correction to the linear spectrum of resonant deep water waves. Two countably infinite families of collisions of eigenvalues with opposite Krein signature which do not lead to instability are presented.
Issue Date: 2012-04
Publisher: Society for Industrial and Applied Mathematics
Citation Info: Akers B, Nicholls DP. Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues. Siam Journal on Applied Mathematics. 2012;72(2):689-711. DOI: 10.1137/110832446
Type: Article
Description: This is a copy of an article published in the SIAM Journal on Applied Mathematics © 2012 Society for Industrial and Applied Mathematics.
URI: http://hdl.handle.net/10027/10516
ISSN: 0036-1399
Date Available in INDIGO: 2013-11-15
 

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