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

Abstract

The spectral stability problem for periodic traveling water 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 case of flat water. We use this extended method of Transformed Field Expansions (which accounts for resonant spectrum) to numerically simulate the evolution of the eigenvalues as the wave amplitude is increased. From this we conclude that all of these configurations are spectrally stable (for waveheight sufficiently small), though our analysis currently excludes the well-known Benjamin-Feir instability. 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.

Modulational instabilities of periodic traveling waves in deep water

“…The predictions of RIT have since been leveraged heavily by numerical methods; the influential works of MacKay and Saffman [6] and McLean [7] led to a taxonomy of water wave instabilities based on RIT. The most recent review article is that of Dias and Kharif [8]; since the publication of this review a number of modern numerical stability studies have been conducted [9][10][11][12][13].…”

“…In a series of recent works, the author and collaborators derived the weakly nonlinear asymptotics of the spectrum in conjunction with the development of boundary perturbation methods, for deep water gravity waves in [13], including surface tension in [28], and with finite depth effects in [29]. Each article in this series considers a two-dimensional fluid, and expands the spectrum in amplitude at a sampling of fixed Bloch parameters; instabilities which have both fixed Bloch parameter and are analytic in amplitude are observed to be rare.…”

“…The leading asymptotics of the traveling wave have been computed numerous times [32][33][34][35]. The asymptotics of the spectrum have been computed for a two-dimensional fluid in [13,28,29], these asymptotics all compute the spectrum with fixed Bloch parameter. In this work, instabilities are computed with Bloch parameters which depend on amplitude, on both a two-dimensional and three-dimensional fluid.…”

“…Both two-dimensional and three-dimensional perturbations are considered. The asymptotic instability locations are compared with the numerical estimates of these locations using the method of Akers and Nicholls [13,28].…”

The stability of capillary waves on fluid sheets

“…Reviews of the stability properties of periodic water waves can be found in Hammack & Henderson (1993) and Dias & Kharif (1999). More recent results on the stability of gravity waves have been obtained by Deconinck & Oliveras (2011) and Akers & Nicholls (2014) for finite depth and Akers & Nicholls (2012) for infinite depth. The stability of gravitycapillary waves in infinite and finite depth was investigated by Djordjevic & Redekopp (1977) and Hogan (1985).…”