Dynamics of Three-Dimensional Gravity-Capillary Solitary Waves in Deep Water

Abstract

A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatially localized in both directions. Branches of both line and lump solitary waves are computed via a numerical continuation method. The stability of each type of wave is examined. The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrödinger equation.The spectral stability of lumps is predicted using the waves' speed energy relation. The role of wave collapse in the stability of these waves is also examined. Numerical time evolution is used to confirm stability predictions and observe dynamics, including instabilities and solitary wave collisions.

Dynamics of gravity–capillary solitary waves in deep water

“…The details of these computations can be found in Akers & Milewski (2010). An interesting particularity is that the physical energy for two dimensional wavepacket solitary waves is predicted to tend to a finite value of 12.04 as the amplitude approaches zero, which we shall verify in the cDtNE model.…”

“…For plane solitary waves, the linear analysis based on NLS (see Akers & Milewski (2010)) shows that the transverse perturbation e ily is unstable, when the wave number l in the y direction satisfies…”

“…Line solitary waves are unstable to transverse perturbations of sufficiently long wavelength. Fully 2D dynamics have been considered by Akers & Milewski (2010) in a one-way simplified model and in a quadratic isotropic model in Akers & Milewski (2009). The main goal of this paper is to study 2D dynamics within a close approximation of the Euler equations.…”

Gravity perturbed Crapper waves

“…Beginning with the Crapper wave, we observe that waves become more localized with increased Bond number. As σ increases, profiles become more solitary than periodic, gaining oscillatory tails similar to those computed in [47,50]. These changes occur at all amplitudes, see figure 1 for the case of y ∞ = 0.2, however, the size of oscillations in the tail decreases with amplitude.…”

“…When the system is discretized with N spatial points, we must then solve for N Fourier modes and the wave speed c. The projection of the partial differential equation into Fourier space gives N equations, to which we append an equation fixing the amplitude to close the system. The resulting nonlinear system of algebraic equations is then solved via Broyden's method, and continuation in amplitude or gravity, similar to [47,48].…”