Transversally periodic solitary gravity–capillary waves

Abstract

When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two-and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity-capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two-and three-dimensional computations of fully localized gravity-capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

Local bifurcation of electrohydrodynamic waves on a conducting fluid

“…The efficiency and accuracy of these numerical models for nonelectric surface water waves (i.e., E p = 0) were discussed by Wang and Milewski 17 and Nicholls and Reitich. 20…”

Non-linear Langmuir waves in a warm quantum plasma

“…28 One of these waves, namely, the long wave, is traditionally named a gravitational wave, whereas the other one, or the short one with a positive dispersion, is called a capillary wave. In one of the recent papers, 29 the authors obtained the profiles of doublefrequency waves based on a numerical solution of non-linear equations describing capillary-gravitational waves.…”

Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

“…Surface elevation of the envelope of gravity-capillary solitary waves perturbed by their most unstable modes corresponding to Ω = 0, q = 1.01 and Ω = 0.10, q = 1.08 are shown in figure 9. These waves which are localized in the direction of propagation and periodic in the transverse direction have been investigated by Milewski & Wang (2014). This new solution found by the latter authors was called transversally periodic plane solitary wave.…”

“…Our purpose is not to consider three-dimensional localized solitary gravity-capillary waves. Akers & Milewski (2009), Akers & Milewski (2010), Wang & Milewski (2012) and Milewski & Wang (2014) used several model equations to investigate numerically the stability of two-dimensional and three-dimensional gravity-capillary solitary waves in deep water. Regarding two-dimensional waves they found that both gravity-capillary solitary waves of elevation and depression are unstable to transverse perturbations.…”