Stable three-dimensional waves of nearly permanent form on deep water

Consider a uniform train of surface waves with a two-dimensional, bi-periodic surface pattern, propagating on deep water. One approximate model of the evolution of these waves is a pair of coupled nonlinear Schrödinger equations, which neglects any dissipation of the waves. We show that in this model, such a wave train is linearly unstable to small perturbations in the initial data, because of a Benjamin-Feir-type instability. We also show that when the model of coupled equations is generalized to include appropriate wave damping, the corresponding wave train is linearly stable to perturbations in the initial data. Therefore, according to the damped model, the two-dimensional surface wave patterns studied by Hammack et al. [J.L. Hammack, D.M. Henderson, H. Segur, Progressive waves with persistent, two-dimensional surface patterns in deep water, J. Fluid Mech. 532 (2005) 1-51] are linearly stable in the presence of wave damping.