On the Transition from Two-Dimensional to Three-Dimensional Water Waves

There are essentially two types of three-dimensional water waves: waves that bifurcate from the state of rest (these waves are commonly called short-crested waves or forced waves), and waves that bifurcate from a two-dimensional wave of finite amplitude (these waves are sometimes called spontaneous waves). This paper deals with spontaneously generated three-dimensional waves. To understand this phenomenon better from a mathematical point of view, it is helpful to work on model equations rather than on the full equations. Such an attempt was made formally by Martin in 1982 on the nonlinear Schrödinger equation, but it is shown here that it is hard to justify his results mathematically because of the hyperbolicity of the nonlinear Schrödinger equation for gravity waves. On the other hand, in some parameter regimes, the nonlinear Schrödinger equation becomes elliptic. In that case, the appearance of spontaneous three-dimensional waves can be shown rigorously by using a dynamical systems approach. The results are extended to the Benney–Roskes–Davey–Stewartson equations when they are both elliptic. Various types of three-dimensional waves bifurcating from a two-dimensional periodic wave are obtained.