ABSTRACT
Surface gravity waves on deep water are necessarily dispersive; as a result, they tend to travel in groups when their wave lengths differ little. During the propagation, while dissipation is negligible, the nonlinear self-interaction among waves in a group, i.e., waves having more or less the same wave length, can be very important, resulting in the gradual but significant evolution of the waves themselves and their group shape. This chapter studies the three-dimensional properties of such nonlinear interactions and some of their applications. It reports three classes of exact solutions of the three-dimensional nonlinear equation governing the evolution of deep water waves under gravity: permanent wave groups, interacting soliton groups; and general solutions for waves having their groups propagating along the characteristic lines. For permanent wave groups the underlying waves have constant wave number and frequency.
