Resonant Four-Wave Interaction in a Dispersive Medium

The resonant four-wave interaction is studied for a nonlinear two-dimensional Klein-Gordon system. A set of dynamical equations describing the slowly varying amplitudes is first derived for a spatially homogeneous case. Exact solutions to the set of equations are then obtained. They are classified into the following four types: (i) periodic solutions, (ii-1) solitary pulse solutions, (ii-2) a phase jump and three solitary pulse solutions, (iii) shock-like solutions. Some of these solutions, especially the type (iii) solutions, are quite different from those of the well-known nonlinear three-wave interaction.