Kinematics and Dynamics of Nonlinear Mode Coupling

A general treatment is given for the interaction of two propagating modes when they are non-linearly coupled by a large-amplitude driving wave. The equations describing the space-time development of the mode amplitudes have a certain symmetry due to the constraints of small-signal energy conservation. The wave solutions, given the vector coupling relations (${\omega }_d={\omega }_a\pm {\omega }_b,k_d=k_a\pm k_b$), are of four possible types, depending on the signs of the energy densities and group velocities of the individual modes. The periodicity imposed by the driving wave is used to construct a spatial-temporal harmonic dispersion diagram in which these interactions are displayed. The results are then shown to be generalizations of the wave-wave couplings of linear theory.