ABSTRACT
In this chapter, the authors derive asymptotic equations for the resonant interaction of small amplitude dispersive waves. In the simplest case, the wave amplitudes satisfy the three wave resonant interaction (TWRI) equations. However, very similar asymptotic expansions can be used to derive nonconservative TWRI equations and coupled Ginzburg-London equations for bifurcation phenomena in systems which are near a point of marginal instability. Linearized theory describes such waves well for a finite time. However, after a long enough interval of time cumulative nonlinear effects lead to a significant change in the wave field. A large amplitude theory is possible for waves that are modelled by completely integrable equations, such as the KdV equation or the NLS equation. Consequently, the energy of the waves produced by the interaction is always much smaller than the energy of the original waves, and linearized theory provides a uniformly valid first approximation to the wave field.
