Evolution in space and time of resonant wave triads - I. The 'pump-wave approximation'

The evolution of weakly nonlinear resonant wave triads is examined for cases where the amplitudes are functions of two spatial coordinates and of time. In this paper, attention is restricted to the ‘pump-wave approximation’ in which one wave is assumed to be much larger than the other two, and to propagate without change in form. The evolution equations for the other two waves ars then linear and so are amenable to solution by classical techniques. Various solutions are described, with particular attention given to the evolution of initially localized disturbances.