Nonlinear three-wave interaction with non-conservative coupling

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Abstract
We consider the problem of three interacting resonant waves with arbitrary (non-conservative) nonlinear coupling. Such coupling arises naturally in the interaction of waves on shear flows, and in interactions between interfacial and gravity waves. We focus on the case where two modes are damped and have identical properties, and the third is linearly unstable. When the damping rates dominate the growth rate, the dynamics evolves on two disparate timescales and it is then possible to reduce the system to a multi-modal one-dimensional map, thus revealing clearly the complex sequence of bifurcations that occurs as the parameters are varied. We also investigate the effect on the equations of small additive noise; this can be simply modelled by a (deterministic) perturbation to the map. It is shown that even at very low levels, the effect of noise can be extremely important in determining the period and amplitude of the oscillations.