On Wave Interactions, I: Explosive Resonant Triads

This paper considers the phenomenon of explosive resonant triads in weakly nonlinear, dispersive wave systems. These nearly linear waves with slowly varying amplitudes become unbounded in finite time. It is shown that such interactions are much stronger than previously thought. These waves can be thought of as a nonlinear instability in the sense that a weakly nonlinear perturbation to some system grows to such a magnitude that the behavior of the system is governed by strongly nonlinear effects. This may occur for systems that are linearly or neutrally stable. This resolution is contrasted with previous resolutions of the problem, which assumed such perturbations remained large amplitude, nearly linear waves. Analytical and numerical evidence is presented to support these claims.