Resonant interactions between waves. The case of discrete oscillations

Abstract

The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.