Resonances in Dispersive Wave Systems

Abstract

Weakly nonlinear wave interactions under the assumption of a continuous, as opposed to discrete, spectrum of modes is studied. In particular, a general class of one‐dimensional (1‐D) dispersive systems containing weak quadratic nonlinearity is investigated. It is known that such systems can possess three‐wave resonances, provided certain conditions on the wavenumber and frequency of the constituent modes are met. In the case of a continuous spectrum, it has been shown that an additional condition on the group velocities is required for a resonance to occur. Nonetheless, such so‐called double resonances occur in a variety of physical regimes. A direct multiple scale analysis of a general model system is conducted. This leads to a system of three‐wave equations analogous to those for the discrete case. Key distinctions include an asymmetry between the temporal evolution of the modes and a longer time scale of as opposed to O(εt). Extensions to additional dimensions and higher‐order nonlinearities are then made. Numerical simulations are conducted for a variety of dispersions and nonlinearities providing qualitative and quantitative agreement.