Resonances in Dispersive Wave Systems

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 (ε t ). 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.