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 genera l class of one-dimensional (1-D) dispersive systems containing weak quadrat ic nonlinearity is investigated. It is known that such systems can possess three-wave resonances, provided certain conditions on the wavenumber and fr equency of the constituent modes are met. In the case of a continuous spect rum, it has been shown that an additional condition on the group velocities is required for a resonance to occur. Nonetheless, such so-called double r esonances occur in a variety of physical regimes. A direct multiple scale a nalysis of a general model system is conducted. This leads to a system of t hree-wave equations analogous to those for the discrete case. Key distincti ons include an asymmetry between the temporal evolution of the modes and a longer time scale of O(epsilon root t) as opposed to O(Et), Extensions to a dditional dimensions and higher-order nonlinearities are then made, Numeric al simulations are conducted for a variety of dispersions and nonlinearitie s providing qualitative and quantitative agreement.