Modulational instability of plasma waves in two dimensions

The nonlinear behavior of whistler waves coupled to either fast magnetosonic waves (FMS) or slow magnetosonic waves (SMS) is investigated in two spatial dimensions. For each branch our investigation is based on a numerical solution of a reduced set of equations consisting of two partial differential equations, of which one, describing the evolution of the whistler wave envelope, is complex of first order in time and the other, describing the slow response of the medium in which the whistler wave is propagating, is real and of second order in time. These equations were solved in a two-dimensional domain with periodic boundary conditions by a fully de-aliased spectral method and using a third-order Adam-Bashforth predictor-corrector method for the time integration. The long-term evolution of the modulational instability for the FMS-coupling shows a quasi-recurrent behavior with a slow spreading of the energy to higher-mode numbers. For the SMS-coupling no recurrent behavior is found, and the energy is gradually leaking to higher-mode numbers while the spatial evolution of the modulation tends to develop small scale ''spikes''. Our results are of general interest for an understanding of the behavior of nonlinear waves in dispersive media.