Remarks on the parallel propagation of small-amplitude dispersive Alfvénic waves

The envelope formalism for the description of a small-amplitude parallel-propagating Alfvén wave train is tested against direct numerical simulations of the Hall-MHD equations in one space dimension where kinetic effects are neglected. It turns out that the magnetosonic-wave dynamics departs from the adiabatic approximation not only near the resonance between the speed of sound and the Alfvén wave group velocity, but also when the speed of sound lies between the group and phase velocities of the Alfvén wave. The modulational instability then does not anymore affect asymptotically large scales and strong nonlinear effects can develop even in the absence of the decay instability. When the Hall-MHD equations are considered in the long-wavelength limit, the weakly nonlinear dynamics is accurately reproduced by the derivative nonlinear Schrödinger equation on the expected time scale, provided no decay instabilities are present. The stronger nonlinear regime which develops at later time is captured by including the coupling to the nonlinear dynamics of the magnetosonic waves.