The nonlinear dynamics resulting from transverse and quasi-transverse instabilities of a finite-amplitude dispersive Alfvén wave propagating along an ambient magnetic field is studied by direct numerical simulations of the three-dimensional Hall-magnetohydrodynamic (Hall-MHD) equations. When the pump wave has a moderate amplitude and a long enough wavelength, one observes the generation of nonlinear structures in the form of helical filaments for the transverse magnetic field intensity and the density fluctuations. An interesting feature is the development of a quasi-incompressible turbulent flow, with a longitudinal characteristic scale large compared to the Alfvén wavelength, that remains spectrally well separated from the wave throughout the evolution. The coexistence of this “reduced MHD” flow with nonlinear Alfvén waves was predicted on the basis of an asymptotic analysis [A. Gazol, T. Passot, and P. L. Sulem, Phys. Plasmas 6, 3114 (1999)] carried out in the long-wavelength limit. Whereas in this regime the generation of the reduced MHD flow is negligible, it becomes significant on a time scale of a few wave periods when dispersion is increased. Increasing the dispersion also leads to a faster destabilization of the wave and to a more rapid dissipation, a remarkable effect due to enhanced instability growth rates. In the case of a larger amplitude pump, or of an Alfvén wavelength close to the ion-inertial length, the helical structures get fragmented and the spectral gap observed at early times between the large-scale flow and the waves rapidly disappears, leading to a fully three-dimensional MHD turbulent flow.