We present hybrid numerical simulations of a small number of low-frequency (much less than the proton gyrofrequency) Alfv¿n waves and, for the first time, show how to produce imbedded rotational discontinuities (RDs) with a small angle (&thgr;Bn) between their asymptotic and normal fields which are stable against dispersion. When the initial waves are linearly polarized and give a waveform whose Fourier components of B2 have wave vectors within ≈10¿ of the background magnetic field B0, the waveform tends to steepen even for small amplitudes (|ΔB|/B≪1). This produces RDs with normals nearly along B0 and nearly constant B between the RDs. However, the waveform is not steady because these RDs widen continually due to the presence of dispersive intermediate or ion cyclotron modes which are permanently attached to their edges. This commonly occurs in simulations of isolated RDs which have small &thgr;Bn. When the initial waveform is modified by including a transverse field component with a wavenumber equaling that from the Fourier component of B2 along B0 so that the field moves on an elliptical (B not constant) arc, the dispersive intermediate modes no longer develop, and the waveform and imbedded RDs evolve toward a steady state with nearly constant B and a circular arc polarization. Spherically polarized waveforms can be made in a similar manner. Only this type of waveform produces RDs without intermediate modes on their edges for all parameters. Oblique fast modes do form, but these have phase speeds which are faster than the Alfv¿n phase speed and so detach from the RD's edge and do not cause the RD to widen continually. The appearance of RDs with small &thgr;Bn varies greatly with the time of development, and this can explain the diversity of hodogram shapes of such RDs in the solar wind. We finally conclude that Alfv¿nic fluctuations with imbedded RDs must evolve through a succession of arc-polarized or spherically polarized waveforms as they travel outward from the Sun: Otherwise, the RDs with small &thgr;Bn in the solar wind would have widen to the point where they are beyond recognition. ¿ 1998 American Geophysical Union