  We study large-amplitude, very oblique Alfv\'en waves at low $\beta$, with
small gradient length scales, comparable to the ion inertial scale $d_i$. Such
waves have large density fluctuations, and slight dispersion from
finite-frequency and finite ion sound radius effects. We derive a weakly
nonlinear evolution equation governing the behaviour of the waves in one
dimension, and categorize the different solitons appearing in different
regimes: the regular solitons involve full rotations of the transverse magnetic
field, similar to modified Korteweg-de Vries (mKdV) solitons (our nonlinear
equation reduces to the mKdV equation in the long-wavelength limit). However,
for sufficiently small soliton widths, some become singular, small-amplitude
solitons with density discontinuities, and are thus expected to become strongly
dissipative in a real plasma. These solutions may be useful in explaining some
aspects of the sharp, ion-scale magnetic field rotations (switchbacks) observed
in the near-Sun solar wind by Parker Solar Probe.
