  We study the nonlinear evolution of very oblique small-scale Alfv\'en waves
with $k_\perp d_i\gtrsim 1$. At these scales, the waves become significantly
compressive, unlike in MHD, due to the Hall term in the equations. We
demonstrate that when frequencies are small compared to the ion gyrofrequency
and amplitudes small compared to unity, no new nonlinear interaction appears
due to the Hall term alone at the lowest non-trivial order, even when $k_\perp
d_i \sim 1$. However, at the second non-trivial order, we discover that the
Hall physics leads to a slow but resonant nonlinear interaction between
co-propagating Alfv\'en waves, an inherently 3D effect. Including the effects
of finite temperature, finite frequency, and electron inertia, the two-fluid
Alfv\'en wave also becomes dispersive once one or more of $k_\perp \rho_s$,
$k_\perp d_e$, or $k_\parallel d_i$ becomes significant: for oblique waves at
low $\beta$ as studied here, this can be at a much smaller scale than $d_i$. We
show that the timescale for one-dimensional steepening of two-fluid Alfven
waves is only significant at these smaller dispersive scales, and also derive
an expression for the amplitude of driven harmonics of a primary wave.
Importantly, both new effects are absent in gyrokinetics and other commonly
used reduced two-fluid models. Our calculations have relevance for the
interpretation of laboratory Alfv\'en wave experiments, as well as shedding
light on the physics of turbulence in the solar corona and inner solar wind,
where the dominant nonlinear interaction between counter-propagating waves is
suppressed, allowing these new effects to become important.
