  Motivated by recent Parker Solar Probe (PSP) observations of "switchbacks"
(abrupt, large-amplitude reversals in the radial magnetic field, which exhibit
Alfv\'enic correlations) we examine the dynamics of large-amplitude Alfv\'en
waves in the expanding solar wind. We develop an analytic model which makes
several predictions: switchbacks should preferentially occur in regions where
the solar wind plasma has undergone a greater expansion, the switchback
fraction at radii comparable to PSP should be an increasing function of radius,
and switchbacks should have their gradients preferentially perpendicular to the
mean magnetic field direction. The expansion of the plasma generates small
compressive components as part of the wave's nonlinear evolution: these are
maximized when the normalized fluctuation amplitude is comparable to
$\sin\theta$, where $\theta$ is the angle between the propagation direction and
the mean magnetic field. These compressive components steepen the primary
Alfv\'enic waveform, keeping the solution in a state of nearly constant
magnetic field strength as its normalized amplitude $\delta B/B$ grows due to
expansion. The small fluctuations in the magnetic-field-strength are minimized
at a particular $\theta$-dependent value of $\beta$, usually of order unity,
and the density and magnetic-field-strength fluctuations can be correlated or
anticorrelated depending on $\beta$ and $\theta$. Example solutions of our
dynamical equation are presented; some do indeed form magnetic-field reversals.
Our predictions appear to match some previously unexplained phenomena in
observations and numerical simulations, providing evidence that the observed
switchbacks result from the nonlinear evolution of the initially
small-amplitude Alfv\'en waves already known to be present at the coronal base.
