Abstract Motivated by recent Parker Solar Probe (PSP) observations of “switchbacks” (abrupt, large-amplitude reversals in the radial magnetic field, which exhibit Alfvénic correlations), we examine the dynamics of large-amplitude Alfvén waves in the expanding solar wind. We develop an analytic model that 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 θ , where θ is the angle between the propagation direction and the mean magnetic field. These compressive components steepen the primary Alfvénic waveform, keeping the solution in a state of nearly constant magnetic field strength as its normalized amplitude δB/B grows due to expansion. The small fluctuations in the magnetic field strength are minimized at a particular θ -dependent value of β , usually of order unity, and the density and magnetic-field-strength fluctuations can be correlated or anticorrelated depending on β and θ . 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 Alfvén waves already known to be present at the coronal base.