  Large-amplitude Alfv\'en waves are observed in various systems in space and
laboratories, demonstrating an interesting property that the wave shapes are
stable even in the nonlinear regime. The ideal magnetohydrodynamics (MHD) model
predicts that an Alfv\'en wave keeps an arbitrary shape constant when it
propagates on a homogeneous ambient magnetic field. However, such arbitrariness
is an artifact of the idealized model that omits the dispersive effects. Only
special wave forms, consisting of two component sinusoidal functions, can
maintain the shape; we derive fully nonlinear Alfv\'en waves by an extended MHD
model that includes both the Hall and electron inertia effects. Interestingly,
these \small-scale effects" change the picture completely; the large-scale
component of the wave cannot be independent of the small scale component, and
the coexistence of them forbids the large scale component to have a free wave
form. This is a manifestation of the nonlinearity-dispersion interplay, which
is somewhat different from that of solitons.
