  Nonlinear waves have been observed in synchrotrons for years but have
received little attention in the literature. While pathological, these
phenomena are worth studying on at least two accounts. First, the formation of
solitary waves may lead to droplet formation that causes significant beam halo
to develop. Secondly, a variety of nonlinear processes are likely involved in
the normal saturation of unstable oscillations, leading to the possibility that
low-level, but potentially broadband fluctuation spectra may develop. Both
features carry indirectly the signature of the machine impedance. In this work
we review a number of observations of nonlinear longitudinal waves made in
Fermilab accelerators, and make a first attempt to develop appropriate
theoretical models to explain these observations.
