  Within the standard perturbative approach of Peierls, a charge-density wave
is usually assumed to have a cosine shape of weak amplitude. In nonlinear
physics, we know that waves can be deformed. What are the effects of the
nonlinearities of the electron-lattice models in the physical properties of
Peierls systems? We study in details a nonlinear discrete model, introduced by
Brazovskii, Dzyaloshinskii and Krichever. First, we recall its exact analytical
solution at integrable points. It is a cnoidal wave, with a continuous
envelope, which may slide over the lattice potential at no energy cost,
following Fr\"ohlich's argument. Second, we show numerically that
integrability-breaking terms modify some important physical properties. The
envelope function may become discontinuous: electrons form stronger chemical
bonds which are local dimers or oligomers. We show that an Aubry transition
from the sliding phase to an insulating pinned phase occurs when the model is
no longer integrable.
