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? A nonlinear discrete model, introduced by Brazovskii, Dzyaloshinskii and Krichever, allows to address this issue beyond perturbation theory. For some special parameters of the model, at integrable points, it has exact solutions: the wave is cnoidal rather than simply harmonic but its envelope function remains continuous. As a consequence, the charge-density wave may slide over the lattice potential at no energy cost, in agreement with the Fröhlich's argument. When the integrability is broken, however, the envelope function may be discontinuous, electrons tend to form some stronger distinct chemical bonds, i.e. local dimers or oligomers. The wave is intrinsically pinned by the lattice. We argue that an Aubry transition from the sliding phase to the insulating pinned phase occurs when the nonlinearities become strong.