I will discuss a series of interesting phenomena that take place
when a nonlinear lattice is harmonically forced.
Depending on the frequency of the forcing, "modulated nonlinear waves"
or "localized excitations" appear. They bear some connection, in the
low amplitude limit, with exact solutions of nonlinear PDE's,
or, in the large amplitude case, with approximate solutions obtained by
truncating exact discrete nonlinear waves solutions.
If instead a sharp pulse is applied at a boundary, "supersonic kinks"
propagate inside the bulk. Their shape can be again obtained with similar
truncations. If the boundary is vibrated with an "out-band" frequency,
a critical amplitude exist below which no trasmission is present, while
above that value trasmission abruptly sets in: this phenomenon has been
called "nonlinear supratransmission". Under that action of noise, multiple
stochastic resonances can appear.