  We study localized traveling waves and chaotic states in strongly nonlinear
one-dimensional Hamiltonian lattices. We show that the solitary waves are
super-exponentially localized, and present an accurate numerical method
allowing to find them for an arbitrary nonlinearity index. Compactons evolve
from rather general initially localized perturbations and collide nearly
elastically, nevertheless on a long time scale for finite lattices an extensive
chaotic state is generally observed. Because of the system's scaling, these
dynamical properties are valid for any energy.
