We study localized traveling waves and chaotic states in strongly nonlinear one-dimensional Hamiltonian lattices. We show that the solitary waves are superexponentially localized and present an accurate numerical method allowing one 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.
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