  The nonlinear aspects of longitudinal motion of interacting point masses in a
lattice are revisited, with emphasis on the paradigm of charged dust grains in
a dusty plasma (DP) crystal. Different types of localized excitations,
predicted by nonlinear wave theories, are reviewed and conditions for their
occurrence (and characteristics) in DP crystals are discussed. Making use of a
general formulation, allowing for an arbitrary (e.g. the Debye electrostatic or
else) analytic potential form $\phi(r)$ and arbitrarily long site-to-site range
of interactions, it is shown that dust-crystals support nonlinear kink-shaped
localized excitations propagating at velocities above the characteristic DP
lattice sound speed $v_0$. Both compressive and rarefactive kink-type
excitations are predicted, depending on the physical parameter values, which
represent pulse- (shock-)like coherent structures for the dust grain relative
displacement. Furthermore, the existence of breather-type localized
oscillations, envelope-modulated wavepackets and shocks is established. The
relation to previous results on atomic chains as well as to experimental
results on strongly-coupled dust layers in gas discharge plasmas is discussed.
