Collective excitations of charged dust grains in dusty plasma lattices

\emph{Dusty Plasmas} (or \emph{Complex Plasmas}) are large ensembles of interacting particles, consisting of electrons, ions and massive, heavily charged, micron-sized dust particulates. The presence of the latter modifies the plasma properties substantially and allows for new charged matter configurations, including liquid-like and solid (quasi-\textit{crystalline}) states (Debye crystals). One- dimensional (1d) dust crystals are formed in discharge experiments, where the electrode \emph {sheath} electric fields and electrostatic interactions constitute a highly nonlinear environment. The nonlinear aspects of horizontal (longitudinal, acoustic mode) as well as vertical (transverse, optical mode) motion of charged dust grains in a (1d) dust crystal are discussed. Different types of localized excitations, predicted by nonlinear wave theories, are reviewed and conditions for their occurrence (and characteristics) in DP crystals are discussed, in a continuum approximation. Dust crystals are shown to support nonlinear \emph{kink-}shaped supersonic solitary excitations, related to longitudinal dust grain displacement, as well as modulated \emph{envelope localized modes} associated with either longitudinal or transverse oscillations. Furthermore, the possibility for highly localized \emph{discrete breather}-type excitations to occur is investigated, for first principles. The relation to known results on atomic chains and on strongly- coupled dust layers in gas discharge plasma experiments is discussed.