  Quasi-crystals formed by charged mesoscopic dust grains (dust lattices),
observed since hardly a decade ago, are an exciting paradigm of a nonlinear
chain. In laboratory discharge experiments, these quasi-lattices are formed
spontaneously in the sheath region near a negative electrode, usually at a
levitated horizontal equilibrium configuration where gravity is balanced by an
electric field. It is long known (and experimentally confirmed) that
dust-lattices support linear oscillations, in the longitudinal (acoustic mode)
as well as in the transverse, in plane (acoustic-) or off-plane (optic-like
mode) directions. Either due to the (typically Yukawa type) electrostatic
inter-grain interaction forces or to the (intrinsically nonlinear) sheath
environment, nonlinearity is expected to play an important role in the dynamics
of these lattices. Furthermore, the coupling between the different modes may
induce coupled nonlinear modes. Despite this evidence, the elucidation of the
nonlinear mechanisms governing dust crystals is in a rather preliminary stage.
In this study, we derive a set of (coupled) discrete equations of motion for
longitudinal and transverse (out-of-plane) motion in a one dimensional model
chain of charged dust grains. In a continuum approximation, i.e. assuming a
variation scale which is larger than the lattice constant, one obtains a set of
coupled modified Boussinesq-like equations. Different nonlinear solutions of
the coupled system are discussed, based on localized travelling wave ansatze
and on coupled equations for the envelopes of co-propagating quasi-linear
waves.
