Nonlinear theory of dust lattice mode coupling in dust crystals

摘要

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