Dynamics of dense lattices

We derive the equations governing the nonlinear dynamics of one-, two-, and three-dimensional lattices in a close to continuum condition (i.e., a dense lattice). The described method correctly captures all terms to a given order in discreteness and, unlike previous approaches, leads to well-behaved partial-differential equations for these problems. In general, the dispersion born out of discreteness counteracts the steepening of waves caused by the nonlinearity and leads to the formation of permanent nonlinear structures.