  The equations governing weakly nonlinear modulations of $N$-dimensional
lattices are considered using a quasi-discrete multiple-scale approach. It is
found that the evolution of a short wave packet for a lattice system with cubic
and quartic interatomic potentials is governed by generalized Davey-Stewartson
(GDS) equations, which include mean motion induced by the oscillatory wave
packet through cubic interatomic interaction. The GDS equations derived here
are more general than those known in the theory of water waves because of the
anisotropy inherent in lattices. Generalized Kadomtsev-Petviashvili equations
describing the evolution of long wavelength acoustic modes in two and three
dimensional lattices are also presented. Then the modulational instability of a
$N$-dimensional Stokes lattice wave is discussed based on the $N$-dimensional
GDS equations obtained. Finally, the one- and two-soliton solutions of
two-dimensional GDS equations are provided by means of Hirota's bilinear
transformation method.
