  We investigate the macroscopic dynamics of sets of an arbitrary finite number
of weakly amplitude-modulated pulses in a multidimensional lattice of
particles. The latter are assumed to exhibit scalar displacement under
pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in
a nonlinear background field. By an appropriate multiscale ansatz, we derive
formally the explicit evolution equations for the macroscopic amplitudes up to
an arbitrarily high order of the scaling parameter, thereby deducing the
resonance and non-resonance conditions on the fixed wave vectors and
frequencies of the pulses, which are required for that. The derived equations
are justified rigorously in time intervals of macroscopic length. Finally, for
sets up to three pulses we present a complete list of all possible interactions
and discuss their ramifications for the corresponding, explicitly given
macroscopic systems.
