  In the work of Colliander et al. (2010), a minimal lattice model was
constructed describing the transfer of energy to high frequencies in the
defocusing nonlinear Schr\"odinger equation. In the present work, we present a
systematic study of the coherent structures, both standing and traveling, that
arise in the context of this model. We find that the nonlinearly dispersive
nature of the model is responsible for standing waves in the form of discrete
compactons. On the other hand, analysis of the dynamical features of the
simplest nontrivial variant of the model, namely the dimer case, yields both
solutions where the intensity is trapped in a single site and solutions where
the intensity moves between the two sites, which suggests the possibility of
moving excitations in larger lattices. Such excitations are also suggested by
the dynamical evolution associated with modulational instability. Our numerical
computations confirm this expectation, and we systematically construct such
traveling states as exact solutions in lattices of varying size, as well as
explore their stability. A remarkable feature of these traveling lattice waves
is that they are of "antidark" type, i.e., they are mounted on top of a
non-vanishing background. These studies shed light on the existence, stability
and dynamics of such standing and traveling states in $1+1$ dimensions, and
pave the way for exploration of corresponding configurations in higher
dimensions.
