  In this manuscript we investigate the Benjamin-Feir (or modulation)
instability for the spatial evolution of water waves from the perspective of
the discrete, spatial Zakharov equation, which captures cubically nonlinear and
resonant wave interactions in deep water without restrictions on spectral
bandwidth. Spatial evolution, with measurements at discrete locations, is
pertinent for laboratory hydrodynamic experiments, such as in wave flumes,
which rely on time-series measurements at a series of fixed gauges installed
along the facility. This setting is likewise appropriate for experiments in
electromagnetic and plasma waves. Through a reformulation of the problem for a
degenerate quartet, we bring to bear techniques of phase-plane analysis which
elucidate the full dynamics without recourse to linear stability analysis. In
particular we find hitherto unexplored breather solutions and discuss the
optimal transfer of energy from carrier to sidebands. Finally, we discuss the
observability of such discrete solutions in light of numerical simulations.
