  Dispersive shock waves (DSWs), which connect states of different amplitude
via a modulated wave train, form generically in nonlinear dispersive media
subjected to abrupt changes in state. The primary tool for the analytical study
of DSWs is Whitham's modulation theory. While this framework has been
successfully employed in many space-continuous settings to describe DSWs, the
Whitham modulation equations are virtually intractable in most spatially
discrete systems. In this article, we illustrate the relevance of the reduction
of the DSW dynamics to a planar ODE in a broad class of lattice examples.
Solutions of this low-dimensional ODE accurately describe the orbits of the DSW
in self-similar coordinates and the local averages in a manner consistent with
the modulation equations. We use data-driven and quasi-continuum approaches
within the context of a discrete system of conservation laws to demonstrate how
the underlying low dimensional structure of DSWs can be identified and
analyzed. The connection of these results to Whitham modulation theory is also
discussed.
