  Dynamical PDEs that have a spatial divergence form possess conservation laws
that involve an arbitrary function of time. In one spatial dimension, such
conservation laws are shown to describe the presence of an $x$-independent
source/sink; in two and more spatial dimensions, they are shown to describe a
topological charge. Two applications are demonstrated. First, a topological
charge gives rise to an associated spatial potential system, allowing nonlocal
conservation laws and symmetries to be found for a given dynamical PDE.
Second,when a conserved density involves derivatives of an arbitrary function
of time in addition to the function itself, its integral on any given spatial
domain reduces to a boundary integral, which in some situations can place
restrictions on initial/boundary data for which the dynamical PDE will be
well-posed. Several examples of nonlinear PDEs from applied mathematics and
integrable system theory are used to illustrate these new results.
