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

Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs