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

Abstract

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.