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 produce a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing non-local conservation laws and symmetries to be found for a given dynamical PDE. This type of potential system has a different form and different gauge freedom compared to potential systems that arise from ordinary conservation laws. Second, when a topological charge arises from a conservation law whose conserved density is non-trivial off of solutions to the dynamical PDE, then this relation yields a constraint 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 results.