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

TL;DR

This paper explores conservation laws involving arbitrary time functions in dynamical PDEs, revealing topological charges and their implications for nonlocal symmetries, boundary conditions, and well-posedness in various dimensions.

Contribution

It introduces a novel class of conservation laws involving arbitrary functions of time, linking them to topological charges and boundary effects in dynamical PDEs.

Findings

Conservation laws with arbitrary time functions describe sources or topological charges.

Spatial potential systems can be derived to find nonlocal symmetries.

Integral of conserved densities reduces to boundary integrals, affecting well-posedness.

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.