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arXiv · 2006.03639

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

Stephen C. Anco·Elena Recio
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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.

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Stephen C. Anco, Elena Recio. 2020-06-05. Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs. https://doi.org/10.1098/rspa.2020.0442

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