Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs. (6th January 2021)

• Record Type: Journal Article
• Authors: Anco, Stephen C.; Recio, Elena
• Is Part Of: Proceedings. Volume 477:Number 2245(2021)
• Publication Date: 2021-01-06
• Subjects: conservation law -- conserved charge -- topological charge -- Cauchy problem -- dynamical PDE; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2020.0442
• Languages: English
• ISSNs: 1364-5021

Abstract

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.