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Transformations, symmetries and Noether theorems for differential-difference equations

Peng, Linyu, Hydon, Peter E. (2022) Transformations, symmetries and Noether theorems for differential-difference equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 478 (2259). Article Number 20210944. ISSN 1364-5021. E-ISSN 1471-2946. (doi:10.1098/rspa.2021.0944) (KAR id:93786)


The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether’s theorem. We state and prove the differential-difference version of Noether’s second theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether’s two theorems. These results are applied to various equations from physics.

Item Type: Article
DOI/Identification number: 10.1098/rspa.2021.0944
Uncontrolled keywords: Differential-difference equations, transformations, symmetries, conservation laws, Noether theorems
Subjects: Q Science > QA Mathematics (inc Computing science) > QA387 Basic Lie theory
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Peter Hydon
Date Deposited: 30 Mar 2022 16:55 UTC
Last Modified: 31 Mar 2022 14:27 UTC
Resource URI: (The current URI for this page, for reference purposes)

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