Transformations, symmetries and Noether theorems for differential-difference equations

Linyu Peng, Peter E. Hydon

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Article number20210944
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2259
Publication statusPublished - 2022


  • Noether's theorems
  • conservation laws
  • differential-difference equations
  • symmetries
  • transformations

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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