Symmetries, Conservation Laws, and Noether's Theorem for Differential-Difference Equations

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Abstract

This paper mainly contributes to the extension of Noether's theorem to differential-difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self-adjointness method is adapted to the computation of conservation laws for differential-difference equations. Several differential-difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

Original languageEnglish
Pages (from-to)457-502
Number of pages46
JournalStudies in Applied Mathematics
Volume139
Issue number3
DOIs
Publication statusPublished - 2017 Oct
Externally publishedYes

ASJC Scopus subject areas

  • Applied Mathematics

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