Symmetries, conservation laws and Noether's theorem for differential-difference equations

Abstract

This paper mainly contributes to the extension of Noether's theorem to differential-difference equations. For that 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 non-variational equations, since Noether's theorem is now available, the self-adjointness method is adapted to the computation of conservation laws for differential-difference equations. A couple of differential-difference equations are investigated as illustrative examples, including the Toda lattice and semi-discretisations of the Korteweg-de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

30 pages

References in corpus (2)

• Non-invertible transformations for the classification of differential-difference equations
• Master symmetry and time dependent symmetries of the differential-difference KP equation

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