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

TL;DR

This paper extends Noether's theorem to differential-difference equations, exploring symmetries, conservation laws, and self-adjointness, with applications to equations like Toda lattice and Volterra.

Contribution

It develops a prolongation formula for continuous symmetries and adapts the self-adjointness method for conservation laws in differential-difference equations.

Findings

Extended Noether's theorem for differential-difference equations.

Derived conservation laws for Toda lattice and Volterra equations.

Established relations between symmetries, conservation laws, and Fréchet derivatives.

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 Fr\'echet 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.