Symmetries, conservation laws and Noether's theorem for differential-difference equations
This work provides a theoretical framework for deriving conservation laws in discrete-continuous systems, which is incremental for the field of mathematical physics.
The paper extends Noether's theorem to differential-difference equations, enabling the computation of conservation laws for non-variational equations via a self-adjointness method. It demonstrates the approach on examples including the Toda lattice and semi-discretizations of the KdV equation.
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.