Self-adjointness and conservation laws of difference equations

TL;DR

This paper establishes a general theorem for deriving conservation laws in difference equations using an adjoint system approach, applicable even without a Lagrangian formalism, and demonstrates how symmetries lead to conserved quantities.

Contribution

It introduces a novel method to find conservation laws for difference equations via an adjoint system, bypassing the need for a Lagrangian, and connects symmetries to conservation laws through a variational principle.

Findings

Theorem for conservation laws in difference equations is proved.

Conservation laws can be derived without a Lagrangian formalism.

Symmetries of the original system lead to conservation laws via the variational principle.

Abstract

A general theorem on conservation laws for arbitrary difference equations is proved. The theorem is based on an introduction of an adjoint system related with a given difference system, and it does not require the existence of a difference Lagrangian. It is proved that the system, combined by the original system and its adjoint system, is governed by a variational principle, which inherits all symmetries of the original system. Noether's theorem can then be applied. With some special techniques, e.g. self-adjointness properties, this allows us to obtain conservation laws for difference equations, which are not necessary governed by Lagrangian formalisms.