Noether-type theorems for difference equations

Abstract

The Noether's type constructions for difference functionals, difference equations and meshes (lattices) are reviewed. It is shown in Dorodnitsyn [J. Soviet Math. 55 (1999) 1490]; [Dokl. Akad. Nauk SSSR 328 (1993) 678] that the invariance of a functional (together with a mesh) does not mean the invariance of the corresponding difference Euler's equation. The stationary value of invariant difference functional is reached on the new difference equations (quasi-extremal equations) which are different for different subgroups. In the present paper the properties of quasi-extremals are considered. Any quasi-extremal equation is invariant under the corresponding subgroup and possesses its own conservation law. Every group operator which commutes with discrete differentiation transforms one quasi-extremal equation into another one, since there exists the group basis of quasi-extremal equations, which corresponds to the basis of conservation laws. It is shown that the intersection of quasi-extremals is invariant with respect to the whole group admitted by difference functional. This intersection has got the full set of difference conservation laws. The last proposition could be viewed as a discrete analog of Noether's theorem; it sufficiently differs from the result early obtained in Dorodnitsyn [Dokl. Akad. Nauk SSSR 328 (1993) 678].