Symmetries and conservation laws of difference equations

Abstract

IntroductionAlgebro-geometric methods provide a natural base for studying symmetries, conservation laws, Lie-Bäcklund maps, Bäcklund transformations, integrability, and other similar properties of systems of partial differential equations. These methods and their applications are detailed in numerous monographs (see, e.g., [1]-[7]). Because of enhanced computer capabilities, lattice models are recently gaining increasingly greater popularity for both numerical simulations and theoretical investigations in mathematics and physics. Symmetries, conservation laws, and other geometric properties of the thus emerging systems of nonlinear difference equations are also of considerable interest, but the appropriate mathematical techniques have not yet been developed, although there are quite a few works in that direction (see, e.g., [8]-[14]).Here, we propose an approach to constructing such a technique based on the notion of a difference jet [15], [16] that allows describing the geometric properties of difference systems from a unified standpoint. We note that the obtained results (Theorems 5-8) differ essentially from similar results for differential equations. This is related to a principal difference between shifts and derivations: shifts specify morphisms of the corresponding algebra of functions, and derivations obey the Leibnitz rule on that algebra. We use the standard notationand let Map(X; Y ) denote the set of all maps from a set X to a set Y . Although this is not stipulated explicitly, all the spaces introduced below have the natural topologies, and all the operations either are assumed to be continuous or are such by construction. In particular, Map(X; Y ) actually means the space of all continuous maps from a topological space X to a topological space Y endowed with the appropriate topology. Algebro-geometric analysis on an integer lattice2.1. The lattice and its difference complex. We briefly describe the difference complex of a lattice (see [15]-[17] for more detail).