Evolution systems on a lattice

Abstract

In the framework of the algebraic geometric approach to differential-difference equations, we study symmetries and conservation laws of evolutionary systems on multidimensional lattices. We describe conservation laws in terms of their characteristics.We here extend results previously obtained in [1] for evolution systems in the Euclidean space to evolution systems on multidimensional integer lattices. For this, we adapt the basic principles of the algebraic geometric (or group) analysis of partial differential equations (see, e.g., [2]-[6]) to the discrete case. We introduce the concept of the difference jet, we study functions on the space of difference jets and operations defined on these functions, and we introduce some elements of the calculus of variations on a lattice. We use the developed formalism to describe conservation laws of evolution systems on multidimensional lattices in terms of their characteristics, which are defined in the same way as for evolution systems in the Euclidean space.In what follows, we assume that all linear operations are over the field R and that all infinite-dimensional vector spaces carry the natural locally convex topology. All linear maps that we introduce are continuous, but we do not prove this explicitly. We use the following standard notation: N = {1, 2, . . . }, Z + = {0, 1, 2, . . . }, and Z = {0, ±1, ±2, . . . }. We let Map(X; Y ) denote the set of all maps from the set X to the set Y . 1. The jet bundle over a lattice 1.1. The lattice. Let D ∈ N. A lattice (more precisely, a D-dimensional lattice) is the set L = Z D . There are two algebraic structures defined on a lattice L: the structure of an Abelian group with respect to addition and the structure of a Z-module. As a Z-module, L has a basis {e 1 , . . . , e D } ⊂ L, where e 1 = (1, 0, . . . , 0), . . . , e D = (0, . . . , 0, 1), and hence L m = (µ 1 , . . . , µ D ) = α µ α e α , µ α ∈ Z, 1 ≤ α ≤ D.For m = (µ 1 , . . . , µ D ) ∈ L, we set |m| = α |µ α | ∈ Z + . For brevity, we write {|m| ≤ p} instead of {m ∈ L; |m| ≤ p}, where p ∈ Z + .The action of the group L is defined on the lattice L, L n → T n ∈ Map(L; L), m → T n m = m + n.The maps T n are called translations.

A differential—Difference bicomplex

“…We note that the algebra A fin (J) and its subalgebra A(V) are tacitly used in the literature on this subject (see, e.g., [7]- [14]). We also note that the situation with the difference variational complex and even more so with the bicomplex is not clear by far, despite the common opinion regarding the similarity of the difference variational complex on an integer lattice to the variational complex on a Euclidean space.…”

Symmetries and conservation laws of difference equations

“…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.…”

“…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.…”