Moving Frames: Difference and Differential-Difference Lagrangians

TL;DR

This paper develops a moving frame theory for partial difference and differential-difference equations, enabling invariant calculus of variations and conservation laws, with applications to integrable systems and semi-discretizations.

Contribution

It introduces a novel moving frame framework for differential-difference equations that preserves the prolongation structure and allows for invariant analysis.

Findings

Developed a theory of projectable moving frames for difference equations.

Applied the theory to Toda-type and nonlinear Schrödinger equations.

Demonstrated the use of invariant calculus of variations and conservation laws.

Abstract

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the nonlinear Schr\"odinger equation.