Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 006, 29 pages      arXiv:2309.09040
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Moving Frames: Difference and Differential-Difference Lagrangians

Lewis C. White and Peter E. Hydon
School of Mathematics, Statistics and Actuarial Science, University of Kent,Canterbury, Kent, CT2 7NF, UK

Received September 19, 2023, in final form January 09, 2024; Published online January 15, 2024

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 Schrödinger equation.

Key words: moving frames; difference equations; differential-difference equations; variational calculus; Noether's theorem.

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