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

SIGMA 20 (2024), 006, 29 pages      arXiv:2309.09040      https://doi.org/10.3842/SIGMA.2024.006
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

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 Schrödinger equation.

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

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References

  1. Boutin M., On orbit dimensions under a simultaneous Lie group action on $n$ copies of a manifold, J. Lie Theory 12 (2002), 191-203, arXiv:math-ph/0009021.
  2. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  3. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  4. Gonçalves T.M.N., Mansfield E.L., On moving frames and Noether's conservation laws, Stud. Appl. Math. 128 (2012), 1-29, arXiv:1006.4660.
  5. Gonçalves T.M.N., Mansfield E.L., Moving frames and conservation laws for Euclidean invariant Lagrangians, Stud. Appl. Math. 130 (2013), 134-166, arXiv:1106.3964.
  6. Gonçalves T.M.N., Mansfield E.L., Moving frames and Noether's conservation laws -- the general case, Forum Math. Sigma 4 (2016), e29, 55 pages, arXiv:1306.0847.
  7. Hydon P.E., Difference equations by differential equation methods, Cambridge Monogr. Appl. Comput. Math., Vol. 27, Cambridge University Press, Cambridge, 2014.
  8. Hydon P.E., Mansfield E.L., A variational complex for difference equations, Found. Comput. Math. 4 (2004), 187-217.
  9. Kim P., Olver P.J., Geometric integration via multi-space, Regul. Chaotic Dyn. 9 (2004), 213-226.
  10. Kogan I.A., Olver P.J., Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Appl. Math. 76 (2003), 137-193.
  11. Kuperschmidt B.A., Discrete Lax equations and differential-difference calculus, Astérisque 123 (1985), 212 pages.
  12. Mansfield E.L., A practical guide to the invariant calculus, Cambridge Monogr. Appl. Comput. Math., Vol. 26, Cambridge University Press, Cambridge, 2010.
  13. Mansfield E.L., Hydon P.E., Difference forms, Found. Comput. Math. 8 (2008), 427-467.
  14. Mansfield E.L., Marí Beffa G., Wang J.P., Discrete moving frames and discrete integrable systems, Found. Comput. Math. 13 (2013), 545-582, arXiv:1212.5299.
  15. Mansfield E.L., Rojo-Echeburúa A., Hydon P.E., Peng L., Moving frames and Noether's finite difference conservation laws I, Trans. Math. Appl. 3 (2019), tnz004, 47 pages, arXiv:1804.00317.
  16. Mansfield E.L., Rojo-Echeburúa A., Moving frames and Noether's finite difference conservation laws II, Trans. Math. Appl. 3 (2019), tnz005, 26 pages, arXiv:1808.03606.
  17. Marí Beffa G., Mansfield E.L., Discrete moving frames on lattice varieties and lattice-based multispaces, Found. Comput. Math. 18 (2018), 181-247.
  18. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Grad. Texts in Math., Vol. 107, Springer, New York, 1993.
  19. Olver P.J., Geometric foundations of numerical algorithms and symmetry, Appl. Algebra Engrg. Comm. Comput., Vol. 11, 2001, 417-436.
  20. Olver P.J., Sapiro G., Tannenbaum A., Differential invariant signatures and flows in computer vision: A symmetry group approach, in Geometry-Driven Diffusion in Computer Vision, Comput. Imaging Vis., Vol. 1, Editor B.M. ter Haar Romeny, Kluwer Academic Publishers, Dordrecht, 1994, 255-306.
  21. Peng L., Hydon P.E., Transformations, symmetries and Noether theorems for differential-difference equations, Proc. A. 478 (2022), 20210944, 17 pages, arXiv:2112.06030.

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