Abstract
Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether’s Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether’s Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka–Volterra equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For simplicity, we restrict attention to scalar equations throughout this paper; the corresponding results for systems are contained in the first author’s Ph.D. thesis [7].
This is one of the oldest branches of geometric integration but, by exploiting the growing power of computer algebra systems, some new strategies for doing this have been developed recently [7].
The corresponding results for systems of difference equations with arbitrarily many independent variables are obtained mutatis mutandis; see [7].
For differential equations on \(\mathbb{R}^{N}\) and difference equations on \(\mathbb{Z}^{N}\), the set of divergence expressions is the kernel of the Euler–Lagrange operator [13].
For a given PDE in Kovalevskaya form, any equation that holds on solutions of the PDE can be pulled back to an identity on the initial conditions; with the above definition, the same is true for PΔEs.
Kovalevskaya form is convenient for proving that the root characterizes each equivalence class of CLaws. However, for any explicit PΔE, the root (with respect to an appropriate set of initial conditions) can also be calculated without transforming to Kovalevskaya form.
References
V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable equations on quad-graphs. The consistency approach, Commun. Math. Phys. 233, 513–543 (2003).
S.C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications, Eur. J. Appl. Math. 13, 545–566 (2002).
S.C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations. II. General treatment, Eur. J. Appl. Math. 13, 567–585 (2002).
L. Arriola, First integrals, invariants and symmetries for autonomous difference equations, Proc. Inst. Math. NAS Ukr. 50, 1253–1260 (2004).
V.A. Dorodnitsyn, A finite-difference analogue of Noether’s theorem, Dokl. Akad. Nauk 328, 678–682 (1993) (In Russian).
V.A. Dorodnitsyn, Noether-type theorems for difference equations, Appl. Numer. Math. 39, 307–321 (2001).
T.J. Grant, Characteristics of Conservation Laws for Finite Difference Equations, Ph.D. thesis, Department of Mathematics, University of Surrey, 2011.
J. Hietarinta, C.-M. Viallet, Searching for integrable lattice maps using factorization, J. Phys. A, Math. Theor. 40, 12629–12643 (2007).
R. Hirota, S. Tsujimoto, Conserved quantities of a class of nonlinear difference-difference equations, J. Phys. Soc. Jpn. 64, 31–38 (1995).
P.E. Hydon, Conservation laws of partial difference equations with two independent variables, J. Phys. A, Math. Gen. 34, 10347–10355 (2001).
P.E. Hydon, E.L. Mansfield, A variational complex for difference equations, Found. Comput. Math. 4, 187–217 (2004).
P.E. Hydon, E.L. Mansfield, Extensions of Noether’s second theorem: from continuous to discrete systems, Proc. R. Soc. Lond. Ser. A 467, 3206–3221 (2011).
B.A. Kupershmidt, Discrete Lax Equations and Differential-Difference Calculus, Astérisque, vol. 123 (Société Mathématique de France, Paris, 1985).
L. Martínez Alonso, On the Noether map, Lett. Math. Phys. 3, 419–424 (1979).
K. Maruno, G.R.W. Quispel, Construction of integrals of higher-order mappings, J. Phys. Soc. Jpn. doi:10.1143/JPSJ.75.123001.
A. Mikhailov, J.P. Wang, P. Xenitidis, Recursion operators, conservation laws, and integrability conditions for difference equations, Theor. Math. Phys. 167, 421–443 (2011).
A.V. Mikhailov, J.P. Wang, P. Xenitidis, Cosymmetries and Nijenhuis recursion operators for difference equations, Nonlinearity 24, 2079–2097 (2011).
E. Noether, Invariante Variationsprobleme, Nachr. D. König. Ges. Wiss. Gött., Math-Phys. Kl. 235–257 (1918). M. A. Tavel, Invariant variation problems (English translation), Transp. Theory Stat. Phys. 1, 186–207 (1971).
P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd edn. (Springer, New York, 1993).
A.G. Rasin, Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method, J. Phys. A, Math. Theor. (2010). doi:10.1088/1751-8113/43/23/235201.
A.G. Rasin, J. Schiff, Infinitely many conservation laws for the discrete KdV equation, J. Phys. A, Math. Theor. (2009). doi:10.1088/1751-8113/42/17/175205.
O.G. Rasin, P.E. Hydon, Conservation laws of discrete Korteweg-de Vries equation, SIGMA (2005). doi:10.3842/SIGMA.2005.026.
O.G. Rasin, P.E. Hydon, Conservation laws of NQC-type difference equations, J. Phys. A, Math. Gen. 39, 14055–14066 (2006).
O.G. Rasin, P.E. Hydon, Conservation laws for integrable difference equations, J. Phys. A, Math. Theor. 40, 12763–12773 (2007).
O.G. Rasin, P.E. Hydon, Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math. 119, 253–269 (2007).
Acknowledgements
We thank the Natural Environment Research Council for funding this research. We also thank the referees for their very helpful recommendations.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grant, T.J., Hydon, P.E. Characteristics of Conservation Laws for Difference Equations. Found Comput Math 13, 667–692 (2013). https://doi.org/10.1007/s10208-013-9151-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10208-013-9151-2