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Characteristics of Conservation Laws for Difference Equations

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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.

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Notes

  1. 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].

  2. 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].

  3. The corresponding results for systems of difference equations with arbitrarily many independent variables are obtained mutatis mutandis; see [7].

  4. 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].

  5. 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.

  6. 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.

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Acknowledgements

We thank the Natural Environment Research Council for funding this research. We also thank the referees for their very helpful recommendations.

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Authors and Affiliations

  1. Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK

    Timothy J. Grant & Peter E. Hydon

  2. Schlumberger Gould Research, High Cross, Madingley Road, Cambridge, CB3 0EL, UK

    Timothy J. Grant

  3. British Antarctic Survey, High Cross, Madingley Road, Cambridge, CB3 0ET, UK

    Timothy J. Grant

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  1. Timothy J. Grant
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  2. Peter E. Hydon
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Correspondence to Peter E. Hydon.

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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

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  • DOI: https://doi.org/10.1007/s10208-013-9151-2

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