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Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube

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Abstract

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable two-component potential Korteweg–de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011). The method is algorithmic and is being implemented in Mathematica.

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Notes

  1. Use the Mathematica function PowerExpand or simply cube the expression.

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Acknowledgements

The research is supported in part by the Australian Research Council (ARC) and the National Science Foundation (NSF) of the U.S.A. under Grant No. CCF-0830783. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of ARC or NSF.

WH is grateful for the hospitality and support of the Department of Mathematics and Statistics of La Trobe University (Melbourne, Australia) where this project was started in November 2007.

The authors thank the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) where the work was continued during the Programme on Discrete Integrable Systems in Spring 2009.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, 80401-1887, USA

    T. Bridgman & W. Hereman

  2. Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia

    G. R. W. Quispel & P. H. van der Kamp

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  1. T. Bridgman
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  2. W. Hereman
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  3. G. R. W. Quispel
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  4. P. H. van der Kamp
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Correspondence to T. Bridgman.

Additional information

Communicated by Elizabeth Mansfield.

At the occasion of his 60th birthday, we like to dedicate this paper to Peter Olver, whose work has inspired us throughout our careers.

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Bridgman, T., Hereman, W., Quispel, G.R.W. et al. Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube. Found Comput Math 13, 517–544 (2013). https://doi.org/10.1007/s10208-012-9133-9

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