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

journal contribution
posted on 28.03.2022, 04:03 by T Bridgman, W Hereman, Gilles QuispelGilles Quispel, Pieter Van Der KampPieter Van Der Kamp
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.

Funding

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.

History

Publication Date

01/08/2013

Journal

Foundations of Computational Mathematics

Volume

13

Issue

4

Pagination

28p. (p. 517-544)

Publisher

Springer

ISSN

1615-3375

Rights Statement

© SFoCM 2012

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