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

T. Bridgman
W. Hereman
G.R.W. Quispel
P.H. van der Kamp
Journal / Anthology

Foundations of Computational Mathematics
Year: 2013
Volume: 13
Page range: 517–544

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PEs) is reviewed. The method assumes that the PEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PEs classified by Adler, Bobenko, and Suris and systems of PEs 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 PEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011). The method is algorithmic and is being implemented in Mathematica.

*Mathematics > Algebra

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